Evolutionary Algorithms based on non-Darwinian Theories of Evolution: The Peircean Perspective

Evolutionary Algorithms based on non-Darwinian

Theories of Evolution:

The Peircean Perspective

PhD Thesis

Junaid Akhtar

2004-03-0019

Advisor: Dr. Mian Muhammad Awais

Co-Advisor: Dr. Basit Bilal Koshul

Department of Computer Science

School of Science and Engineering

Lahore University of Management Sciences

Dedicated to everyone and everything that brought me together

Lahore University of Management Sciences

School of Science and Engineering

CERTIFICATE

I hereby recommend that the thesis prepared under my supervision by Junaid Akhtar titled Evo-

lutionary Algorithms based on non-Darwinian Theories of Evolution: The Peircean

Perspective be accepted in partial fulfillment of the requirements for the degree of doctor of phi-

losophy in computer science.

Dr. Mian M. Awais (Advisor)

Recommendation of Examiners’ Committee:

Name Signature

Dr. Mian Muhammad Awais ——————————————

Dr. Basit B. Koshul ——————————————

Dr. Asim Karim ——————————————

Dr. Shafay Shumail ——————————————

Acknowledgements

It would not be unfair to thank Allah foremost at this point. I believe I am grateful to Him

mostly for giving me ideas that guide me. I want to thank my parents and family, that is the

least I can do in return for giving me their full support and peace of mind; my wife for motivating

me towards the completion of this milestone; my advisors, Dr. Mian Muhammad Awais and

Dr. Basit Koshul, for their most valuable advises over the years. And my friends for making

the time so memorable, especially Faheem, Umar Suleman, Zeeshan, Jahan, and Saqib. Higher

Education Commission, Pakistan, and Lahore University of Management Sciences, Pakistan, are

also acknowledged for the funding.

Contents

Abstract xiii

1 Introduction 1

1.1 Investigation 1: Research Directions within EC . . . . . . . . . . . . . . . . . . . . . 2

1.2 Investigation 2: Challenges Facing EC . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Investigation 3: New Developments within Evolutionary Sciences . . . . . . . . . . . 8

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Thesis Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Darwinian Evolutionary Algorithms: From Natural Reality to Problem Solving 13

2.1 Darwin’s Stature and Contribution in Science . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Darwin, the Forward-looking Genius . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Darwin, the Backward-looking Intellect . . . . . . . . . . . . . . . . . . . . . 16

2.2 The Road from Darwin to Post Modern Synthesis . . . . . . . . . . . . . . . . . . . 20

2.3 From Evolutionary Theory to Evolutionary Computation . . . . . . . . . . . . . . . 23

2.3.1 Classical Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Classic Algorithms in EC are Foundationally Darwinian . . . . . . . . . . . . 25

2.4 Instantiation of an EA with an Optimization Problem . . . . . . . . . . . . . . . . . 27

2.5 Non-Classical Variant EAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 Non-Darwinian-type Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.2 Peirce-type Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

v

3 Peircean Theory of Evolution 37

3.1 Understanding Peirce’s Framework through Evidence . . . . . . . . . . . . . . . . . . 40

3.1.1 Evidence for Peirce’s Thirdness in the biological world . . . . . . . . . . . . . 40

3.1.2 Evidence for Peirce’s Firstness in the biological world . . . . . . . . . . . . . 44

3.2 Understanding Peirce’s Evolutionary Framework through Definitions . . . . . . . . . 45

3.2.1 Firstness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Secondness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.3 Thirdness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Understanding Peirce’s Evolutionary Framework through a Possible Synthesis . . . . 48

3.3.1 Definition :: Peircean Evolutionary Theory . . . . . . . . . . . . . . . . . . . 50

3.4 Nature of Chance and Laws in Light of Modern Physics and Cosmology . . . . . . . 50

3.4.1 Heisenberg on the Nature of Reality . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.2 Weyl on the Historical understanding of Reality . . . . . . . . . . . . . . . . 51

3.4.3 Holton Concludes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Peircean Framework for Evolutionary Algorithms 57

4.1 Principles derived from Peirce’s theory for Evolutionary Algorithms . . . . . . . . . 57

4.1.1 Firstness as Spontaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.2 Secondness as Necessity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.3 Thirdness as Generalizing tendency . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.4 Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.5 Reality at Multiple Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Peircean Evolutionary Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Comparison of Peircean EA with Related EA . . . . . . . . . . . . . . . . . . . . . . 67

4.3.1 IDEA 1 :: Diversity Retaining Measures . . . . . . . . . . . . . . . . . . . . . 68

4.3.2 IDEA 2 :: Distributed Population Dynamics . . . . . . . . . . . . . . . . . . 68

4.3.3 IDEA 3 = IDEA 1 + IDEA 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

vi

5 Experimental Evidence 71

5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Results on an Extended Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Analyses 85

6.1 Stagnation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 Schema Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3.1 Clusters as Schema - Effects on Disruption Analysis . . . . . . . . . . . . . . 90

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7 Future of EC & our Conclusions 93

7.1 Hans-Paul Schwefel’s Future Directions for EC . . . . . . . . . . . . . . . . . . . . . 93

7.1.1 Equivalence between Peircean Framework and Schwefel’s Future Directions . 95

7.2 Kenneth De Jong’s Agenda for the 21stCentury and Peirce . . . . . . . . . . . . . . . 98

7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A Cooperation vs. Competition in Evolution 103

A.1 Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A.1.1 Hypothesis 1: Only selfishness is real, cooperation is not . . . . . . . . . . . . 105

A.1.2 Hypothesis 2: Cooperation is as real as Selfishness . . . . . . . . . . . . . . . 107

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List of Figures

2.1 Flow chart for Classical Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . 25

2.2 Image registration: A 2-D problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Representation of EA Individuals for Image Registeration Problem . . . . . . . . . . 28

2.4 Crossover Operator’s Procedure - Parents 1 and 2 produce Children 1 and 2 . . . . . 30

4.1 Hypothetical 2D search space depicting a cluster formation within the population . . 61

4.2 Algorithm 1: Pseudo code for Peircean Evolutionary Algorithm . . . . . . . . . . . . 64

4.3 Algorithm 2: Pseudo-code for intra-cluster evolution . . . . . . . . . . . . . . . . . . 66

4.4 Algorithm 3: Pseudo-code for inter-cluster evolution . . . . . . . . . . . . . . . . . . 67

5.1 Multiple views of function f2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73



5.4 Convergence comparison: P-EA vs. D-EA . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5 Extended Benchmark Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.1 Cluster Analysis: Effect of clustering on stagnation using Schwefel’s function . . . . 88

ix

List of Tables

4.1 Relationship of Peircian principles at different levels of algorithmic reality . . . . . . 63

5.1 Benchmark mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Parameters for P-EA and D-EA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Compiled results for 50 runs of D-EA on f1, f2, f3 and f4, with stop-count = 500 . . 77

5.4 Compiled results for 50 runs of P-EA on f1, f2, f3 and f4, with stop-count = 500 . . 78

5.5 P-EA on f4 with varying stop-counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6 P-EA compared with other EA on f3 [30-D] . . . . . . . . . . . . . . . . . . . . . . . 80

5.7 P-EA compared with other EA on f4 [30-D] . . . . . . . . . . . . . . . . . . . . . . 81

5.8 Comparison between Peircean-EA and Darwinian-EA . . . . . . . . . . . . . . . . . 83

6.1 Number of generations before the population stagnates,20 experiments . . . . . . . . 86

7.1 Relation between Peircean framework and Schwefel’s future challenges . . . . . . . . 97

xi

Abstract

Historically, Evolutionary Algorithms (EA) have been important for the Evolutionary Computa-

tion (EC) community for two primary reasons: 1) As a simulation of evolutionary processes, the

way they happen in the natural world, and 2) using them to solve computationally hard problems,

including optimization. With the passage of time EA have become increasingly focused on function

optimization. Given this narrowing of vision in the EC community, it is worth revisiting a paper

written in 1997 by Hans-Paul Schwefel on the future challenges for EC. In that paper the author

argues that the more an algorithm models natural evolution at work in the universe, the better

it will perform (even in terms of function optimization). The present dissertation tests Schwefel’s

hypothesis by designing an EA based on the evolutionary theory of Charles Sanders Peirce, founder

of American school of thought known as pragmatism. Peirce’s theory not only accounts for bio-

logical evolution on earth (as other theories of evolution do) but also offers an account of global,

cosmological and universal evolution. In going beyond just biological evolution, Peirce’s theory of

evolution meets the criteria suggested by Schewefel in his 1997 paper. We follow this course be-

cause of three primary reasons. Firstly, Peirce has not been seriously tested in EC, although there

have been EA based on other theories and sub-theories. Secondly, Peirce’s universal theory, by

not being restricted to biological evolution alone, qualifies for Schwefel’s hypothesis, perhaps more

than most other theories that have already been modeled algorithmically. But most importantly

because, in experimental terms, it lets us make an original claim that Peirce’s insights are useful in

improving the existing EA in computer science, as Peircean EA can potentially solve some of the

major problems in this area such as the loss of diversity, stagnation, or premature convergence. In

this thesis, we have provided a novel algorithm based on Peirce’s theory of evolution, initiated it

xiii

for the class of optimization problems, and tested it extensively against a benchmark set of math-

ematical problems of varying dimensions and complexity. Comparative results with classical and

advanced EA form another significant part of the thesis, and help in strengthening the viability of

Schwefel-Peirce hypothesis for EC. Besides the experimental and empirical findings, the theoretical

aspects of Peircean EA’s performance are also brought under discussion in the light of Holland’s

schema theorem and disruption analysis. It is then claimed that Peircean arrangements for an EA

help minimize the disruptions potentially caused by the crossover operator and consequently help

in the survival and continued evolution of good schemas.

xiv

Chapter 1

Introduction

We had been in correspondance with Hans-Paul Schwefel, one of the founders of Evolution Strate-

gies, regarding the design of Evolutionary Algorithms (EA) based on Charles Sanders Peirce’s

theory of evolution. All of a sudden he asks about the relevance of a “19th” century mathemati-

cian and evolutionary philosopher for the 21stcentury evolutionary sciences, especially Evolutionary

Computation (EC).1 The answer to Schwefel’s pertinent question seemed deceptively simple at first,

however, this entire thesis, in many ways, became an address to that sole question.

The simple answer is that Peirce is perhaps more necessary for evolutionary sciences today than

he was in the 19thcentury. But to reach to that conclusion or to answer the question satisfactorily,

there are three separate paths that would need to be investigated and developed towards an eventual

convergence. One path investigates the historical developments within EC, to make a legitimate

room for exploring Peircean ideas for an EA. Second path investigates the challenges faced by the

EC community as noted in the literature, and shows how Peircean framework has the potential

to act as a legitimate solution to those challenges. Having worked out the relevancy of Peirce

for 21stcentury EC specifically, the third path investigates the developments in the evolutionary

sciences generally, especially the cutting edge fields of biology. It is earnestly hoped that towards the

end of these three investigations it can be concluded, that the relevant research communities within

1Actual correspondance text: “I assume that not many people (and reviewers) are familiar with Pierce’s ideasabout evolution that may seem to many people some kind of outfashioned. He is better known as mathematician,mainly having done his work in the 19thcentury (we are in the 21stnow).”

1

the broader evolutionary sciences and especially EC are ripe for evolving their theoretical principles

to that of the Peircean evolutionary framework, rather than sticking to even older 19thcentury ones.

1.1 Investigation 1: Research Directions within EC

John Holland’s motivation behind his pioneering Genetic Algorithms (GA) model was to simulate

biological adaptive systems (Holland, 1975). In other words, Holland sought to model biological

evolution as proposed in Darwin’s theory. After Holland though, his students became increasingly

focused on designing GA for solving optimization problems (Goldberg, 1989). While the practical

need for optimization in GA is indeed important, it was only a marginal concern in Holland’s

original GA. Because his GA sought to model evolution in the natural world, Holland had to keep

in view the fact that for complex adaptive systems “improvement is usually much more important

than optimization.” This is an important point to keep in mind becasue - as (De Jong, 1993) notes:

“There is a subtle but important difference between ‘GAs as function optimizers’ and ‘GAs are

function optimizers’.” De Jong goes on to point out that there are important insights to be had

when this difference is understood and its implications are taken into account in developing GA

(and we may add EA in general).

This sentiment has been echoed in slightly different terms by two other pioneers in EA; (Schwefel,

1997) notes that “organic evolution certainly does not only aim at finding static optima just once

and with ultimate precision. Organic evolution happens within an ever-changing environment,

where evolvability is more important than precision”; Lawrence J. Fogel, pioneer in Evolutionary

Programming notes in (Back et al., 1997a, Section H1.2) that even though, “the solution of complex

engineering problems is important, but the use of evolutionary algorithms need not be restricted

to mere function optimization. The methods can also be used to gain an understanding of how

competitive or cooperative agents may interact given a variety of different available resources and

purposes.”

David B. Fogel aptly notes in the introduction of his Handbook of Evolutionary Computation

that “efforts in evolutionary computation commonly derive from one of four different motivations:

improving optimization, robust adaptation, machine intelligence, and facilitating a greater under-

2

standing of biology” (Back et al., 1997a, Section A1.1). On the one hand, this clearly indicates that

the spirit of EC is multi-faceted and cannot be reduced in its entirety to function optimization, or

any of the other three motivations for that matter. But at the same time it can lend itself to the

view that there is an either-or situation for the EC community–either one can be in EC to improve

function optimization or to understand the processes of natural evolution better. The two tasks

appear to be independent of each other and do not seem to be meaningfully related.

Perhaps it is because of this sectional view of EC that over the years the practical focus in EC is

increasingly “reduced” to factors such as efficiency, engineering applications, and standardizations.2

A careless reading of the foregoing could be taken as a suggestion that the importance of function

optimization or its efficiency in EA is being trivialized. We are, however, arguing something

very different! What is being suggested is that function optimization and its efficiency can be

enhanced by recognizing that there is a direct relation between understanding the natural processes

of evolution in greater detail and improvement in the working of evolutionary algorithms. In the

words of David B. Fogel:

Our challenge is, at least in some important respects, to not allow our own biases

to constrain the potential for evolutionary computation to discover new solutions to

new problems in fascinating and unpredictable ways. However, as always, the ultimate

advancement of the field will come from the careful abstraction and interpretation of

the natural processes that inspire it. (Back et al., 1997a, Section A1.1.6)

While the empirical reality of evolution occurring in nature is equally accessible to the entire

scientific community, yet slightly different understanding and perspective in theorizing the natural

processes have had different consequences. This consequential effect can be illustrated using data

from EC as well. Memetic algorithms are based on the notion of cultural evolution (Caraffini et al.,

2013; Moscato, 1989), inspired by Richard Dawkins’ proposal of meme as a unit of information that

reproduces itself as people interact and exchange ideas (Dawkins, 1976). Horizontal gene transfer

that takes place in bacterial organisms has inspired the design of specialized crossover operators

2Task Force on Future Directions in Evolutionary Computation (FDEC) had been part of the EvolutionaryComputation Technical Committee (ECTC), IEEE Computational Intelligence Society (CIS). The TF held an annualWorkshop as part of the IEEE Congress on Evolutionary Computation

3

that have spawned a new class of evolutionary algorithms called pseudo-bacterial genetic algorithms

(Balazs and Koczy, 2012; Nawa and Furuhashi, 1998). Lamarckian conception of evolution and

Baldwin effect on learning have been used by hybrid EA, and are reportedly more efficient than

their Darwinian counterparts (Fuhrmann et al., 2010; Mitchell and Taylor, 1999; Yuan et al.,

2010). Island Model EA are based on Eldridge and Gould’s punctuated equilibria theory (Segura

et al., 2011; Skolicki, 2005; Srinivasa et al., 2007), where subpopulations independently evolve while

infrequent migrations take place using some communication topology. There are Diffusion Model

EA as well which are based on population genetics idea of local (not global) selection and crossover.

Genetic makeup spreads from individual to population levels based on the diffusion model (Back

et al., 1997a; Jaimes and Coello, 2009). They are designed for parallel processing with one individual

per processor where the selection of parents is dependent on the underlying parallel architecture,

usually SIMD (Single Instruction, Multiple Data streams). The Genetic Algorithms itself is John

Holland’s “adaptation” of Darwin’s theory of evolution (Holland, 1975). This brief detour serves to

underscore the point that a different interpretation of natural evolution results in a different model

of EA.

According to Jacques Monod, Darwin’s theoretical explanation for evolution is an exquisite

mix of “chance and necessity” (Monod, 1971). In non-philosophical terms it is a combination of

a variety of chances and a variety of laws. In order for Darwinian evolution to work it takes as a

given, not only these two agents, but ironically the first batch of replicating life as well. Being a

naturalist, Darwin did not make an attempt to try and relate the two apparently warring agents

(chance & necessity), or how they “evolved” themselves before playing a role in the evolution of

the universe and its living forms. However, there is one man that did that after Darwin.

Charles Sanders Peirce, the 20th century evolutionary pragmaticist, has made major contribu-

tions to numerous fields such as logic and philosophy of science, formal and mathematical logic,

topology, linguistics, epistemology and semiotics. When Peirce looked at the empirical effects of

evolution, he came to the conclusion that there are three (not two) types of phenomenon, three basic

categories that are operative in the universe. First, chance; Second, necessity; Third, habit-taking.3

3Peirce’s use of habits is different from its Lamarckian usage: “For Peirce, habits are not provisional adaptiveresponses to fluctuating environmental conditions; they are steps on the universal road from indeterminacy to law,

4

From this, Peirce built a philosophically intricate system, and for its heart he installed semiosis, his

theory of signs, which attempts at describing the inter-relationship of his three categories. Stressing

unconditionally on the irreducibility of his semiotic triad, he said:

....by “semiosis” I mean, an action, or influence, which is, or involves, a cooperation of

three subjects, such as a sign, its object, and its interpretant, this tri-relative influence

not being in any way resolvable into actions between pairs. (Peirce, 1958, 5.484)

As far as the relation between a sign and its object is concerned, a sign, for Peirce, does not

properly function or signify outside the specific context of this triadic relation. This conception is

at odds with theories of signification that rely on a dyadic (or two part) relationship between signs

and the objects they signify. For Peirce, semiosis of processes of signification operate only when a

sign is considered in its triadic form, i.e. when a sign is a representation of its object, such that it

produces or modifies its interpretant.4 This effectively turns the interpretant into another sign of

the same object, and thus helps enable further interpretations. This dynamic semiotic process is

the cause for evolution and growth of meaning. It implies that organic matter learns to engage with

the world, not simply by forming ideas in response to stimuli, but by forming habits of responding

to non-organic material. For Peirce, habits are the ultimate interpretants of the world’s signs, and

he viewed habit-formation as the physiological manifestation of the sign-taking capacity (Ochs,

1993, pg. 68).

Mapping Peirce’s semiotic language on to his phenomenology it can be argued that evolution-

ary growth requires a cooperation of three evolutionary agencies; and for Peirce absolute chance,

mechanical necessity, and tendency to take habits are a sign that his three categories, Firstness,

Secondness and Thirdness are severally operative in the cosmos. Peirce shows that in logic they are

represented as beginning, end, and process. In psychology they are feelings, reaction-sensations,

and thought or reason. In the case of biology Peirce extends the almost linear two step evolutionary

process of random variation followed by natural selection into a non-linear triadic process. He says,

a road traveled by objects as well as by organisms...Habit-taking is a plastic faculty. The peculiar characteristic ofhabit is: “not acting with exactitude” ” (Menand, 2001, pg. 365)

4For Peirce, the interpretant can mean more than one thing: (a) The context in which a Sign relates to an Object(b) The relationship that an Object is able to establish with its context (Interpretant), because of Objects relationshipwith Sign (c) The potential consequences of the relationship of Sign with Object

5

First is the principle of individual variation or sporting; Second, the principle of heredity transmis-

sion; and Third, the process whereby the accidental characteristics become fixed (including, but

not limited to the elimination of unfavorable characters by natural selection.) (Peirce, 1958, 6.32)

Does the foregoing discussion have any implications for the EC community? In other words,

when the underlying theory of evolution evolves from a serial two-ness to a dynamic three-ness,

how does that affect the EC models consequently? Our work is an answer to this question. Taking

the lead from the pioneers in EC, our goal is to develop a framework for EA based on a theory

that offers a more intricate and explanatory account of the natural evolutionary process. Towards

this end, we are proposing a new framework for EA based on Peirce’s semiotic theory of evolution.

We have tested it on a benchmark of mathematical problems and compared the results with other

EAs.

1.2 Investigation 2: Challenges Facing EC

It is slightly ironic that our search for answering Schewefel’s concern regarding Peirce’s relevance

to EC leads us straight to Schwefel’s own research agenda. He fully understands the necessity

of developing a model which is closer to natural reality. In his essay “Challenges to and Future

Development of EA” he says:

Current evolutionary algorithms are certainly better models of organic evolution. Nev-

ertheless, they are still far from being isomorphic mappings of what happens in nature.

In order to perform better, an appropriate model of evolution would have to comprise

the full temporal and spatial development on the earth (a real global model) if not

within the whole universe. We must be more modest in order to understand at least a

little of what really happens – as always within natural sciences. (Schwefel, 1997)

We will refer to this passage as Schwefel’s hypothesis. Our call for revisiting Schwefel’s

research agenda is not because we are interested in the fulfillment of the initial promises of EC,

in and of themselves, or that we are not inclined towards looking at EA as function optimizers.

On the contrary, we bring to attention those unfulfilled promises only since they can potentially

6

help improve the function optimization capabilities of EA as well. The reason we chose Schwefel

to make a case is because of the clarity with which he has laid bare the relation between the two

things that are apparently viewed as being largely unrelated - performance of an EA and natural

understanding of reality. Some of the other future challenges according to Schwefel are:

1. Evolvability more important that precision

2. Dynamic interaction of agents

3. Multiple selection criteria

4. Cooperation as important as competition

5. Incorporation of social learning and epigenetic factors

If Schwefel had left out some open challenges to EA, Kenneth De Jong, the founding editor

of the premier Evolutionary Computation journal, while presenting a history of the field of GA

concludes with an extended “agenda for the 21stcentury” (De Jong, 2005). Some of the highlights

of this agenda are:

1. Developing a more general EC/EA framework

2. Decentralized and Speciation models

3. Self adapting and coevolutionary systems

4. Incorporating more biology into EA, especially Lamarckian ideas

5. Using EA to further our evolutionary understanding

In general our work is about revisiting Schewefel’s and De Jong’s research agenda for the

21stcentury. In particular it is about testing Schwefel’s hypothesis. It is in the pursuit of this end

that we introduce the evolutionary theory of Charles Sanders Peirce–the evolutionary philosopher,

mathematician, semiotician, and scientist par excellance (Peirce, 1958). As we saw above, Schwe-

fel’s hypothesis asks for a model of evolution that goes beyond biological evolution and encompasses

7

global, cosmological and universal evolution. Typically evolution has been confined to biological

processes, which help only explain the last few billion years of development within the universe.

Being a through-going evolutionary philosopher (and not a mere naturalist) Peirce sought to un-

derstand not only biological life, but also the emergence of all inanimate and animate matter as

well as the laws of nature shaping their behavior, in evolutionary terms. This dissertation tries

to demonstrate how Peircean principles have the potential to fit the bill for the above-mentioned

21stcentury research agenda for EA in general, and solve the problem of stagnation in particular,

caused by the loss of diversity in the population at early stages of classical EA paradigms (Deb,

2001; Fogel, 1994). Our claim is that the Peircean framework for EC achieves this naturally, with-

out having to resort to arbitrary and artificial arrangements that are commonly employed just

for demonstrating an improvement in the results (Lozano et al., 2008; Mahfoud, 1995; Sareni and

Krahenbuhl, 1998). In other words, Peirce can potentially provide the much needed theoretical

consistency to the future developments within EC.

1.3 Investigation 3: New Developments within Evolutionary Sci-

ences

Lawrence J. Fogel reminds us of the greatest challenge and promise the EC community had to

fulfill:

Perhaps the greatest challenge facing evolutionary computation is its use as a means

for gaining a greater understanding of natural evolution. This has been the promise of

the efforts of artificial life, but like many other such promises throughout the course of

computer science, they have been left mainly unfulfilled. (Back et al., 1997a, Section

H1.2)

Moreover, identifying reductionism as a problem for scientific understanding of complex pro-

cesses, he said:

An important step forward could be realized if attempts to perform such credit assign-

ment and related schema analysis in complex systems were abandoned in favor of more

8

holistic understandings of how selection acts on complex sets of behaviors in concert,

rather than in isolation. Just as no general understanding of the physics of flight can

come from assigning credit to feathers or flapping wings, no general understanding of

complex adaptive systems can come from piecemeal analysis of their ‘genes’. Refocus-

ing attention on ‘organisms’ rather than ‘genes’ represents a compelling and promising,

although old, direction for further investigation.

There is a surprising similarity between what Fogel had to say to his own community and what

(Goldenfeld and Woese, 2007), veterans in biology had to say to their own:

In the last several decades we have seen the molecular reductionist reformulation of

biology grind to a halt, its vision of the future spent, leaving us with only a gigantic

whirring biotechnology machine. Biology today is little more than an engineering dis-

cipline. Thus, biology is at the point where it must choose between two paths: either

continue on its current track, in which case it will become mired in the present, in

application, or break free of reductionist hegemony, reintegrate itself, and press for-

ward once more as a fundamental science. The latter course means an emphasis on

holistic, “nonlinear,” emergent biology–with understanding evolution and the nature of

biological form as the primary, defining goals of a new biology.

Fogel and Woese share with their respective community the potential consequences of working

on a reductionist paradigm in evolutionary sciences. Physics, the mother of natural sciences has

an advantage of hindsight that it can lend to the relatively younger evolutionary biology and even

younger EC. Having witnessed the transition from deterministic Newtonian physics to probabilistic

Quantum mechanics, the physicist-philosopher (Bohm, 1969) makes an interesting observation:

It does seem odd ... that just when physics is ... moving away from mechanism, biology

and psychology are moving closer to it. If the trend continues ... scientists will be

regarding living and intelligent beings as mechanical, while they suppose that inanimate

matter is too complex and subtle to fit into the limited categories of mechanism.

9

Fortunately, biology has already begun to see a movement away from the classical mechanistic

conceptions. Works of cutting edge groups like epigenetics, systems biology, symbiogenesis, emer-

gent biology and biosemiotics are at the frontiers and taking the same direction as seen in the

movement within physics. Unfortunately there is no equivalent EC group working in tandem with

the above-mentioned counterparts in biology. We would like to introduce this research work as an

effort to bring EC at par with the frontiers of evolutionary biology.

This new inertia demands explanation of old phenomena using a new perspective. We introduce

the framework of Charles Sanders Peirce as a non-reductionist explanation for evolution. The

classical Darwinian conception and its modern synthesis are based on two agents at the fundamental

level: blind chance and mechanistic laws. While Peirce acknowledges the role of both of these agents

in the evolutionary process, he identifies one additional agent, i.e. generalizing tendency. The

introduction of this third agent appropriates subtle but significant changes in our understanding

of the character of both chance and laws. In Peirce’s triadic scheme chance does not always follow

a blind uniform distribution but can follow distributions similar to those followed by laws of large

numbers. Laws on the other hand, are not encapsulated by the metaphor of machines, perfect and

precise in their output, but show tendencies of habits and hence variation and exceptions in their

delivery. This allows Peirce to describe the emergence of both chance and law as the outcomes

of an evolutionary process in the universe rather than as eternal givens outside the universe -

that work mysteriously. According to Peirce the main flaw of Darwinian and other philosophical

systems has been to either reduce this third regulatory element to one of the other two, or to miss

it altogether (Peirce, 1958, 6.303). It is surprising to see how many contemporary debates within

evolutionary sciences have a chance to be settled if this third Peircean agent is incorporated within

the evolutionary paradigm. The scope of this research work does not allow to venture into the

resolution of those debates. However we shall use the Peircean framework to show how it fares

in the wake of some of the new developments within the cutting edge evolutionary biology, in

general. In particular, we show how Peircean framework fits into the ongoing debate within the

evolutionary sciences regarding one of the most critical issue, that of an evolutionary explanation

for the empirical evidence of ’cooperation’.

10

1.4 Contributions

A brief overview of the main contributions of this work is listed below:

1. Bringing philosophy and (computer) science into a relationship

2. Realizing through experiments and theory the significance of seeing ‘EAs as function opti-

mizers’ rather than ‘EAs are function optimizers’

3. Highlighting the role of evolvability over precision for evolution, and the factors that con-

tribute towards it in an EA

4. Exploring the relationship between the efficiency of function optimization in an EA and the

interpretation of natural processes and theories of evolution

5. Significance of Charles Sanders Peirce and his work for the 21stcentury science

6. Presenting Peircean theory of evolution as a synthesis, and making it converse with contem-

porary developments withing evolutionary sciences

7. Bringing EC at par with its counterparts in other evolutionary sciences

8. Readying Peirce for EC by extracting a Peircean EA out of his philosophical volumes

9. Mapping Peircean EA’s characteristics with the future challenges listed by Schwefel and De-

Jong

10. Possible presentation of Peirce as a unifying framework within EC, that could help many

variant EAs

11. Testing Peircean EA on extensive benchmark mathematical functions

12. Comparing the results with Classical and advance EAs

13. Presenting a Peircean solution to a major problem in EC - stagnation

14. Demonstrating how Peirce can help resolve Darwinian controversies, especially ‘Cooperation’

11

15. Putting to test Peirce’s evolutionary hypotheses, and helping Peircean scholarship too, bring-

ing philosophy and science into a badly needed two-way relationship

16. Identification of further possibiliies to be explored by the EC community

1.5 Thesis Roadmap

The disseration roadmap is as follows: Chapter 2 shows how EAs solve optimization problems, but

more importantly, how they travel the road from Nature to theorization to algorithmic modeling.

The chapter also discusses the intellectual environment at the time of Darwin, and the number of

things that went in to making his theory. Finally we map various components of Darwin’s theory

to the respective elements of the classical EA (C-EA) model.

In chapter 3 the discussion is repeated, but for Peirce; mainly, Peircean conception of the

universe and Peircean evolutionary framework are discussed using evidence from Darwin, biology,

theoretical definitions, and quantum physics.

The discussion is finally brought home in chapter 4 where the ground work laid in the previous

chapter is used to build a general framework for EC, as well as a specific Peircean EA (P-EA).

Chapter 5 captures the experimental details of our work. An important segment is the compar-

ison of C-EA and P-EA over an extensive set of benchmark mathematical optimization functions.

The discussion then moves beyond the empirical grounds in chapter 6. The chapter investigates

the theoretical understanding of the effect of Peircean principles in an EA’s convergence. Some

interesting results are drawn using the classical schema theory of John Holland and the associated

disruption analysis.

Note: The three separate investigations (in response to Schwefel’s opening question) presented

in this introduction chapter are not given a serial treatment within this dissertation. They are rather

introduced intermittently as sub-discussions within different chapters, as well as in the appendices,

and also winded up in the conclusion - chapter 7.

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Chapter 2

Darwinian Evolutionary Algorithms:

From Natural Reality to Problem

Solving

The objective of this chapter is to introduce the elements that go into the making of an evolutionary

algorithm. Notwithstanding the recent plethora of EA variants that hardly seem to even try and

engage with any evolutionary theory, classically speaking, EAs have had one design route: evolution

factually occurs in the natural reality, there are theories to explain evolutionary phenomena, and

that EAs are computational models of those theories that attempt to explain the processes or sub-

processes of natural evolution. This is a well-established reading and construction of an EA; and it

shortlists one of the defining objectives of the scientific domain known as Evolutionary Computation

- to understand the natural reality.

In revisiting the roots of EC, we hope to achieve an understanding of not just the history and

philosophy of EC, but also by instantiating the above-mentioned general definition through John

Holland’s classical model of GA, we intend to further understand the elements that go into the

making of the Darwinian paradigm - the grand theory upon which Holland based his adaptive com-

putational model. This chapter also aims to introduce Darwinian paradigm through the lens of its

historical development, reconstructing the theory’s truly evolutionary character and consequently

13

deconstructing any false finalistic images of it. We shall begin from the beginning then.

2.1 Darwin’s Stature and Contribution in Science

Charles Darwin is rightfully placed in Westminster Abbey near Newton’s last abode. Though

centuries apart in time, both men fundamentally set forth new foundations for science, one in biology

and the other in physics, respectively. In the wake of the success physics enjoyed after Newton,

what makes Darwin even greater is his resistance against not just creationism (the established

belief of his time that all the variety of species have always been a separate creation;) but also

against reductive and deterministic mechanism (the belief that perfect laws control the outcome

of every event perfectly, and that these universal laws could be perfectly captured by reducing

them to elements at much basic/fundamental level) - the established framework in the physics of

his era. In such a terrifying intellectual environment, for someone to almost single-handedly carve

out space for a theory that could not be sold as either naturally theistic or causally mechanistic,

was a feat which only a man like Darwin could have performed. At a time when most scientists

would have wanted to win legitimacy for their work by following the route of the physical sciences

(mathematical rigour + deterministic laws), Darwin fearlessly showed an independent scientific

legitimacy for biology and his ‘Origins of Species’ proved that the language of biology did not need

to be embedded with mathematical equations to present a valid scientific theory (Mayr, 2004).

Darwin managed to show that the living matter that naturalists (biologists) studied, did not or

could not lifelessly obey any set of universal laws, while at the same time showing that it was still

possible to study the regularities within the diversity of life without resorting to transcendental

theistic arguments ultimately. For this two-edged victory, Darwin will rightly stand out for his

contributions to science, perhaps always.

The above paragraph is not a mere lip-service! Then why is it that we want to explore non-

Darwinian evolutionary theories as a foundation for EA?1 To answer this question we will have

to show two tendencies - as if running in parallel - in Darwin’s thoughts and work. One is his

1Note that non-X does not necessarily mean anti-X, even if the force of culture pushes in favour of this interpre-tation!

14

forward looking creative genius tendency, and most people are already aware of it. The other

one is a backward tendency; where according to our reading, and in the hindsight of the scientific

development within biology, we show where Darwin slipped because of the remnants of the biases of

his intellectual era in his framework. The only legitimacy in choosing an alternative theory would

be, if that theory adopted the forward tendencies of Darwin’s theory, while making corrections in

the backward tendencies. Our claim is that Charles Peirce has that potential (substantiated in the

next chapter).

2.1.1 Darwin, the Forward-looking Genius

Darwin’s Origin of Species demolished the premise that God had immutably created all

the variety of species during the 6 days of His Creativity - and that it has been this way

ever since - as the 17thcentury churchmen had so carefully formulated. (Mayr, 2004)

There is no doubt that Darwin’s main address every now and then in his ‘Origin of Species’ is

towards the issue of independent creation of all varieties and species. He ends many of his illustrative

ideas by stating that independent creation hypothesis could not explain this or that particular

phenomenon. He resolves the issue through his theory of evolution by Natural Selection

preserving the fitter modified descendents w.r.t adaptation to their respective living

conditions; and this slow and continuous accumulation of fitter characteristics giving

rise to ever-divergent variations, that over time come to be classified as different

species.

There is little doubt left today that, in contrast to the belief in independent and fixed species of

creationism, evolution is the empirically proven scientific stance. We say ‘little’ because Intelligent

Design schools and Christian Creationists still exist today, staring coldly away from the ever-

accumulating evidence in support of evolution. We think their fate has been sealed by Darwin’s

theory.

The other, lesser explicit forward-looking tendency in Darwin’s work is his divergence from

the methods of reductive-mechanical physical sciences. This in fact earned his book the infamous

critique of “the law of the higgledy-piggledy” from John Herschel, the English matheatician and

15

astronomer Darwin really looked up to (Darwin, 1995, page 220). This thought has been developed

further by Ernst Mayr, considered by some as “the Darwin of the 20thcentury”, in his catalogue of

essays titled, “What Makes Biology Unique?” According to (Mayr, 2004), Darwin manages to do

this by being a true naturalist; furnishing a complete theory without feeling an impluse to detail a

single mathematical equation, and by being nuanced about almost every scenario, giving examples

of exceptions wherever possible, and by avoiding universal and absolutist language most of the

times. Here are a few examples to illustrate the point:

I am convinced that Natural Selection has been the most important, but not the exclu-

sive, means of modification.

This nuanced approach of Darwin has been ignored by the neo-Darwinists of today. Another

aspect that got lost in the transition to neo-Darwinism is the fact that even though Darwin did not

agree with Lamarck’s account in its entirety, yet he did not shy away from using the Lamarckian

language:

Changed habits produce an inherited effect, as in the period of the flowering of plants

when transported from one climate to another. With animals the increased use or disuse

of parts has had a more marked influence;

Before we get to the where and why of the departures of the proponents of the Modern Synthesis

from Darwin’s original theory, we would like to complete the section by discussing now the seemingly

backward looking intellectual aspects in Darwin.

2.1.2 Darwin, the Backward-looking Intellect

Notwithstanding all the true geniuses of the man, Darwin also belonged to the previous scientific

era, where certainty and determinism were the scientific ideals to be achieved. For instance, Darwin

often spoke of chance in a mechanistic sense:

I have hitherto sometimes spoken as if the variations so common and multiform in

organic beings under domestication, and in a lesser degree in those in the state of

16

nature - had been due to chance. This, of course, is a wholly incorrect expression, but

it serves to acknowledge plainly our ignorance of the cause of each particular variation.

(Darwin, 1859, page 131)

T. H. Huxley, who became the mouth-piece for Darwin, while clearing the allegations of chance

off Darwin’s work had this to say:

It is not a little wonderful that such an accusation as this should be brought against

a writer who has, over and over again, warned his readers that when he uses the word

“spontaneous,” he merely means that he is ignorant of the cause of that which is so

termed; and whose whole theory crumbles to pieces if the uniformity and regularity of

natural causation for illimitable past ages is denied. But probably the best answer to

those who talk of Darwinism meaning the reign of “chance,” is to ask them what they

themselves understand by “chance”? Do they believe that anything in this universe

happens without reason or without a cause? Do they really conceive that any event has

no cause, and could not have been predicted by any one who had a sufficient insight into

the order of Nature? If they do, it is they who are the inheritors of antique superstition

and ignorance, and whose minds have never been illumined by a ray of scientific thought.

The one act of faith in the convert to science, is the confession of the universality of

order and of the absolute validity in all times and under all circumstances, of the law

of causation. This confession is an act of faith, because, by the nature of the case, the

truth of such propositions is not susceptible of proof. But such faith is not blind, but

reasonable; because it is invariably confirmed by experience, and constitutes the sole

trustworthy foundation for all action.

If one of these people, in whom the chance-worship of our remoter ancestors thus

strangely survives, should be within reach of the sea when a heavy gale is blowing,

let him betake himself to the shore and watch the scene. Let him note the infinite vari-

ety of form and size of the tossing waves out at sea; or of the curves of their foam-crested

breakers, as they dash against the rocks; let him listen to the roar and scream of the

shingle as it is cast up and torn down the beach; or look at the flakes of foam as they

17

drive hither and thither before the wind; or note the play of colours, which answers a

gleam of sunshine as it falls upon the myriad bubbles. Surely here, if anywhere, he will

say that chance is supreme, and bend the knee as one who has entered the very pene-

tralia of his divinity. But the man of science knows that here, as everywhere, perfect

order is manifested; that there is not a curve of the waves, not a note in the howling

chorus, not a rainbow-glint on a bubble, which is other than a necessary consequence of

the ascertained laws of nature; and that with a sufficient knowledge of the conditions,

competent physico-mathematical skill could account for, and indeed predict, every one

of these “chance” events. (Huxley, 2010)

Affirming and reasoning about the regularities in nature is one thing, but this pre-modern

tendency to ascertain perfection in knowledge and also to place this certainty in its foundations,

while leaving no room for real chance or fallibilism is what we term as a backward tendency in

Darwin and most scientific men of his intellectual era. This apologetic defence brings Darwin back

into the folds of the very physical cartesianism which he intended to naturally break from.

Previously we mentioned the nuanced characteristics of Darwin’s observations positively, and

he was remarkable especially in collecting the examples in nature that acted as exceptions to his

otherwise general observations. Darwin seemed to be comfortable with the fact that the biological

entities do not seem to need to conform to any universal code or law like the way dead matter

does. Strangely this nuanced element is left behind when it comes to independence between laws

of variation and selection. Gould has summarized this Darwinian “necessity” aptly:

Textbooks of evolution still often refer to variation as “random.” We all recognize this

designation as a misnomer, but continue to use the phrase by force of habit. Darwini-

ans have never argued for “random” mutation in the restricted and technical sense of

“equally likely in all directions,” as in tossing a die. But our sloppy use of “random”

does capture, at least in a vernacular sense, the essence of the important claim that we

do wish to convey namely, that variation must be unrelated to the direction of evolution-

ary change; or, more strongly, that nothing about the process of creating raw material

biases the pathway of subsequent change in adaptive directions. This fundamental

18

postulate gives Darwinism its “two step” character, the “chance” and “necessity” of

Monod’s famous formulation – the separation of a source of raw material (mutation,

recombination, etc.) from a force of change (natural selection). (Gould, 2002, page 144)

Darwin also displays quite an uncharacteristic single-mindedness when it comes to presenting

natural selection as the only creative agency in evolutionary nature, and competition among both

inter and intra-species being the only real mode of relationship across the living world. Sometimes

this clearly lends itself on to the racial and imperial English worldview of his time.

At some future period, not very distant as measured by centuries, the civilised races

of man will almost certainly exterminate and replace throughout the world the savage

races. (Darwin, 1871)

Nietzsche, the great German philosopher has picked in Darwin another English tendecy, and

put it in his trademark style saying that European scholars have not been able to completely

out-grow their social conditions, and that these remain a part of almost each scholar’s intellectual

idiosyncrasy:

The wish to preserve oneself is the symptom of a condition of distress, of a limita-

tion of the really fundamental instinct of life which aims at the expansion of power

and, wishing for that, frequently risks and even sacrifices self-preservation. It should

be considered symptomatic when some philosophers–for example, Spinoza who was

consumptive–considered the instinct of self-preservation decisive and had to see it that

way; for they were individuals in conditions of distress.

That our modern natural sciences have become so thoroughly entangled in this Spinozis-

tic dogma (most recently and worst of all, Darwinism with its incomprehensibly onesided

doctrine of the “struggle for existence”) is probably due to the origins of most natural

scientists: In this respect they belong to the “common people”; their ancestors were

poor and undistinguished people who knew the difficulties of survival only too well at

firsthand. The whole of English Darwinism breathes something like the musty air of

19

English overpopulation, like the smell of the distress and overcrowding of small people.

But a natural scientist should come out of his human nook; and in nature it is not con-

ditions of distress that are dominant but overflow and squandering, even to the point

of absurdity. The struggle for existence is only an exception, a temporary restriction of

the will to life. The great and small struggle always revolves around superiority, around

growth and expansion, around power – in accordance with the will to power which is

the will of life. (Nietzsche, 1974)

Nietzsche was no naturalist, but he had a sharp eye for dogmas of all forms and shapes. In light

of Lynn Margulis’ work on symbiosis in nature (Margulis and Sagan, 2002), Nietzsche’s criticism

of Darwin does seem to be on the mark; as shall be discussed in appendix A. The objective behind

listing these backward tendencies is to hope for a construction of a truly modern evolutionary

theory. If we are to claim that Peirce’s theory has that potential, this dissertation would have to

show how Peirce manages to avoid these Darwinian pitfalls, while retaining the positives.

2.2 The Road from Darwin to Post Modern Synthesis

The Darwinian explanation of the evolutionary process, as popularly understood, is based on two

main factors: 1) variation through blind chance providing the raw material and 2) mechanistic

laws, primarily natural selection, sorting through the raw material to select the fittest; shaping the

evolutionary course. Darwin assumes that these two characteristics, blind chance and mechanistic

laws, have existed in the universe from eternity and will continue to exist for all eternity without any

change or modification (without any evolution). Even if this philosophical oversight is corrected and

a non-evolutionary basis is added to the otherwise evolutionary framework of Darwin, a growing

body of empirical evidence suggests that there are several shortcomings in the Darwinian theory of

evolution.

Historically speaking, the simple fact that a neo-Darwinian modern synthesis had to be proposed

suggests that there are at least some oversights in classical Darwinism. But the journey from

Darwinism to this modern synthesis was not a simple matter of rediscovery and incorporation of

Mendel’s genetic principles into the classical Darwinian theory. It is worth noting that it was

20

the very geneticists who rediscovered Mendel who challenged one of the fundamental principles

of classical Darwinism. The analysis of these post-Mendel geneticists suggested that the laws of

Mendelian genetics and the newly discovered phenomenon of discrete mutations did not fit well with

the Darwinian notion of blending of inheritance. A direct consequence was that due to discrete units

of traits being inherited, the gradual evolutionary process did not seem possible. These saltationists

eclipsed Darwinism (Bowler, 1983, page 14) until the introduction of the idea of population genetics.

It was shown that the gene pool of a population rather than an individual’s genome follows smooth

bell-curved distributions, and hence Mendelian genetics was reconciled with Darwinism. Effectively,

the architects of the modern synthesis performed three tasks: 1) they preserved some of the key

ideas of Darwin, such as gradualism, infinitesimal random variation, and adaptation through natural

selection; 2) they introduced Mendelian mechanisms of genetic inheritance, along with ideas of

population genetics; and 3) they discredited all alternative ideas at the time including Darwin’s

flirtations with use and disuse of organs, Darwin’s hereditary mechanisms, Lamarckian ideas of

inheritance, and the ideas of mutationists/saltationists (Mayr, 1993). The fact is that the modern

synthesis became the established framework in evolutionary biology:

The major tenets of the evolutionary synthesis, then, were that populations contain

genetic variation that arises by random (i.e., not adaptively directed) mutation and

recombination; that populations evolve by changes in gene frequency brought about by

random genetic drift, gene flow, and especially natural selection; that most adaptive

genetic variants have individually slight phenotypic effects so that phenotypic changes

are gradual (although some alleles with discrete effects may be advantageous, as in

certain color polymorphisms); that diversification comes about by speciation, which

normally entails the gradual evolution of reproductive isolation among populations; and

that these processes, continued for sufficiently long, give rise to changes of such great

magnitude as to warrant the designation of higher taxonomic levels (genera, families,

and so forth). (Futuyma, 1986, page 12)

Since then almost all of these tenets of the modern synthesis have been challenged as being either

inaccurate or incomplete. They include gradualism, externalism, gene centrism, and selfish com-

21

petition. For instance, the saltationists’ scholarship was revived by Gould’s theory of punctuated

equilibrium. This presents an open challenge to the gradualism of Darwin (Gould 1984). Evidence

for non-random mutations has been accumulating, as well (Shapiro, 2005). The revisionist muta-

tionists now claim that the Darwinian way of describing the introduction of novel variants merely as

“random” is inadequate. Clearly, the generation of variation by mutation-and-altered-development

is spontaneous in many ways, but it also exhibits habits or tendencies. These tendencies impose

biases on the outcome of evolution. The failure to recognize that such “internal” tendencies have

an impact on evolution is a key failure of Darwinism according to neo-mutationist scholarship

(Yampolsky and Stoltzfus, 2001). Similarly, molecular biology and epigenetics have studied cellu-

lar functioning in far more depth than was possible at the time the modern synthesis was originally

proposed. It is now confirmed that inheritance takes place through extra-genetic mechanisms as

well as through genetic processes (Jablonka and Lamb, 2008). At the very least, the emphasis in

genotype-phenotype mapping has shifted from individual genes to networks of genes (Brem and

Kruglyak, 2005; Ehrenreich et al., 2010; Pigliucci and Muller, 2010). Cellular environment, tran-

scription and regulation mechanisms are being given as much significance as was previously given

solely to genes and their associated proteins. Systems biologists, who work on dynamic multi-level

selection systems, claim that reduction of such a system to any one level, such as genes, is bound

to give an incomplete picture of reality (Noble, 2006). On another front, Lynn Margulis’s work on

symbiogenesis proves that processes other than natural selection (for instance, symbiosis) have also

acted as creative evolutionary agents, especially for the evolution of eukaryotes from prokaryotes

(Margulis, 1981; Margulis and Sagan, 2002). This symbiosis, being a rather long-term cooperative

process, provides quite some challenge to the ‘selfish’ understanding of evolution. The horizon-

tal gene transfer between organisms that ranges from the entire genome (through hybridization,

symbiosis and parasitism) to partial exchange has resulted in another significant challenge to the

strictly vertical classification in the Darwinian ‘Tree of Life’ (Boto 2010). New metaphors such

as ‘Web of Life’ are being understood and developed by biologists in light of the accumulating

evidence for lateral gene transfers (Goldenfeld and Woese, 2007). In response to these numerous

challenges to the fundamental tenets of the modern synthesis, on the one hand there are calls for

22

extensions in the neo-Darwinian modern synthesis (Jablonka and Lamb, 2005; Pigliucci and Muller,

2010), and on the other there is a growing community of evolutionists who are working towards

non-Darwinian explanations of evolution especially in light of the new biological findings (Rose and

Oakley, 2007; Shapiro, 1997). We represent the latter group!

2.3 From Evolutionary Theory to Evolutionary Computation

Darwin’s theory has taken its fair share of blows but more importantly it has withstood the test

of time. One of the reasons is that it is so easy to summarize its thesis. Everyone has their own

version; here is our rephrasing of Darwin’s argument:

Evolution by Natural Selection preserves the fitter modified descendents

w.r.t. adaptation to their respective living conditions; and this slow and

continuous accumulation of fitter characteristics gives rise to ever-divergent

variations, that over time come to be classified as different species.

One of the key ideas in Darwin’s theory is adaptation. Credit must go to John Holland for taking

this Darwinian understanding of evolution through natural adaptation seriously and converting it

into an eligible design for artifical adaptive systems (Holland, 1975). Though Holland’s initial

algorithm had little or nothing to do with optimization, his students started applying what came

to be known as Genetic Algorithms (GA) or classical evolutionary algorithm (C-EA) to problems

of optimization (Goldberg, 1989). We shall show here the recipe that maps an evolutionary theory

on to a much simplified C-EA. The ingredients include:

• An individual encoding and representing a candidate solution to the problem

• A population of individuals

• An objective function that maps each individual to its fitness value

• A selection function that can introduce bias in the population towards the survival of the

fitter individuals

23

• A binary function to introduce variation through crossing ‘selected’ individuals to (re)produce

descendents

• A unary function to introduce mutation based variations to descendents

• A stopping criteria

2.3.1 Classical Evolutionary Algorithms

To be sure, living beings evolve and grow, but evolution and growth does not necessarily solve any

problems, and even if it could be said that natural evolution solves a problem, it does so only across

the span of millions of years. These factors are in contrast with the “fast buck,” efficient engineering

mindset of our “information age.” On the other hand, evolution has sustainably designed and

produced a tremendous variety of species capable of living in and adapting to extreme environments.

If engineering algorithms can take some cues from the natural processes of evolution, perhaps many

non-trivial, computationally hard problems can be heuristically solved in a reasonable time, helped

by the ever-increasing computational capacity and speed of computer processors. So the basic idea

is to mimic natural evolution in how it has evolved complicated and better-fitted organisms from

their very rudimentary origins.

In the evolutionary algorithms of computer science, instead of going to the very origins, a basic

encoding representation of the problem to be solved is assumed to be in place, and the population

of solutions is initialized randomly (within the bounds of the problem domain). Furthermore, there

must be a function which serves as the meter for fitness. Now some of the randomly initialized

individuals will have a higher fitness value against their encoded solutions than others. The as-

sumption in classical evolutionary algorithms, inspired by Darwin’s theory, is that if we have a

selection function which can bias the next generation of individuals in the direction of the fitter

individuals in the population, then the fitter individuals will have a greater chance to participate in

generating the next batch of individuals that make up the population; and over many generations,

the population will gradually improve in its average fitness (because of fitter individuals crossing

over) and move closer to the global optima. The selection function serves as an exploitation fac-

tor, while crossover and mutation functions serve as exploration factors within the solution/search

24

space. The continual exploitation and exploration by the evolving candidate solutions heuristically

converges onto the global optima, as shown in Figure 2.1.

Figure 2.1: Flow chart for Classical Evolutionary Algorithms

2.3.2 Classic Algorithms in EC are Foundationally Darwinian

(Beyer et al., 2002) have summarized the origin and basic working principles of Evolutionary

Algorithms with brialliant brevity:

Evolutionary algorithms (EA) form a class of probabilistic optimization methods that

are inspired by some presumed principles of organic evolution. .... The general frame of

EP, GA, and ES is essentially the same and very simply summarized by a loop over par-

tially randomized variation and selection operators steering exploration and exploitation

(or chance and necessity) and, in contrast to traditional optimization procedures, acting

upon a set of search points in the decision variable space.

The above quote also highlights the dyadic formulation of EA models i.e. chance and necessity.

We have already mentioned in the introduction that Darwin’s conception of evolution primarily

revolves around two fundamentals: blind chance and mechanistic or necessitarian laws. This fact

also gets translated into its EA models. In the later chapters we will see how Peircean concep-

tion of evolution is triadic and how its translated model (Peircean-EA) also acquires this third

characteristic.

A review of literature in the field lays bare the fact that almost all the classical evolutionary

algorithms (including genetic algorithms, evolutionary programming and evolution strategies) have

25

their roots in the Darwinian theory of evolution. One does not even need to survey the literature

to establish this fact; some of the leading experts in EC explicitly acknowledge it themselves:

The GA works on the Darwinian principle of natural selection....Whether the specifica-

tions be nonlinear, constrained, discrete, multimodal, or even GA is entirely equal to

the challenge. (Reeves, 2002)

A leading textbook in the field of AI writes:

Genetic Algorithms begin with a population of candidate problem solutions. Candidate

solutions are evaluated according to their ability to solve problem solutions: only the

fittest survive and combine with each other to produce the next generation of possible

solutions. Thus, increasingly powerful solutions emerge as in a Darwinian universe.

It is oddly fitting that these approaches should seek the origins of intelligence in the

same processes that, it may be argued, gave rise to life itself.... Certainly one of the

strongest models of learning we have may be seen in the human and animal systems that

have evolved towards equilibration with the world. This approach to learning through

adaptation is reflected in genetic algorithms, genetic programming, and artificial life

research. (Luger, 2002)

We believe that there is a high level of correlation between the three techniques, as they are

based on the same principles and have the same underlying architecture. Therefore, the evidence

we found for GA being Darwinian should in principle be enough for generalizing it to the broader

spectrum of EA:

The majority of current implementations of evolutionary algorithms descend from three

strongly related but independently developed approaches: genetic algorithms, evolu-

tionary programming, and evolution strategies. (Back et al., 1997b)

Having shown the evidence that the main EA in EC are Darwinian in their foundations, we

now turn to the last leg of this chapter. The more initiated readers of EC can of course skip the

following section.

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2.4 Instantiation of an EA with an Optimization Problem

In Digital Image Processing, image registration is the process of “match[ing] two or more pictures

taken, for example, at different times, from different sensors, or from different viewpoints” (Brown,

1992, page 325).In this process a smaller template is to be searched within a larger image. Usually

the scaling and rotation and other transformations of the template are involved in the search, but

in order to simplify this illustration for readers outside the computer science community, we limit

the search space to two dimensions and do not add those transformations.

(a) Search Image (b) Template Image (c) Template Matching

Figure 2.2: Image registration: A 2-D problem

Figure 2.2(a) shows the search image, Figure 2.2(b) shows a scaled up template image, while

in Figure 2.2(c) the white box shows the smaller template image matching the larger search image

at a specific coordinate. Hence, it is a problem of searching for that x and y (or row and column)

coordinates where the template makes a maximum correlation with the image at that location.

Correlation coefficient ranging from -1 to +1 becomes the fitness value. This is how a searching

problem has been translated into an optimization language in this case. We discuss the ingrediants

of solving this optimization problem through an EA now:

Representation

For any problem to be solved through an EA, we need to encode it in a specific representation. The

template matching, without the scaling, rotation and other linear or non-linear transformations

can be represented as a 2D problem, as mentioned above. The complete candidate solution to

the image registration through template matching problem is just a 2D vector, where one axis

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needs to cover the maximum number of rows of pixels that the search image has, and the second

axis covers the maximum number of columns. This (row #,col #) pair can completely represent

the solution to this specific problem, as the template image can then be placed at this specific

(row #,col #) coordinate and its solution can be validated through correlation. Had scaling or

rotation been involved, more fields would sure need to be added to the representation along with

the location vector. Usually EA use binary encoding to represent individuals and populations, as

this helps in crossover and mutation operators, as it shall become obvious shortly, but there are

other representations possible as well. In the case of binary representation, to cover the search

space of 1024 x 512 rows and columns a 19 bit long binary representation would be required - 10

bits to cover the 1024 rows of the search image, and further 9 bits to cover the 512 columns of the

search image, as shown in Figure 2.3.

Figure 2.3: Representation of EA Individuals for Image Registeration Problem

Population

Instead of a point search, EA are a population based searching algorithm, where the individuals

in the population interact with each other to get to the global optima in the search space. In the

case of image registeration the population of the EA would comprise of (row #,col #) pairs, how

many such pairs, or the size of the population is a variable and is one of the design parameters,

28

but is usually fixed. The only problem is that even if the size of the population is known to the

algorithm (let us say it is 100), what are the exact values of these 100 (row #,col #)? Classically,

EA populate these using uniform random distribution, in our case the first axis ranges from 1 to

1024, and the second ranges from 1 to 512.

Fitness Evaluation

Once the EA has the initial population, it needs to pass it on to the fitness evaluation function,

which in this case simply takes the template image, places it on the specific (row #,col #) of the

search image and calculates the correlation value between the two (in simpler terms, it calculates

how much the overlapping parts of the image match each other). This correlation value becomes

the fitness value against this individual. A high fitness value of an individual means that a high

percentage of the template image matches with the underlying search image when placed at the

specific (row #,col #) represented by this individual. The same process needs to be repeated for

the entire population.

Selection

Selection of the fittest is carried out based on the fitness values. The objective of the selection

function is to introduce bias within the population, meaning that in the next generation of the

randomly-initialized population, all individuals should not have an equal chance of representation.

Those individuals that turned out to be fitter should have a higher chance of participating in the

process of generating the next batch of individuals, and this selection is done for every generation.

There are various selection functions that the community has devised and keeps on devising. We

will illustrate only tournament selection here. Tournament size is again a design choice; for instance,

if the tournament size is 4 and the population size is 100, then the tournament selection function

will have to perform 100 tournaments in each of which 4 random individuals are picked from the

population, and the fittest among them wins that tournament. Appropriate tournament size is a

tradeoff. If the tournament size is too large, there is a chance that only a few fittest individuals

would survive until the next generation, and if the tournament size is too small, then too small a

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selection pressure might result in a random search, and this selection function might not fulfill its

objective of exploiting or introducing the needed bias for the search. Once the 100 individuals have

been selected, they are to generate the next generation by using variation operators. The two most

classical ones are crossover and mutation.

Crossover

Crossover is a binary operator that takes two parents and produces two children by crossing them

at randomly chosen crossover points. Here is where the binary representation comes in handy.

The process is extensilvely explained in Figure 2.4. It shows the power of crossover operator as an

exploratory tool, as the least significant bits of two parents (P1 & P2) across the crossover point

get swapped. This crossover produces two children (C1 & C2) or two new search points that are a

variation of parent search points. This crossover operator is applied to all parents with a certain

probability which is termed “crossover probability.”

Figure 2.4: Crossover Operator’s Procedure - Parents 1 and 2 produce Children 1 and 2

Mutation

Mutation is a unary operator that is applied to a single individual with a very small mutation

probability. All it does is to flip a bit from 0 to 1, or 1 to 0, and hence produces a variation in the

population. Together with crossover, it gives exploratory powers to an EA.

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Stopping Criteria

Once the population of the EA is initialized, selection, fitness and variation operators keep on

acting to produce generation after generation, unless one of the following usual stopping criterion

is met:

• Desirable fitness (correlation value >= 0.95) is achieved by an individual in the population

• Maximum number of generations have been exhausted (usually 10,000)

• No significant improvement in fitness has occurred since past 500 or so generations

2.5 Non-Classical Variant EAs

We have chosen classical EA to open up our case - the objective being the understanding of Peircean

EA in contrast. There are otherwise ‘countably infinite’ varities of EA it seems (pun intended,)

and making an exhaustive survey of advance varieties would almost be a meaningless activity for

this dissertation2.

2.5.1 Non-Darwinian-type Ideas

(Deb et al., 2002) propose a non-dominated sorting genetic algorithm for multi-objective functions

that uses a specialized selection operator incorporating a mathematical cuboid function to preserve

the spread of population. The use of such mathematical cuboids to achieve selection is hard to

argue for, both in nature and in the Darwinian accounts. (Runarsson and Yao, 2000) propose a

stochastic ranking system for selection using a bubble sort like procedure. Although this is a basic

algorithm in computer science, it is nowhere to be seen in the natural or Darwinian world. (Salomon,

1997) proposed a deterministic genetic algorithm saying that it is due to random application of

variation operator that genetic algorithms perform non-optimally. Such deterministic applications

of mutations are nowhere to be seen in natural as well as Darwinian world. (Herrera and Lozano,

2000) use triangular probability distributions for fuzzy recombination, and a hypercube topology

2We have ensured that there is no related work when it comes to Peirce and an evolutionary algorithm directly.

31

for exploration and exploitation. It is hard to find any example in the natural and Darwinian world

for both these phenomena.

These were some of the variants of classical EA, examples of which are available neither in

natural nor Darwinian world. Of greater relevance to our central claim are those variants of EA,

examples of which can be found in natural accounts, but provisions for such deviations are not to

be found in a Darwinian model. Their motivation has either been some social phenomenon or, as

is with most cases, merely the empirical improvement in the accuracy or efficiency from function

optimization point of view (Mantere, 2006; Beyer, 1996; Fogel and Chellapilla, 1998; Ray and Liew,

2003; Amor and Rettinger, 2005).

2.5.2 Peirce-type Ideas

Without the pretence of a survey, here is list of a few EA variants, which carry Peirce-type ideas

within their construction. We will briefly describe them here; their details can be had through the

references. We compare and contrast them with Peircean EA at the end of chapter 4 - where the

comparison makes more sense.

IDEA 1 :: Diversity Retaining Measures

Loss of diversity within the population means, for all practical purposes, the ‘death of evolution.’

Since crossover will not produce a new variety, and mutation rates are kept very low, there cannot

be further possible exploration of the search space, once diversity has been lost. This often means

a premature convergence to a local minima for an EA. To help avoid this, different operators have

been developed which help retain diversity, or in other words, help prevent stagnation due to loss

of diversity.

Mostly this loss of diversity is triggered because of the presence of a relatively very fit individual

in the population, and if the selection function is not balanced enough, in a few generations, this

one individual replicates itself, replacing the entire population. Keeping population in niches, and

the fitter individuals in every niche sharing their fitness with the rest of the sub populations is one

way to suppress the triggering phenomenon (Sareni and Krahenbuhl, 1998) in the selection phase.

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In some EA especially those in which the population size is to remain fixed, it has to be determined

which individuals out of the new and old lot will remain and which will be replaced. During

this replacement phase, one technique that helps avoid stagnation is that the child which is most

similar to its parent, and with a higher fitness replaces the parent (Mahfoud, 1995). Other hybrid

formulations also have been proposed in which the individuals who explicitly do not contribute

towards increasing or maintenance of diversity are replaced (Lozano et al., 2008).

IDEA 2 :: Distributed Population Dynamics

The class of EA that evolves the population distibutively and comes closer to P-EA is Island Model

EA (IMEA) and Diffusion Model EA (DMEA), also called Cellular models. This section briefly

outlines their salient features.

Island Model EA

1. Based on Eldridge and Gould’s punctuated equilibria theory (Gould, 2002).

2. Mostly implemented in parallel processes/processor, with a certain communication topology

(Segura et al., 2011; Skolicki, 2005; Srinivasa et al., 2007).

3. Each subpopulation/deme is assigned a different processor, and usually the population is

evenly distributed.

4. Each subpopulation evolves independently.

5. Migration usually takes place after regular intervals (generations).

Diffusion Model EA

1. Based on population genetics idea of local (not global) selection and crossover.

2. Genetic makeup spreads from individual to population based on diffusion model (Back et al.,

1997a; Jaimes and Coello, 2009).

3. Designed for parallel processing with one individual per processor.

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4. Selection of parents dependent on the underlying parallel architecture, usually SIMD (Single

Instruction, Multiple Data streams).

IDEA 3 = IDEA 1 + IDEA 2

Some EAs are a hybrid, having both the distributed/hierarchical population dynamics as well as

mechanisms of introducing fresh supply of individuals within the population - to avoid stagnation.

(Hu et al., 2003) combine two ideas for multi-objective EA: maintaining a hierarchical or-

ganization of repositories of individuals according to different fitness ranges, and the continual

introduction of new genetic material at the bottom level fitness repositories. They claim that this

is a sustainable exploratory model that avoids stagnation in the long-term.

(Amor and Rettinger, 2005) propose a GA and a self-organizing maps (SOM) as a hybrid

combination that tackles stagnation. Their GA-SOM maintains search histories keepting track

of the frequencies of neurons in the lattice of the SOM. This way the population is uniformaly

divided, as the lattice covers the search space uniformly. A reseeding operator ensures diversity

by introducing new individuals around those neurons (equivalently, search regions) which have had

the least frequent activations. (Kubota et al., 2005) have also designed a SOM based reproduction

operators.

Summary

This chapter has basically tried to achieve three objectives: a) to place Darwin’s theory in its

intellectual context, highlighting some of its achievements, and also listing some shortcomings that

were “naturally” ignored at that point in time in history due to the intellectual environment, but

which perhaps need to be revisited and repaired in light of contemporary scientific understanding,

if possible; b) to establish a relation between Darwin’s complex theory and John Holland’s adaptive

system as its simpler EA model; c) to instantiate the classical EA for an interesting digital image

processing optimization problem of image registration through template matching; and d) to enlist

some of the divergent non-classical EAs. This chapter also sets the tone for the next chapter, where

Charles Sanders Peirce’s evolutionary framework is explored for its potential contributions within

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EC.

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Chapter 3

Peircean Theory of Evolution

Peirce offers an evolutionary theory that embraces the advancements in classical theories while

avoiding some of their most glaring shortcomings. The notion of evolutionary growth is at the

heart of Peirce’s philosophy of pragmaticism. Building on the insights of (Spencer, 1867), Peirce

notes that “evolutionary growth” is not a mere increase; rather, it has certain characteristics that

must be accounted for in any theory of evolution. Evolution is a two-fold passage from homogenous

forms to heterogeneous on the one hand, and from unorganized to organized systems on the other.

The first passage leads to an increase in diversity, while the second movement leads to an increase in

regularity or uniformity. Given the fact of evolutionary growth (and its defining characteristics) the

question emerges: “What is the cause of this growth? Is it pure chance or is it mechanistic law?” An

adequate theory of evolution has to account not only for the fact of diversity and uniformity in the

universe but also for the fact that there is a process of “growth” in both phenomena. Peirce notes

that neither the randomness of pure chance nor the determinism of mechanistic law can account

for the simultaneous increase in diversity and uniformity (Peirce, 1958, 1.174)1. The following

observation by Peirce identifies the three elements that are at work in evolutionary growth:

[D]iversification is the vestige of chance spontaneity; and wherever diversity is increas-

ing, there chance must be operative. On the other hand, wherever uniformity is in-

creasing, habit must be operative. But wherever actions take place under an established

1Standard way to cite Peirce is (Volume number. Paragraph number). For example (1.174) means volume 1,paragraph 174. This format will be followed wherever Peirce is cited.

37

uniformity, there, so much feeling as there may be, takes the mode of a sense of reaction.

(Peirce, 1958, 6.267)

Summarily stated, we can say:

1) Diversity is the result of chance

2) Uniformity is the result of mechanistic reaction

3) Growth is the result of habit

For Peirce, biological evolution and its mechanisms did not suddenly come into existence in

the universe some four billion years ago with the emergence of the first forms of self-replicating

RNA. It is a process that is at least fourteen billion years old – or as old as the universe itself.

Biological evolution is only a specific manifestation of a more general phenomenon: evolution at the

cosmological level. A natural problem arises regarding the terms that should be used to describe

the basic elements for such a universal evolutionary theory. If biological terms are used, they

are instantly rendered useless at the level of physics, chemistry and cosmology, and vice-versa.

Hence, Peirce uses the technical terms “Firstness,” “Secondness,” and “Thirdness” (which he calls

“categories”) to describe chance, reaction, and habit respectively. Peirce’s use of the triadic scheme

illustrates the fact that his theory of evolution is entangled with other aspect of his philosophy, such

as phenomenology, metaphysics, logic, semiotics, etc. In non-technical language he has described

different manifestations of the three categories in these terms:

Firstness: chance, spontaneity, feeling, possibility

Secondness: reaction, mechanical necessity, force, competition

Thirdness: habit, generalizing tendency, law, cooperation

For Peirce, wherever there is evolutionary growth in the universe, there is dynamic interaction

between chance, necessity, and habit. Evolutionary growth, in the most general sense, is a movement

from pure chance or spontaneity (Firstness) to necessity (Secondness) by the gradual growth of habit

(Thirdness). The following quote summarizes Peirce’s hypothesis of evolution at the cosmological

38

level:

This theory is that the evolution of the world is hyperbolic, that is, proceeds from one

state of things in the infinite past, to a different state of things in the infinite future.

The state of things in the infinite past is chaos, tohu bohu, the nothingness of which

consists in the total absence of regularity. The state of things in the infinite future is

death, the nothingness of which consists in the complete triumph of [mechanistic] law

and absence of all spontaneity. Between these, we have on our side a state of things

in which there is some absolute spontaneity counter to all law, and some degree of

conformity to law, which is constantly on the increase owing to the growth of habit.

The tendency to form habits or tendency to generalize is something which grows by its

own action, by the habit of taking habits itself growing. Its first germs arose from pure

chance. (Peirce, 1958, 8.137)

In short, the triadic relationship between Firstness, Secondness, and Thirdness (the relationship

between chance, necessity, and habit) is the basic principle underpinning evolutionary growth. By

making biological evolution a subset of his theory of cosmological evolution, not only can the

question of biological origins be stated in more precise terms, but also the principles underpinning

Peirce’s theory of evolution can become open to scientific scrutiny, which is not the case in the

classical theories.

[T]he problem of how genuine triadic relationships first arose in the world is a better,

because more definite, formulation of the problem of how life first came about; and no

explanation has ever been offered except that of pure chance, which we must suspect

to be no explanation. (Peirce, 1958, 6.322)

Perhaps owing to Darwin’s efforts, Peirce did not need to waste his energies in conversing with

the creationists’ arguments. While Darwin’s main issue was to deal with the question of creation

of fixed species, Peirce’s problem is readily and visibly different. He is intellectually engaged in

conversation with the philosophical school of nominalism on the one hand (who advocates that there

are no universals or generals in the working of our universe, only specifics, implying that laws are a

39

mere fiction) and on the other, mechanical philosophers (who advocate such a narrowing conception

of cause and effect, that these mechanistic laws render spontenaity, freedom and chance a mere

lifeless illusion). Peirce proposed pragmaticism as a simultaneous response to both nominalist and

mechanical philosophers, for according to Peirce, the natural processes of evolution as well as the

very human processes of scientific inquiry are not possible while remaining faithful to any of those

two philosophical frameworks. But without making this disseration too absorbed in philosophy, our

objective is to highlight the fact that Darwin had a different set of issues and questions to address

than Peirce. Academic experience suggests that when the questions are different, the answers

ought to be different as well. The intelligibility of answers, at times, depends on the ingenuity of

the questions posed or the issues pursued.

3.1 Understanding Peirce’s Framework through Evidence

Many of the intricate philosophical issues facing Peirce were settled by showing how his category of

Thirdness was actively operative in the natural world, and how because of its realness the presumed

definitions of both chance and laws would have to be evolved by the scientific community. This

chapter sets the following three goals for itself 1) define Peirce’s Firstness, Secondness and Third-

ness, 2) show how they are operative in the biological world, and 3) show how the movement in the

modern physics is supportive of Peircean ideas rather than the doctrines of 19thand 20thcenturies.

3.1.1 Evidence for Peirce’s Thirdness in the biological world

We now step into Darwin’s Origin of Species to highlight how the tendency to take habits (Third-

ness) plays a significant role in the biological evolution at times. Please bear with us for the long

quotes, but understanding Darwin’s findings is important for understanding Peirce’s Thirdness:

Exhibit A: From Origin of Species, Chapter 6

I will now give two or three instances both of diversified and of changed habits in the

individuals of the same species. In either case it would be easy for natural selection

to adapt the structure of the animal to its changed habits, or exclusively to one of

40

its several habits. It is, however, difficult to decide, and immaterial for us, whether

habits generally change first and structure afterwards; or whether slight modifications of

structure lead to changed habits; both probably often occurring almost simultaneously.

Of cases of changed habits it will suffice merely to allude to that of the many British

insects which now feed on exotic plants, or exclusively on artificial substances. Of

diversified habits innumerable instances could be given: I have often watched a tyrant

flycatcher (Saurophagus sulphuratus) in South America, hovering over one spot and

then proceeding to another, like a kestrel, and at other times standing stationary on

the margin of water, and then dashing into it like a kingfisher at a fish. In our own

country the larger titmouse (Parus major) may be seen climbing branches, almost like

a creeper; it sometimes, like a shrike, kills small birds by blows on the head; and I have

many times seen and heard it hammering the seeds of the yew on a branch, and thus

breaking them like a nuthatch. In North America the black bear was seen by Hearne

swimming for hours with widely open mouth, thus catching, almost like a whale, insects

in the water.

As we sometimes see individuals following habits different from those proper to their

species and to the other species of the same genus, we might expect that such individuals

would occasionally give rise to new species, having anomalous habits, and with their

structure either slightly or considerably modified from that of their type. And such

instances occur in nature. Can a more striking instance of adaptation be given than

that of a woodpecker for climbing trees and seizing insects in the chinks of the bark?

Yet in North America there are woodpeckers which feed largely on fruit, and others

with elongated wings which chase insects on the wing. On the plains of La Plata, where

hardly a tree grows, there is a woodpecker (Colaptes campestris) which has two toes

before and two behind, a long pointed tongue, pointed tail-feathers, sufficiently stiff

to support the bird in a vertical position on a post, but not so stiff as in the typical

woodpeckers, and a straight strong beak. The beak, however, is not so straight or so

strong as in the typical woodpeckers, but it is strong enough to bore into wood. Hence

41

this Colaptes in all the essential parts of its structure is a woodpecker. Even in such

trifling characters as the colouring, the harsh tone of the voice, and undulatory flight,

its close blood-relationship to our common woodpecker is plainly declared; yet, as I can

assert, not only from my own observations, but from those of the accurate Azara, in

certain large districts it does not climb trees, and it makes its nest in holes in banks! In

certain other districts, however, this same woodpecker, as Mr. Hudson states, frequents

trees, and bores holes in the trunk for its nest. I may mention as another illustration

of the varied habits of this genus, that a Mexican Colaptes has been described by De

Saussure as boring holes into hard wood in order to lay up a store of acorns.

Petrels are the most aerial and oceanic of birds, but in the quiet sounds of Tierra del

Fuego, the Puffinuria berardi, in its general habits, in its astonishing power of diving,

in its manner of swimming and of flying when made to take flight, would be mistaken

by any one for an auk or a grebe; nevertheless it is essentially a petrel, but with many

parts of its organisation profoundly modified in relation to its new habits of life; whereas

the woodpecker of La Plata has had its structure only slightly modified. In the case

of the water-ouzel, the acutest observer by examining its dead body would never have

suspected its sub-aquatic habits; yet this bird, which is allied to the thrush family,

subsists by diving – using its wings under water, and grasping stones with its feet. All

the members of the great order of Hymenopterous insects are terrestrial, excepting the

genus Proctotrupes, which Sir John Lubbock has discovered to be aquatic in its habits;

it often enters the water and dives about by the use not of its legs but of its wings, and

remains as long as four hours beneath the surface; yet it exhibits no modification in

structure in accordance with its abnormal habits.

He who believes that each being has been created as we now see it2, must occasionally

have felt surprise when he has met with an animal having habits and structure not

in agreement. What can be plainer than that the webbed feet of ducks and geese are

formed for swimming? Yet there are upland geese with webbed feet which rarely go

2Another incidence where Darwin’s main addressee being Creationists is highlighted

42

near the water; and no one except Audubon has seen the frigate-bird, which has all its

four toes webbed, alight on the surface of the ocean. On the other hand, grebes and

coots are eminently aquatic, although their toes are only bordered by membrane. What

seems plainer than that the long toes, not furnished with membrane of the Grallatores

are formed for walking over swamps and floating plants? – the water-hen and landrail

are members of this order, yet the first is nearly as aquatic as the coot, and the second

nearly as terrestrial as the quail or partridge. In such cases, and many others could be

given, habits have changed without a corresponding change of structure. The webbed

feet of the upland goose may be said to have become almost rudimentary in function,

though not in structure. In the frigate-bird, the deeply scooped membrane between the

toes shows that structure has begun to change.

He who believes in separate and innumerable acts of creation may say3, that in these

cases it has pleased the Creator to cause a being of one type to take the place of one

belonging to another type; but this seems to me only re-stating the fact in dignified

language. He who believes in the struggle for existence and in the principle of natu-

ral selection, will acknowledge that every organic being is constantly endeavouring to

increase in numbers; and that if any one being varies ever so little, either in habits or

structure, and thus gains an advantage over some other inhabitant of the same country,

it will seize on the place of that inhabitant, however different that may be from its own

place. Hence it will cause him no surprise that there should be geese and frigate-birds

with webbed feet, living on the dry land and rarely alighting on the water; that there

should be long-toed corncrakes, living in meadows instead of in swamps; that there

should be woodpeckers where hardly a tree grows; that there should be diving thrushes

and diving Hymenoptera, and petrels with the habits of auks.

These few paragraphs of Darwin exhibit that sometimes habits of individuals/species change

first and the structural/bodily changes follow. In the time that the structural changes have not

taken place, and that must be a slower process than the individual’s choice of change of habit, this

3and again...

43

changed habit could not have been passed on to the next generation through the genetic-based

laws of inheritance, but there must be other social means of learning and teaching through which

these new and individual habits are to generalize and spread within the population of a specie. As

is evident, Darwin’s struggle to settle the creationists’ case makes him miss the gaps being created

between his own arguments. Only because he is busy speaking to a creationist, he misses the point

that over here natural selection takes a back seat; what is categorically more important here is

the willful spontaneous change of habit (Firstness) and a conscious sustaining/fixation of the habit

(Thirdness), even though the habits do not have the structural support as of yet; which becomes

secondary for that individual’s evolutionary growth at the moment and can come in later through

natural selection preserving those variations in structure that are in the direction of well-sustained

and taught habits of the species (Secondness).

So for Evolution to take place in reality, sometimes evolution by habit-taking becomes

more important than the ‘mechanisms’ of natural selection.

3.1.2 Evidence for Peirce’s Firstness in the biological world

Even though Darwin had initially insisted on the absolute independence of the processes of variation

and selection, in his later editions he conceded about the “variations which seem to us in our

ignorance to arise spontaneously. It appears that I formerly underrated the frequency and value of

these latter forms of variation, as leading to permanent modifications of structure independently

of natural selection” (Darwin, 1872, page 421).

This, in Darwin, opens up multiple possibilities:

a) that of a relation between the processes of variation and selection, as indicated by recent molecu-

lar biology findings where the rates of mutation as well as genomic rearrangement for an individual

get affected by stress under ecological challenges, often resulting in positively evolving the genome

(Shapiro, 2011)

b) redefining chance as a blind uniform distribution that does not get effected by anything outside

of it, to other distributions such as Gaussian/Normal. This would mean that chance has its own

44

habits as well, and that the outcomes of events involving randomness are not necessarily distributed

in a blind and habit-less way, all equally likely and uniformly. Fortunately we do not have to explain

this part as Probability is an established and well understood domain of mathematics now.

c) legitimizing the saltation sholarship, in which large genomic changes have historically taken

place under special circumstances in quick jumps, as opposed to a strictly neo-Darwinian slow and

continuous accumulation of one advantageous change at a time, naturally selected over long raging

times.

d) legitimizing neo-mutationist scholarship which ascribes the role of creativity to processes of

spontaneity as well, previously strictly associated with natural selection.

So for Evolution to take place in reality, sometimes evolution by spontaneity becomes

more important, sometimes evolution through natural selection, and sometimes evolu-

tion through habit-taking. If that is what the empirical evidence suggests, and that is

also what Peirce forwards as his evolutionary conception, then surely Peirce can help

take out the controversy in Darwin. Same is the case with evolution only through competition

in Darwin vs. evolution also through cooperation in Peirce (see Appendix A). Having seen Peirce’s

categories working in the biological world, let us now turn to their technical definitions.

3.2 Understanding Peirce’s Evolutionary Framework through Def-

initions

This section details how Peircean theory is precisely a synthesis of the above-mentioned three

different types of phenomena, in which the inter-relations of Firstness, Secondness and Thirdness

is what defines what evolution is, not a reduction to any one of the three - universally, and under

all conditions.

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3.2.1 Firstness

Out of the womb of indeterminacy we must say that there would have come something,

by the principle of Firstness. (Peirce, 1958, 1.412)

The classical view of the world assumes that all specifications of the universe go back to the

beginning, where “chance” played its utmost part to put in place all the perfect values to all the

respective variables that shall ever need to operate. The Peircean framework gives an explanation

in which spontaneity plays a dynamic role not only in the initial event but also in continual growth.

By thus admitting pure spontaneity or life as a character of the universe, acting always

and everywhere though restrained within narrow bounds by law, producing infinitesimal

departures from law continually, and great ones with infinite infrequency, I account for

all the variety and diversity of the universe, in the only sense in which the really sui

generis and new can be said to be accounted for. (Peirce, 1958, 6.59)

3.2.2 Secondness

Where the idea of Firstness is characterized by independence from everything else, the idea of

Secondness is precisely that which is necessitated by some First. The units of Firstness did not

spring up in isolation, but “in accidental reaction upon one another, and thus into a kind of

existence.” (Peirce, 6.199) Peircean Secondness is encapsulated by functions such as reaction,

another, compulsion, negation.

Generally speaking genuine Secondness consists in one thing acting upon another, –

brute action. I say brute, because so far as the idea of any [regularity] or reason comes

in, Thirdness comes in. When a stone falls to the ground, the law of gravitation does

not act to make it fall. The law of gravitation is the judge upon the bench who may

pronounce the law till doomsday, but unless the strong arm of the law, the brutal sheriff,

gives effect to the law, it amounts to nothing. True, the judge can create a sheriff if

need be; but he must have one. The stone’s actually falling is purely the affair of the

stone and the earth at the time. This is a case of reaction. (Peirce, 1958, 8.330)

46

Although Darwinism argues that the process of evolution reflects the balance between chance

and laws, Peirce rejected the idea that such a balance offers any adequate explanation of the physical

universe and the natural world as we know it. He demonstrates that Thirdness is a prerequisite for

evolution and eventually for life.

3.2.3 Thirdness

The mode of being which consists in the fact that future facts of Secondness will take

on a determinate general character, I call a Thirdness. (Peirce, 1958, 1.26)

Where Second is always instantaneous, Third shall always have a process involved. As soon as

the idea of relation, pattern, logic, reason or law manifests itself in any phenomenon, it is due to

the presence of Thirdness. It is precisely because of this Third or tendency to generalize or take

habits that the above mentioned ideas are possible in reality.

[The tendency to take habits] is a generalizing tendency; it causes actions in the future

to follow some generalization of past actions; and this tendency is itself something

capable of similar generalizations; and thus, it is self-generative. We have therefore

only to suppose the smallest spoor of it in the past, and that germ would have been

bound to develop into a mighty and over-ruling principle, until it supersedes itself

by strengthening habits into absolute laws regulating the action of all things in every

respect in the indefinite future. (Peirce, 1958, 1.409)

According to Peircean conception then, three elements are active in the world:

first, chance; second, reactive mechanisms; and third, habit-taking. It is with the

interrelation of these three elements that Peirce explains the evolution of the world

from a state of nothingness to the one that we experience now.

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3.3 Understanding Peirce’s Evolutionary Framework through a

Possible Synthesis

The following observation by Peirce shows the relationship between his triadic logic of three cate-

gories, non-biological evolution and biological evolution in the universe.

Three conceptions are perpetually turning up at every point in every theory of logic,

and in the most rounded systems they occur in connection with one another. They

are conceptions so very broad and consequently indefinite that they are hard to seize

and may be easily overlooked. I call them the conceptions of First, Second, Third....

In psychology Feeling is First, Sense of reaction Second, General conception Third,

or mediation. In biology, the idea of arbitrary sporting is First, heredity is Second,

the process whereby the accidental characters become fixed is Third. Chance is First,

[Selection] is Second, the tendency to take habits is Third. Mind is First, Matter is

Second, Evolution is Third. (Peirce, 1958, 6.32)

For Peirce, the most important philosophical implication of Thirdness is that there is meaning

and intelligibility in the universe (Peirce, 1958, 1.366). If evolution is restricted to the biological

domain then both meaning and intelligibility are severely undermined, if not completely obliterated.

As is the case with evolution at the non-biological levels, evolution at the biological level cannot be

understood or explained only with the aid of two categories. This means that Peirce is critiquing

both Darwin’s theory (which considers evolution to be the product of only chance and necessity)

and Lamarck’s theory (which considers evolution to be the product of necessity and habit). Peirce

recognizes that there have been three primary characteristics for the explanation of natural process

of evolutionary growth: Neo-Darwinism has primarily relied on chance (Firstness), Spencer on

mechanical principles (Secondness), and Lamarck on habitual use and disuse of organs (Thirdness).

According to Peirce they all suffer from one acute problem: by reducing the explanation of evolution

to one, two or any combination of two categories, the theory that thus forms cannot independently

and consistently explain the process of evolution4. Cosmological, geological and biological evidence

4This argument can benefit from a discussion on Peirce’s Reduction Thesis as well, but due to limitations of spaceit is left out as a future possibility.

48

supports Peirce’s claim.

Biological development cannot be separated from geological developments on earth. The geo-

logical record shows the same pattern or habit identified by Peirce for the cosmos: progressively in-

frequent catastrophic geological changes have affected the otherwise stabilizing biological life forms.

Citing (King, 1877), a leading geologist of his day, Peirce says: “the testimony of monuments and

of rocks is that species are unmodified or scarcely modified, under ordinary circumstances, but are

rapidly altered after cataclysms or rapid geological changes. Under novel circumstances, we often

see animals and plants sporting excessively in reproduction, and sometimes even undergoing trans-

formations during individual life, phenomena no doubt due partly to the enfeeblement of vitality

from the breaking up of habitual modes of life, partly to changed food, partly to direct specific

influence of the element in which the organism is immersed.”

If evolution has been brought about in this way, not only have its single steps not been

insensible, as both Darwinians and Lamarckians suppose, but they are furthermore nei-

ther haphazard on the one hand, nor yet determined by an inward striving on the other,

but on the contrary are effects of the changed environment, and have a positive general

tendency to adapt the organism to that environment, since variation will particularly

affect organs at once enfeebled and stimulated. This mode of evolution, by external

forces and the breaking up of habits, seems to be called for by some of the broadest and

most important facts of biology and paleontology; while it certainly has been the chief

factor in the historical evolution of institutions as in that of ideas; and cannot possibly

be refused a very prominent place in the process of evolution of the universe in general.

(Peirce, 1958, 6.17)

Peirce’s theory does not portray itself as a thesis for cataclysm or as an antithesis of Darwin

or Lamarck. His triadic theory only embraces the advancements in the abovementioned classical

theories, while avoiding some of their most glaring shortcomings. Not surprisingly, Peirce’s view of

biological evolution fits well with the fossil record as well as with the latest contentions of the evolu-

tionists mentioned in the introduction. For instance, Gould’s conception of punctuated equilibrium

which is at stark contradiction with Darwinian gradualism, sits well with Peircean cataclysm. The

49

Peircean provision, that under specific conditions the mutations have a tendency to adapt as op-

posed to being pure blind chance universally, is in line with the revisionist mutationists’ claims.

Instead of reducing the entire focus to genes, this theory suggests a multiplicity of mechanisms

from environment to metabolism, which is what the molecular findings of epigentics and systems

biology demand. Generalizing tendency and relationship, which are the manifestation of Peircean

Thirdness, are precisely those terms that Lynn Margulis is looking for in an evolutionary theory

that could support the evidence of symbiogenesis (see Appendix A). Philosophically speaking, what

takes the controversy out of evolution is the fact that for Peirce real chance has patterns and real

laws have exceptions.

3.3.1 Definition :: Peircean Evolutionary Theory

One might still be wondering as to what is the equivalent Peirce’s single sentence definition for the

theory of evolution. After all, we did give our version of Darwin’s theory in the last chapter. Here

is our encapsulated Peircean synthesis:

The processes of growth and selective adaptation to its environment are

characterized by the formation of new or the modification of old habits.

This relational definition first of all uses such general language that it avoids major controversies

especially in the wake of the modern biological developments (as listed in section 2.2). But more

importantly, this has room for both freedom and consciousness as exhibited in the higher life

forms, as well as the mechanisms exhibited by the lower life forms. Lastly, it brings into relation

spontaneity, adaption/selection and habits; or Firstness, Secondness and Thirdness in Peircean

terms.

3.4 Nature of Chance and Laws in Light of Modern Physics and

Cosmology

We now turn our gaze from biology to physics - the mother of natural sciences. The world Darwin

belonged to was still coordinated by Newtonian physics, and the assumption that the universe is a

50

perfect Cartesian machine was at work in most theories of 19thcentury, including that of Darwin’s.

We have already seen his conception of chance, how it was not an active agency, but only filled the

gaps of our ignorance for the time being. The other side of the same coin then is the assumption

that the laws are immutable and eternally perfect. This section has two objectives, a) to show

the transition in scientific understanding of these two categories, chance and laws, and b) to show

how Peirce had anticipated such a universe and how his categories of Firstness, Secondness and

Thirdness fit the description provided by the 21stcentury physics. Since we are not physicists

ourselves, we will bring in three quantum physics authorities to present our case.

3.4.1 Heisenberg on the Nature of Reality

The mechanistic world view has an operational assumption, that any event can be precisely worked

out by first breaking down our understanding to the most basic and elementary particles and then

putting it all together “accurately” in a bottom-up fashion. Heisenberg exclaims that the nature

of reality is not this simple any more:

What is an elementary particle?...If one wants to give an accurate description of the

elementary particle and here the emphasis is on the word ‘accurate’ the only thing which

can be written down as description is a probability function. But then we can see that

not even the quality of being... belongs to what is described. It is a possibility for being

or a tendency for being. (Heisenberg, 2007, pg. 44)

The probability function does unlike the common procedure in Newtonian mechanics

not describe a certain event but, at least during the process of observation, a whole

ensemble of possible events. (Heisenberg, 2007, pg. 28)

3.4.2 Weyl on the Historical understanding of Reality

Hermann Weyl, being a theoretical physicist as well as a historian of mathematics, shows in his

primer how the human understanding of laws of nature has moved from the search of determinism

to that of statistical regularities:

51

The idea of functional law, to which science seems to reduce causality, is not altogether

unproblematic. Twice in its history physics believed that it had overcome in principle

the decomposition of the world into individual systems (individual events and their

elements, which after all are only approximately isolated from one another) and had

grasped the world as “a whole in which all is interwoven.” The physics of central

forces and later the pure field physics seemed for a moment to have reached that goal.

Causal law here took the following form: the derivaties with respect to time of the

state quantities at a world point are mathematical functions of the state quantities

themselves and their spatial derivatives at that point. Consequently, the state of the

world at any moment would determine the state at the immediately following moment

by means of differential laws. Thus only the world’s state at a single moment would

remain ‘arbitrary’ or ‘accidental,’ and from it the world’s whole past and future could

be computed by integration of the ‘Laplacean world formula.’ (Weyl et al., 2009, pg.

190-191)

The classical definition of quantitative probability coined by Laplace – the quotient of

the number of favorable cases over the number of all possible cases – emphasizes the

objective aspect. Yet, this definition presupposes explicitly that the different cases are

equally possible. Thus it contains as an aprioristic basis a quantitative comparison of

possibilities. ...It is because of the arbitrariness of such a measure that Laplace, from

his consistently deterministic conception of nature, is in the end unable to ascribe to

probability anything but a subjective meaning; it deals with events whose premisses

are incompletely known, and thus is “relative to this our knowledge and ignorance.”

Laplace therefore calls two events equally possible if we are equally undecided as to

their occurence. (Weyl et al., 2009, pg 195)

In the fourth part of his Ars conjectandi, Jacob Bernoulli throws the bridge from the

subjective to the objective conception of probability by means of his “law of large

numbers.” According to this objective interpretation the probability calculus serves

to establish regularities expressed in the mean values of many similar events rather

52

than in the individual event. Beside the strictly valid causal laws we thus have

regularities of a statistical nature. (Weyl et al., 2009, pg 196)

As long as one believes in strict causality, statistics must find its proper foundation in

a reduction to strict law. If, however, there should be a ‘primary probability’ for the

individual atomic events that cannot be reduced to causal laws – and such seems to

be the case according to the most recent development of physics – then we seem to be

forced to introduce into the natural laws as an original factor either that probability

itself or some quantity connected with it. (Weyl et al., 2009, pg 198)

Most of the physical concepts, especially those concerning matter with its atomic struc-

ture (e.g. the density of gas), are not exact but statistical, that is, they represent mean

values affected with a certain degree of indeterminancy. Similarly most of the usual

physical ‘laws,’ especially those concerning matter, must not be construed as strictly

valid laws of nature but as statistical regularities. (Weyl et al., 2009, pg 199)

At any rate, in the actual conduct of physical research, statistics today plays at least as

important a part as the strict law. Attempts to reduce one to the other have gradually

fallen back behind the independent building up of a statistical thermodynamics. Of

the two laws which are of universal significance for all physical phenomena, the laws

of conservation of energy and the law of continuously increasing entropy, one is the

prototype of a strict law, the other of a statistical law. ...From all that has been said

it will be clear how little contemporary physics, based as it is half on laws and half on

statistics, can pose as a champion of determinism. (Weyl et al., 2009, pg 202-211)

This historical account puts an end to the world as known by Darwin. We would let Gerald

Holton conclude - a research professor of physics as well as a historian of science at Harvard.

3.4.3 Holton Concludes

Max Born was awarded the Nobel Prize in Physics in 1954, primarily for his statistical

interpretation of quantum mechanics. The fact that the award came so long after his

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original publication (1926) indicates that this interpretation was not at once regarded

as an indubitable, basic “discovery,” but that it gradually came to be accepted as a

fruitful viewpoint in physical theory. Insofar as many scientists do accept the notion

that natural processes are fundamentally random rather than deterministic, we have to

admit that the “Newtonian world machine,” or the mechanistic viewpoint advocated

by Descartes and Boyle in the seventeenth century, has been abandoned. (Holton and

Brush, 2001)

Summary

What becomes blindingly obvious after reading the above sections is how the quantum physics of

21stcentury and Charles Sanders Peirce are almost on the same page, and how this new conception

of reality is different from that of Darwin’s. Mainly because chance has an element of law at work,

and laws have an element of chance at work:

a) Peirce embedded in his evolutionary framework such a notion of chance that is actually an active

agency rather than the old notions where chance only represented one’s ignorance,

b) For Peirce, laws of nature are not immutable, universal and unintelligible as imagined in the

past. Instead they are characteristically fallible, just like our interpretations about them.

c) But more importantly, Peirce managed to establish a relation between the creative and spon-

taneous chance and increasingly regular laws by placing a Third active agent between the two:

habit-taking tendency of generalization.

With this triadic evolutionary framework, Peirce brings natural laws into the umbrella of evo-

lutionary inquiry as well, and becomes readily relevant for the contemporary and future human

understanding and explanation of the universe (Smolin, 2013). This explanatory power over the

other evolutionary accounts, which are divorced from any extra-biological development, not only

gives Peirce’s theory a scientific superiority in principle, but also gives us a warrant to venture

into evolutionary algorithms based on Peirce’s theory of evolution. Lastly - and we hope Schwefel

agrees by now - Peirce’s theory goes a long way to meet the conditions that Schwefel foresees for a

universal theory of evolution:

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Current evolutionary algorithms are certainly better models of organic evolution. Nev-




within the whole universe. (Schwefel, 1997)

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Chapter 4

Peircean Framework for Evolutionary

Algorithms

This chapter begins with a discussion on the principles derivable from Peirce’s theory of evolution

from EC’s point of view, and ends with a description of a detailed Peircean evolutionary algorithm.

The objective of this chapter is thus, to make the transition possible; the transition from a complex

and philosophical evolutionary theory to a much simpler, pseudo-codable algorithm - just like John

Holland once managed to do for GA.

4.1 Principles derived from Peirce’s theory for Evolutionary Al-

gorithms

In basic terms, the standard EA implementation is a two-pronged strategy that appears in EA

literature under different names: “exploration-exploitation,” “variation-selection,” and “chance-

necessity” (Beyer et al., 2002). If we could rewrite that in Peircean terminology, it would roughly

translate into “Firstness-Secondness.” The foremost difference is that the Peircean EA would have

a Third (generalizing) element working simultaneously, an element that we implement in the form

of clustering. A second difference is that for Peirce the meaning of “Firstness” is not confined

to variation, and the meaning of “Secondness” is not captured entirely by selection, either. But

57

thirdly and most importantly, the relationship and interplay between the three elements of evolution

build an entirely different system, and hence those elements get translated into an entirely different

evolutionary algorithm. This section elaborates upon the different meanings that can be derived

from the Peircean concepts from the standpoint of evolutionary computation. Next section explains

the new Peircean EA.

4.1.1 Firstness as Spontaneity

Unlike the classical EA, where random population is only generated once - at the beginning, in

Peircean EA, Firstness is an ever present phenomenon which means that there has to be a provision

for the random individuals to be generated every iteration. This could raise the possibility of

population explosion in case the number of iterations increases. We could easily work our way

around that, as shall be explained in the next section (in case keeping a constant population is an

objective).

4.1.2 Secondness as Necessity

Through Secondness, Peirce gives us many different operators that can be employed to change the

population or to generate the next generation of individuals under the brute forces of reactions.

In Peircean language the operators are: Compulsion, Dependence, Negation, Another. If we were

to categorize them, we would find that some function of crossover points could be employed to

develop the operator ‘Another’. The selection operator fits Compulsion but it could also be used for

affecting the population through the external, biased input. We will discuss that shortly. Different

functions of mutation could be used to actualize operators like Independence and Negation. In

short, the standard operators of recombination, mutation and selection can be extended to devise

the Secondness operators.

4.1.3 Thirdness as Generalizing tendency

The most striking feature of Thirdness is the generalizing tendency translating into collaboration

and cooperation among organisms. This new paradigm gives us two directions. First and foremost,

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it strongly suggests that the algorithm has an inclination towards collaboration between individuals

i.e., after Firstness has taken place and new individuals have been randomly initialized, they interact

with each other using Secondness operators but brought into relation by clustering (Thirdness).

Clustering distributes the population into smaller sub-populations, each cluster representing the

generalization of its constituent individuals. Clustering represents Thirdness in that, in Peircean

EA, intra-cluster level operators like crossover and selection (Secondness) are taking place but

only on those individuals (Firstness) that are being related with each other through the long-term

bonds of each cluster. Thirdness is that which relates Firstness and Secondness, and clustering is

doing precisely that and cannot be reduced to Secondness. Secondly, this paradigm opens up new

possibilities for survival; instead of a singleton ‘survival of the fittest,’ the Peircean framework gives

us another epitome of survival: ‘collaboration.’ The more an individual is part of a collaboration,

the greater its chances of surviving. The criterion for clustering is not fitness-based; rather, it is

spatial, binding each cluster to its own contextual environment more naturally.

4.1.4 Dynamic System

Peirce acknowledges the individual organism as actively taking part in the evolutionary process.

Hence the system is not closed like the Darwinian system. This gives us four openings in the field

of Evolutionary Computation, and further establishes the explanatory power of Peircean framework:

1. Interactive Evolutionary Computations, where the fitness or objective function is scored by

a human operator, is now justified in this dynamic framework, which had to be incorporated

in classical EA by moving outside the Darwinian model - just to achieve results.

2. Unlike in classical EA where there is no role for external bias or prior information, Peircean

EA provides a semiotic framework where, external bias is acknowledged, accounted for, and

allowed. It could be initially taken as an input, which, through the operator of Compul-

sion acts as a guide for the development of future generations. In the absence of any prior

information about the problem at hand, the algorithm should work unaffected.

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3. Schwefel’s meliorization becomes a possibility now due to individual’s Firstness. We do not

know how to implement it into an EA for now, but it is a possibility nevertheless for the future,

where the agent can have the will and consciousness to shape the present in accordance to a

future in mind.

4. Peirce recognizes evolution taking place at multiple levels of reality in parallel instead of

reducing the entire process to the genetic or individual level. In the Peircean framework

Firstness, Secondness and Thirdness work at all levels of natural (or algorithmic) reality and

in our case at Individual, Cluster and at Global level.

4.1.5 Reality at Multiple Levels

For Peirce, evolution by Firstness, Secondness and Thirdness takes place at multiple levels of reality

in parallel instead of solely at the genetic or individual level. Figure 4.1 allows us to see a snapshot

of the search space at one hypothetical generation along with the key agents/levels involved in

the Peircian development of an Evolutionary Algorithm. In the Peircian framework Firstness,

Secondness and Thirdness have to work at all levels of natural reality and in our case at Individual,

Cluster and Global level. For a complete understanding of this new paradigm we shall like to

establish the exact roles played by Firstness, Secondness, Thirdness on all levels of our algorithmic

reality, namely: Individual, Cluster and Global levels as seen in Figure 4.1. This would eventually

turn into a 3 x 3 matrix of relationships shown in Table 4.1. This exercise will help us to extract

principles from Peirce’s theory and to give a final shape to our EA in the next section.1

1But is this whole cluster scheme not analogous to Darwin’s group selectionism? The analogy fits only at asuperficial level. If we take the analogy at face value we cannot explain the emergence of clusters to begin with.Clusters are those phenomena in empirical reality where individuals cooperate with each other and pass on acquiredhabits to the next generation. Based on the accumulated (and accumulating) evidence in biology, selection is awoefully inadequate concept to explain the reality of cooperation, habit formation, etc., in nature. It is obviously thecase that selection does not disappear once clusters emerge, and selection is important in helping us to understandthe survival of clusters once they have emerged-but selection is a blinder that obscures some of the key dynamicsthat are at work during the origin/emergence of clusters. In a way, our main contribution is the working out of therelationship among Firstness, Secondness and Thirdness at individual, cluster and global levels of our algorithmicreality. It is out of this exercise that we then devise the algorithm. Now at this level the analogy of our proposedalgorithm with a group selectionist model does not hold.

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Figure 4.1: Hypothetical 2D search space depicting a cluster formation within the population

Firstness at Individual level

Spontaneity, chance, and complete freedom are some of the ideas associated with Firstness. At the

individual level, the mutation operator introduces this spontaneity to bring about something novel.

Firstness at Cluster level

The cluster center is the individual which represents its respective cluster and is thus distinct

from other members of that cluster. A mutated cluster center would result in the introduction of

spontaneity at the cluster level.

Firstness at Global level

Each cluster has its own fittest individual that can also be called the cluster ideal. The set of these

fittest individuals from each cluster form the global representations. Mutated cluster ideals are

responsible for the introduction of spontaneity at the global level. Another mode of spontaneity at

the global level is that of cataclysmic changes, such as those that take place at the first generation,

61

when totally new elements are introduced in the search space, and these cataclysmic new elements

could also be introduced in later generations with a low probability.

Secondness at Individual level

Keywords that correspond to Secondness are “necessity” and “compulsion”. At the individual level

Secondness manifests itself in the form of genetic heredity - the individual is compelled to receive

its genetic blueprint from its parents and then to pass it on to the next generation. Crossover will

play the role of the Secondness operator and bring two individuals to react and produce offspring.

Secondness at Cluster level

At the cluster level various elements are involved at the same time to encapsulate Secondness. First,

the selection operator is responsible for retaining the fittest individuals or cluster ideals. In turn,

these cluster representatives play an active role in crossover for the next generation. Cluster-based

survival is also an indication of Secondness at cluster level.

Secondness at Global level

Just as intra-cluster interactions are the key aspect of Secondness at the cluster level, at the global

level inter-cluster dynamics play their part. Cluster representatives or the cluster ideals exchange

information across the clusters using the Secondness operators. This information exchange, between

geographically or functionally fittest individuals, becomes the cause for making the entire cluster

move across the search-landscape. It could result in either cluster unification or diversification in

the long run. Selection towards the global ideals is also a part of Secondness at the global level.

Thirdness at Individual level

Thirdness is best described by the keywords “representation,” “relationship,” “continuity,” “habit,”

“generalization,” “pattern,” “regularity,” etc. At the individual level Thirdness appears as the

representation of a chromosome or individual’s size, and all other parameters dictated by the

problem that the EA is going to solve or the function that is being optimized using the Peircean

62

EA.

Thirdness at Cluster level

Thirdness at the cluster level gives us another technique for a mating operator. The more the

strength or objective function’s value of an individual, the more is the effective radius of that

individual, and within that radius there is a higher probability of it crossing with other individuals.

However, this has not been tested in Peircean EA in this study.

Thirdness at Global level

At the global level Thirdness takes the form of generalization. The very idea of clustering stems

from this generalization. All the parameters for cluster radii or number of clusters at any time

are specified dynamically at this level. Other than that, all parameters that help optimize the

evolutionary algorithm are dynamically set by the Thirdness factor, for example, crossover or

mutation rates, prior information or bias, efficiency improvement measures, memorization of states,

etc.

Table 4.1: Relationship of Peircian principles at different levels of algorithmic realityIndividual Cluster Global

Firstness mutation mutated cluster center mutated cluster ideals

Secondness crossover cluster based survival,selection towards clus-ter ideals

inter-cluster interac-tion, selection towardsglobal ideals

Thirdness meaningful representa-tion, problem specificparameters

radius based mating op-erators

cluster formation, priorinput bias, optimizationparameters

4.2 Peircean Evolutionary Algorithm

The preceding section has laid down the general framework for designing evolutionary algorithms.

The Thirdness principle’s most important contribution is that the population shall cooperate and

survive in cluster-based communities. The Secondness principle dictates the terms under which

individuals and clusters interact with each other through various operators of recombination and

63

selection. The Firstness principle introduces and retains novelty in the population through various

operators of variation. Figure 4.2 lays out the basic algorithm in the form of a pseudo code.

Figure 4.2: Algorithm 1: Pseudo code for Peircean Evolutionary Algorithm

A variety of algorithms could be devised based on multiple interpretations of the Peircean

principles listed in the previous section. This multiplicity is not an issue as long as the interpretation

or the algorithm does not violate the spirit of Peirce’s evolutionary framework. While stressing

the need for a synthesis within the EC community, our claim is that this general framework has

the potential to act as a theoretical foundation for many variant EAs, especially co-evolutionary

and the like. We have only devised one instance of the otherwise general Peircean framework for

carrying out the experiments.

Algorithm 1 when stated simply is this: Until the stopping criteria are met, 1) distribute the

population in clusters. 2) First each cluster internally generates its next generation. 3) Next each

cluster’s fittest individuals make an inter-cluster information exchange. 4) Finally a small number

of new individuals also get introduced into the population. However, it shall be helpful if we take

a step by step tour of this P-EA:

Line 1: Initialization In case we do not have prior knowledge, the population is initialized in

the standard way, using uniform random distribution in the first generation. If we have any prior

bias available that we want to add to the first generation, we do that by either generating population

through a prior distribution (instead of using uniform distribution) or by simply inserting a few

individuals which encode the prior information along with other randomly generated individuals.

64

Line 2: Stopping criteria There are three stopping criteria for the algorithm: 1) minimum

accepted fitness value achieved by any individual in the population, 2) maximum number of gen-

erations exhausted, which is 10,000 generations, or 3) if the minimum value of the fitness of the

population does not improve for a certain number of generations beyond the first 1000 generations.

This certain number will be referred to as stop-count from now onwards.

Line 3: Parameter tuning While the other parameters that are listed in Table III remain

constant, we have experimented with two parameters: Cluster radius and mutation rate. Very

simple heuristics are employed. Mutation rate is logarithmically decreased from 0.2 to 0.02 in

about the first 1000 generations. There on, it becomes static at 0.02. In the case of cluster

radius, it starts from a constant number. In case the number of clusters become less than 2, the

cluster radius is decreased linearly relative to the previous generation, and in case the number

of clusters exceed 6, the cluster radius is relatively increased. While these are not precisely self

adaptive parameters, and the numbers for minimum and maximum bounds might be arbitrary, but

nevertheless Peircean EA has this potential that is to be exploited for future work.

Line 4: Cluster analysis We have used subtractive clustering which takes cluster radius

for input instead of specifying the number of clusters and returns the number of clusters and

cluster center representatives from the population (Chiu, 1994). It is a single pass algorithm but

notice how adding the procedure of clustering to the iterations of an EA does not add much to its

running time complexity as clustering is a polynomial time operation with respect to population

size. Since precise clustering is not the problem we want to solve, we could use any efficient linear

time algorithm available for that matter (Han et al., 2006). In any case it does not add to the

number of function calls to the objective function.

Line 6: Intra-cluster evolution This procedure is the heart of P-EA and its pseudo-code

is expressed as algorithm 2 in Figure 4.3 as well. Its function is to take one cluster at a time and

produce the next generation while identifying each cluster’s fittest individuals. One thing to note

here is that no special selection operator is involved. Parents are chosen using a uniform random

distribution which does introduce a very small selection pressure (as a handful of cluster individuals

get selected more than once), but other than that no special selection is done. Even though we

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did mention the possibility of experimenting with various Secondness operators, we have restricted

to the standard multi-point recombination operator and standard mutation operator. This assures

that any improvements in the results over a classical EA are attributed to the Peircean framework

and not to any complex operators. Since there is always a chance of a few individuals getting

selected more than once in line 3 of algorithm 2, the following relationship holds:

Size of (parents + non-parents) ≥ Size of (Input cluster)

Size of (parents) == Size of (children)

Size of (children + non-parents) ≥ Size of (Input cluster)

Hence to keep the cluster population size constant, in line 9 of algorithm 2, the fittest among

children and non-parents are selected to represent the next generation of the cluster. Notice this is

not elitism, since parents are completely replaced with their children, and there is always a chance

of losing the previous fittest individual. The minus 1 at the end of the above-mentioned line is to

counter the effect of line 9 of algorithm 1. It keeps the total population size in place.

Figure 4.3: Algorithm 2: Pseudo-code for intra-cluster evolution

Line 8: Inter-cluster evolution Inter-cluster evolution is in many respects similar to Intra-

cluster evolution. The difference is that instead of a single cluster, representatives from each cluster

recombine and mutate to produce the next batch and the fittest are chosen in an elitist selection.

The pseudo-code is exhibited in Figure 4.4 as algorithm 3.

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Figure 4.4: Algorithm 3: Pseudo-code for inter-cluster evolution

Line 9: Add random individuals to population Last but not the least, through Firstness,

in every generation new individuals come into being and become part of the current population.

They are of two types. Some of the individuals are generated absolutely randomly, the way the

first generation comes about. Some random individuals are mutated versions of cluster centers

and cluster fittest. We keep a uniform distribution between the two types. To avoid population

explosion we devised a mechanism in which (x * total clusters) new individuals are added to the

population in each generation, and likewise x weakest individuals farthest from their respective

cluster centers are eliminated. This is another definition of survival of the fittest: The more an

individual is a part of a cluster, the more its chances of survival.

4.3 Comparison of Peircean EA with Related EA

This section is the counterpart of section 2.5. While in chapter 2 some alternatives to classical

Darwinian EA were presented, here some of them are to be discussed in light of the Peircean EA.

One thing to note is that the P-EA is a very minimanlist algorithm. It is not advance, in the sense

of having outrageously “unnatural” components to it. This aspect of the P-EA contrasts with the

advance (non-Classical) EA variants discussed in section 2.5, and we leave the discussion at just

that, and get to the Peirce-related EAs.

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4.3.1 IDEA 1 :: Diversity Retaining Measures

Most diversity maintaining techniques have come to the community as an afterthought - after

the problem of stagnation had been reported from various pockets of EC community. Hence, the

techniques, when embedded into the EAs seem forceful and artificial. P-EA on the other hand,

does not go for the stagnation-avoidance per se. It is because of the persistent presence of Firstness

as a real entity in a Peircean world-view, that raw genetic material gets continually introduced

within the population. In other words, stagnation gets tackled as a result of the natural translation

of Peirce’s theory into an algorithm, not because it is a special challenge within the EC community

that needs to be dealt with.

4.3.2 IDEA 2 :: Distributed Population Dynamics

This section outlines the differences of Island Model EA (IMEA) and Diffusion Model EA (DMEA)

with P-EA.

1. All three are based on different underlying conceptions of evolution that translate into their

respective implementations.

2. Elaborate set of parameters, on top of the standard selection, recombination and mutation, are

required in IMEA such as distribution of subpopulations, communication topology, migration

scheme, number of migrants, assimilation scheme etc. As opposed to that current P-EA only

requires radius based clustering and Firstness operator parameters additionally. Hence P-EA

is really a very simple EA model in comparison.

3. As the parallel processing of EA matured it made sense to do an independent component-

wise analysis of IMEA and DMEA, for instance, by keeping everything else the same and

varying the number of subpopulations. The simple P-EA currently does not have independent

components in the same sense. For instance, we could show the difference in results by turning

off the Firstness at one level, but really the Firstness at other levels would still be operative.

4. IMEA as well as DMEA are designed to exploit the efficiency of parallel processing, hence

global selection is avoided as much as possible since it hurts the efficiency and the whole point

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in parallelization of EA. P-EA are not designed with such bias in mind. In that sense it is

more general.

5. Localization in IMEA or DMEA are artificially constrained by the architecture whereas P-EA

promises a natural yet dynamic division of population based on clustering, within which the

local selection and recombination etc. make more natural sense.

The fact that IMEA is based on Gould’s theory and not the classical Darwinian gradualism

makes our case easier to understand, which is: A different theory or conception of natural evolution

would result in a different EA model which would have different powers. It is still possible for

someone to state that IMEA is purely Darwinian. However, as evident, that is not the case.

Similarly P-EA might have similarities with many models, but it is based on a different theory of

evolution.

4.3.3 IDEA 3 = IDEA 1 + IDEA 2

(Hu et al., 2003) have incorporated both ideas, but they have extended a hierchical fair competition

(HFC) model (as exhibited in sub-cultures and sub-populations within societies, e.g. school systems

(Hu and Goodman, 2002)) for parallel implementation of multi-objetive evolutionary algorithms

(MOEA). They explicitly say that this phenomenon of fair competition between the individuals

having a similar fitness range, not with individuals outside that range, is demonstrated in some

parts of biological and some parts of societal life; not applicable at any wider scale. The P-EA is

general in two respects: 1) It is based on Peirce’s universal theory of evolution, and 2) P-EA is a

general framework which can even provide a theoretical basis for HFC based multi-objective EAs

- this relation does not hold true the other way around.

(Amor and Rettinger, 2005) propose that a successful EA should have three characteristics:

1) good exploration early on, 2) good exploitation later on, and 3) an introduction of novelty

within the population throughout. They use a special type of artificial neural networks called

self-organizing maps (SOM) to help achieve the three tasks. Our contention is that this choice of

SOMs and explicit search histories is not backed up by any natural evolutionary theory. Peirce can

provide them that!

69

Conclusion

(Hu et al., 2003) state in their conclusion that the “current MOEAs still suffer from their convergent

nature inherited from the conventional EA framework.” We go one rational step further and say

that the conventional EA frameworks inherit the problem from the conventional dyadic theory

of evolution. This additional Thirdness and a real Firstness in Peirce brings the balance in the

exploration-exploitation equation.

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Chapter 5

Experimental Evidence

This is a good time for a re-cap. Darwin’s biological theory of evolution has been successfully estab-

lished, to the extent that it has even found its use in computer science in the shape of evolutionary

algorithms. However, and in time, some of the scientists working in evolutionary sciences seemed

to have hit the wall, with no way out of their problems, while using a strictly Darwinian toolbox.

By now what was once called a modern synthesis has broken down into varying directions which

seem less and less Darwinian in the strictest sense of the word. These are groups with ideas such

as epigenetics, systems biology, neo-mutationists, symbiogenesis, etc. One major problem now in

evolutionary biology is how to bring a new evolutionary synthesis to hold all these varying ideas

together. We have briefly presented Peircean framework as a potential synthesis for these diverging

groups in section 3.3.

Similarly, engineers working with classical evolutionary / genetic algorithms also hit the wall

when they encountered the problem of stagnation. Some have proposed engineering solutions to

varying degree of success, but often at the cost of getting divorced from the underlying natural

theories of evolution (Lozano et al., 2008; Mahfoud, 1995; Sareni and Krahenbuhl, 1998). We

propose a route once followed at the beginning by John Holland, and quite recently reinforced by

Schwefel. We believe that the solution for the problems of EC, inlcuding stagnation, are to come

from nature. We have thus far defined Peirce’s theory of universal evolution and established how

it is more modern and more explanatory than Darwin’s theory. We have also described how a

71

Peircean EA can be carved out of Peirce’s theory of evolution. It is now time to test the strength

of our interpretation of Peircean ideas.

5.1 Experimental Setup

The objective of our experiments is to compare the results of classical Darwinian EA (D-EA)

and Peircean EA (P-EA) on a benchmark suite of problems. The criterion of fairness has been

a priority while designing the experiments for such a comparison. Although we are aware of the

consequences that ‘No free lunch’ theorem (Wolpert and Macready, 1995) would have on any

comparative matrices used between D-EA and P-EA, yet there are limitations to the application

of this theorem as well, especially in coevolutionary algorithms (Wolpert and Macready, 2005).

However, we are still not going to make any claim that our Peircean EA would out-do Darwinian

EA on any given data even though coevolution has similarities with Peircean Thirdness, but any

further development of this thought is beyond the scope of this research work. Another consequence

of this theorem is that there is no unanimously acknowledged set of functions or problems that are

an established benchmark for testing of EA (English, 1996). Literature shows many types of

benchmarks, ranging from purely mathematical functions with known or unknown global minima

(Hwang and He, 2006; Leung and Wang, 2001; Koziel and Michalewicz, 1999) to real world problems

(Hwang and He, 2006; Han and Kim, 2004). The test set used by us (Salomon, 1995) has 4

mathematical optimization functions of different characteristics shown in Table 5.1. The main

reason behind using this benchmark is that it covers a variety of functions, while still being concise.

An elaborate function set is explored in appendix ??.

Table 5.1: Benchmark mathematical functions

Function Limits Value N

f1(x) =∑n

i=1 10i−1x2i −10.0 <= xi <= 10.0 0 5

f2(x1, x2) = 100(x2 − x21)2 + (1− x1)2 −1.5 <= xi <= 1.5 0 2

f3(x) = n+∑n

i=1(x2i − cos(2πxi)) −5.12 <= xi <= 5.12 0 100

f4(x) =∑n

i=1−xi sin(√|xi|) −500 <= xi <= 500 -12569.4537 30

Function f1 is a simple 5 dimensional quadratic parabola with different eigen values along each

72

axis. For optimization the convergence speed usually depends upon the ratio between the smallest

and largest eigen values. Function f2 was proposed by Rosenbrock. The difficult part of this

function is the narrow curved valley that contains the minimum at x = (1, 1). Figure 5.1 shows the

2D function from multiple views.

Figure 5.1: Multiple views of function f2

Function f3 is generalized form of Rastrigin’s function. It is highly multimodal having its

minimum at x = 0. The probability of making progress towards the global minimum decreases as

we move closer to it because of presence of it being surrounded by multiple local minima on all

sides. This characteristic makes it a very interesting function. Figure 5.2 shows different views of

the 2D forms of f3.


73

Last function f4 was proposed by Schwefel, and we are using a 30-dimensional form of it instead

of 20D used in the benchmark just to introduce more rigor. The difficulty posed by this function

is that the global minimum is reached when, ∀i, xi = 420.9687. This makes it an unconventional

function for any EA. Figure 5.3 shows multiple views of 2D form of Schwefel’s function. Detailed

references about these functions are listed in (Salomon, 1995).


This section also illustrates the fairness in our experimental setup. If there is any bias, it is by all

means towards helping D-EA perform optimally, and not the other way around. For instance we did

extensive experiments just to find out the best parameter settings for D-EA on f4, and used that for

quoting the results. Furthermore, we then exported the same parameters to P-EA. The parameters

optimized experimentally include: Crossover probability, mutation probability, number of crossover

points, selection pressure. Final implementation of D-EA is taken from canonical algorithm of (Back

et al., 1997a) with tournament selection having selection pressure of 5, 4-point crossover operator,

and 0.02 mutation rate. Representation and parameter values are listed in Table 5.2. Although

Peircean theory does not restrict us from exploring or inventing new operators, we strictly used

74

the same techniques for both mutation and crossover that we incorporated in the Darwinian setup.

Initial population and stopping criteria are taken from the literature and obviously same for both

Darwinian and Peircean experiments (Leung and Wang, 2001; Salomon, 1995).

Table 5.2: Parameters for P-EA and D-EA

Population size 200

Crossover rate 0.7

Number of crossover points 4

Starting cluster radius 3

Mutation rate 0.02

Maximum generations 10,000

Stop-count 500

Representation 32 bit signed fixed point IEEE format

On average 50 experimental runs for each function were done to have a confidence on the

average values quoted for number of generations to converge and minima attained for the respective

function. For comparative purposes number of function evaluations is also used in the literature

frequently but in our case number of function evaluations is a function of number of generations, as

in each iteration the D-EA evaluates the entire population, whereas the P-EA evaluates the entire

population too but cluster-wise, making no difference. So for the mere sake of aesthetics we chose

to quote the number of generations rather than its equivalent, though numerically large, number

of function evaluations per experiment. The setting for most of the experimental environment is

taken from the literature related to benchmark problems listed previously in this section. There

are three stopping criteria: 1) Minimum accepted threshold for function value met, 2) Maximum

number of generations exhausted, which is 10,000 generations, or 3) If the minimum value of the

function being optimized does not improve beyond the first 1000 generations. We will refer to this

as stop-count from now onwards. In our experiments the stop-count will be 500 unless specified

otherwise. Crossover and mutation probabilities are kept at 0.7 and 0.02 respectively. Tournament

selection is used for D-EA with selection pressure of 5. Initial population of 200 individuals is kept

for all functions except for f3, the reason for which is explained in the results section. The only

variables that naturally get added to P-EA are the ones used for clustering, intra and inter-cluster

75

functions. We’ve tried radius-based clustering. For intra as well as inter-clustering we have used

the same settings and implementation for crossover and mutation operators that was used in D-EA.

Table 5.2 lists the parameters used and their values, including the common representation scheme.

5.2 Results

This section presents the detailed results for multiple experimental runs against each benchmark

function listed in Table 5.1. Against a total of 50 experiments, four statistics for the following three

aspects are calculated:

value converged at,

euclidean distance of point of convergence from the global optimum point,

and number of generations it took to converge/stop.

The four statistics for 50 experiments against each of the above-mentioned aspects are: minimum

(or best), maximum (or worst), mean and standard deviation. Table 5.3 shows these 12 statistics

when D-EA is tested on the four benchmark functions while Table 5.4 shows the same for P-EA.

Function f1 is the simplest out of the lot. It’s a simple quadratic with a smooth gradient towards

the single minimum in the search space. The only complexity is perhaps added by its 5 dimensions.

Both D-EA and P-EA show the same performance in terms of the number of generations as shown

in the column titled f1 in tables 5.3 and 5.4, while both P-EA and D-EA manage to converge

sometimes at the global minimum fitness value of 0 shown by best value. However, this is the only

similarity between the two. The main difference is that P-EA achieves this global minimum value

with a very small standard deviation. The other difference is that the worst value achieved by P-EA

(0.00169373) is still several times better than the mean value achieved by D-EA (0.00723572). Both

these statistics show the consistency of P-EA.

f2 is a 2 dimensional problem but apart from the difficult and narrow contour leading to the

global minimum, literature shows a small population size of only 15 rather than 200 against this

function, which adds to the complexity. While D-EA without exception (std. generations = 0) stops

because of stop-count after 1500 generations, P-EA gets consistently close to the global optimal

value and location. The fact of consistency is captured by the difference in the standard deviation

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for the converged values for P-EA and D-EA respectively.

Table 5.3: Compiled results for 50 runs of D-EA on f1, f2, f3 and f4, with stop-count = 500

f1 f2 f3 [100-D] f4 [30-D]

Best value 0 0.03668213 17.20363923 -11073.16103647

Worst value 0.15411377 20.54306412 36.23110552 -9289.66431338

Mean value 0.00723572 2.79967430 26.60560718 -10187.67149048

Std value 0.02220748 4.32435847 4.08085164 404.02023424

Min distance 0 0.03125 4.03112533 1883.30742526

Max distance 2.00000763 3.83051584 5.84107118 3271.54978445

Mean distance 0.04645569 2.53123548 4.98577523 2755.63172195

Std distance 0.28198878 0.86250604 0.38016095 314.59786089

Min generations 17 1500 3005 1693

Max generations 1500 1500 4751 2906

Mean generations 1329.66 1500 3955.72 2275.38

Std generations 466.39 0 418.0183 283.18868665

The complexity in function f3 is due to the fact that as we move closer to the global minimum,

there are a huge number of local minima surrounding it on all sides which makes it highly probable

for an EA to get stuck. D-EA got affected by these pockets of local minima, as shown by the mean

distance from the global minimum it stopped at, while P-EA consistently converged. The difference

between the mean values in both tables is significant for f3. At the same time, P-EA converges to

acceptable minimum value in far lesser number of generations as compared to D-EA.

Schwefel’s function or f4 in our case is perhaps the most complex of this benchmark. Likewise

the gap between the mean values converged at by D-EA and P-EA is widest in this case. While

D-EA barely crosses the -10,000 mark on average and never touches the global minima once, P-EA

shows a good mean value of -12,500, and reaches the known global minimum value of -12,569.4 at

least once in the 50 experiments, as shown in Table 5.4. However, there is a trade off, the P-EA

converges to acceptable minimum value in greater number of generations as compared to D-EA.

But on a closer inspection, the reason why D-EA stops comparatively earlier is because of meeting

the third stopping criterion: the stop-count, as it fails to meet the first two stopping criteria. Since

P-EA in its evolutionary run keeps on exploring and finding better individuals, hence it does not

77

stop because of stop-count or by reaching the maximum limit on the number of generations. It

often stops because of the fact that the minimum accepted threshold for function value had been

evolved, as evidenced by mean value (and distance) close to global minimum and in comparison to

D-EA, a very low standard deviation value.

Table 5.4: Compiled results for 50 runs of P-EA on f1, f2, f3 and f4, with stop-count = 500

f1 f2 f3 [100-D] f4 [30-D]

Best value 0 0 0.00043512 -12569.48476389

Worst value 0.00169373 0.00107193 0.00998695 -12217.85535537

Mean value 0.00024567 0.00004288 0.00691935 -12500.62990597

Std value 0.00052553 0.00021219 0.00169906 92.42908979

Min distance 0 0 0.00390625 0.12129379

Max distance 0.00676582 0.13975425 0.02194551 1384.65925883

Mean distance 0.00339717 0.01257788 0.01744841 364.52082151

Std distance 0.00280038 0.03367504 0.00546443 450.18003363

Min generations 72 6 1856 2875

Max generations 2484 5953 3087 8310

Mean generations 1352.24 1198.9 2308.22 5552.24

Std generations 571.26 1370.97 303.98 1400.34

Another series of experiments for Schwefel’s function were done keeping stop-count as variable

for P-EA. The objective was to see the trend of average minimum value found at the cost of

increasing average generations allowed. P-EA consistently reached around the desired minimum

in less than 10,000 generations (which is the max generations allowed) as shown in Table 5.5. An

interesting pattern in P-EA is the increasing accuracy in terms of mean minimum value attained

as the stop-count condition is relaxed without compromising much in terms of mean number of

generations. No such meaningful pattern was found for D-EA. The best figures for average minimum

value were attained for P-EA when bias tests were conducted. As discussed in section 4, Peircean

framework allows us to use aprior bias or information that we have regarding the problem. Assuming

we took advantage from the shape of Schwefel function, and placed a couple of individuals in the

initial random population having a bias such that those two individuals had for all xi = 400. Last

column in Table 5.5 shows the results where in less than 3000 generations on average, P-EA gets

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closest to the globally known minima each time. Although this performance was expected with

such an initial bias, but the important factor here is that Peircean framework has a provision for

bias.

Table 5.5: P-EA on f4 with varying stop-counts

Stop-count = 500 Stop-count = 1500 Max gen Bias test

Experiments 50 50 50 10

Best value -12569.4847 -12569.4856 -12569.4858 -12569.4849

Mean value -12500.6299 -12548.1603 -12560.2230 -12568.8151

Mean distance 364.5208 106.3488 66.6988 1.4409

Mean generations 5552.24 5903.9 6382.12 2930.2

Figure 5.4 shows the convergence of fitness for the 1000 generations against all four benchmark

functions. The solid line depicts P-EA while the dotted one shows D-EA convergence graph. It

shows that as the functions become progressively difficult the gap between P-EA and D-EA widens,

and eventually P-EA converges around the global optimum, whereas D-EA gets stuck in some local

optima.

0 200 400 600 800 1000 1200−50

0

50

100

150

200

250

300

350Convergence of P−EA vs C−EA on F1 (1000 gen.)

P−EAC−EA

(a) f1

0 200 400 600 800 1000 1200−2

−1

0

1

2

3

4

5


P−EAC−EA

(b) f2

0 200 400 600 800 1000 12000

50

100

150

200

250

300


P−EAC−EA

(c) f3

0 200 400 600 800 1000 1200−14000

−12000

−10000

−8000

−6000

−4000

−2000


P−EAC−EA

(d) f4

Figure 5.4: Convergence comparison: P-EA vs. D-EA

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Finally we compare P-EA with other EA in Table 5.6 and Table 5.7 for functions f3 and f4

respectively. The list includes: Orthogonal Genetic Algorithm with Quantization (OGA), Fast

Evolution Strategy (FES), Particle Swarm Optimization (PSO), Evolutionary Optimization (EO),

two versions of Quantum-inspired Evolutionary Algorithm (QEAH and QEAR), a Fast Evolutionary

Programming (FEP), and a pair of conventional/classical Genetic Algorithms and Evolutionary

Programming (CGA and CEP). Results of OGA and CGA are referred from (Leung and Wang,

2001); FES from (Yao and Liu, 1997); PSO and EO from (Angeline, 1998); QEAH, QEAR, FEP

and CEP from (Han and Kim, 2004). The last two rows of the two tables show the results for our

implementation of D-EA and P-EA (with stop-count = 500). A point to note here is that some of

the above-mentioned EA are optimized in ways that the P-EA is not. For instance, OGA, which

apparently gives the best figures, has been optimized to explore the solution space along orthogonal

dimensions. Still, the rudimentary P-EA fares well, as it converges near the optimal location (only

OGA is better) for 30 dimensional case of f3, that too in the minimum number of generations. For

the more complicated Schwefel’s function, its performance is still comparable.

Table 5.6: P-EA compared with other EA on f3 [30-D]

Mean Value Std Value Mean Function Evaluations

OGA 0 0 224,710 feval

FES 0.16 0.33 500,030 feval

PSO 47.1354 1.8782 250,000 feval

EO 46.4689 2.4545 250,000 feval

CGA 22.967 0.78 335,993 feval

QEAwH 0.039 0.19 5000 gen

QEAwR 18.7 7.4 5000 gen

FEP 0.046 0.012 5000 gen

CEP 89 23.1 5000 gen

D-EA 5.1638 2.5315 1549.4 gen / 309,880 feval

P-EA 0.0087 0.0006 445.32 gen / 89,064 feval

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Table 5.7: P-EA compared with other EA on f4 [30-D]

Mean Value Std Value Mean Function Evaluations

OGA -12569.4537 6.447 x 10−4 302,166 feval

CGA -8444.7583 65.7326 458,653 feval

FES -12556.4 32.53 900,030 feval

QEAwH -12569.48662 3 x 10−9 9000 gen

QEAwR -12353.447 163.8 9000 gen

FEP -12554.5 52.6 9000 gen

CEP -7917.187 634.5 9000 gen

D-EA -10187.6715 404.0202 2275.38 gen

P-EA -12500.6299 92.4291 5552.24 gen / 1,111,0448 feval

5.3 Results on an Extended Benchmark

Another set of results on an extended benchmark of 12 mathematical functions of up to 100 di-

mensions is listed in this section. Figure 5.5 lists the forms of the benchmark functions. For the

lack of space refer to Leung and Wang (2001) for detailed description of each function. Against

most of the functions the 30 dimensional form has been used, while F7 and F9 are 100-D. Table 5.8

lists the comparative results of both the Peircean (P-EA) and the classical Darwinian evolutionary

algorithm (D-EA) when they are run for 50 times against each function. For 50 independent runs,

what the table lists against each function is how P-EA and D-EA perform statistically in terms

of, 1) the average number of generations it took the EA to converge, 2) the average of the best

function values that the EA converged at, and 3) the standard deviation of those values.

The most interesting observation is that apart from one exception (F14), P-EA, by the time it

stops, is almost always very close to the global minimum value for each function, that too with a

small standard deviation value. This is not the case for D-EA. But more important than that is

the fact that for P-EA evolvability, improvement and growth seem to be vital. In terms of number

of generations (or function evaluations equivalently) P-EA seems to be more efficient or at par with

D-EA in most of the cases. In some cases where D-EA makes an early stop (e.g. F7,F9,F10,F14) it

is always the case where D-EA has stopped because of premature convergence to a local minimum

due to stagnation of its population. This fact is evident by looking at the respective mean function

81

value columns. In all of these tough specific cases, P-EA keeps on evolving its population and gets

much closer to the global minimum before stagnating. The results would make more sense when

the above-mentioned analysis is coupled with the stagnation analysis and cluster analysis, detailed

in the next chapter.

Figure 5.5: Extended Benchmark Functions

82

Table 5.8: Comparison between Peircean-EA and Darwinian-EA

F.Mean number ofgenerations

Mean functionvalue (standarddeviation)

Globalminfunc.value

P-EA D-EA P-EA D-EA

F1 2895.2 2286.48-12569.3(0.6195)

-10962.6(305.88)

-12569.4

F2 2308.22 3955.720.0069(0.0017)

26.61(4.08)

0

F3 1502.82 1513.740.2730(0.0324)

2.3272(0.2686)

0

F4 1618.92 1689.120.0198(0.0233)

0.4264(0.4000)

0

F7 8984.52 2656.7-95.08(0.8304)

-73.93(1.8858)

-99.2784

F9 8414.04 3826.02-78.32(0.0087)

-58.68(1.5735)

-78.3324

F10 7840.6 1764.0652.71(30.02)

1899.87(900.19)

0

F11 1502.74 1527.760.0570(0.0097)

0.0780(0.1436)

0

F12 1620.76 1606.70.5241(0.2964)

0.5969(0.3046)

0

F13 1500 1647.40.0146(0.0023)

0.0274(0.1416)

0

F14 10000 2770.183307.14(2946.47)

325689.5(92779.9)

0

F15 2750.94 3120.780.4978(0.5710)

29.1570(13.3290)

0

83

Chapter 6

Analyses

The empirical data on the benchmark functions suggests that P-EA holds promise for the EC

community even from the perspective of function optimization, as it has good exploratory powers.

The question however still remains as to what causes the P-EA to keep on exploring till it converges

near the global optima. One intuitive claim is based on Schwefel’s hypothesis, that P-EA performs

well because it mimics the natural processes of evolution better than the classical EA. In this

chapter there are three aspects that need to be analyzed for P-EA: 1) how long before the diversity

of the population is lost; 2) what is the effect of number of clusters on diversity; and, 3) the effect

of clusters on Holland’s schema theorem.

6.1 Stagnation Analysis

Stagnation has been a major problem noted in the literature with respect to population based

EA (Deb, 2001; Whitley, 1994). Allegations are leveled against selection and the fittest or best

individual taking over the entire population. Consequently, there are various stagnation avoidance

techniques employed in the literature as well (Friedrich et al., 2008; Sareni and Krahenbuhl, 1998).

It is hypothesized that the problem might be rooted in the Darwinian paradigm itself, hence any

cosmetic solution will either be artificial or not from within the Darwinian framework. Peircean

EA however, does not use any special technique to avoid stagnation. Whatever it is that causes the

delays in its population getting stagnated comes from within the Peircean framework - for instance

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its cluster based retention of population (Thirdness) and Firstness principles of spontaneity. For

monitoring the stagnation a simple analysis was made. At the end of each generation or iteration,

the percentage difference between the fittest individual and the entire population was calculated.

There could be other metrics for calculating stagnation but the choice of this metric is justified by

the fact that in D-EA stagnation is primarily caused by the fittest individual replicating itself in

the next generation while getting selected with most probability (Deb, 2001; Friedrich et al., 2008;

Whitley, 1994).

The results in Table 6.1 show the average generations for both D-EA and P-EA against each

function up till which the population had stagnated by 99%. D-EA stagnates very early on and

this fact is translated in all the results as seen above. This behavior is consistently shown for each

function. P-EA on the other hand shows greater resilience towards stagnation and the average

number of generations before it stagnates is in quite contrast to that of D-EA. This gives the P-EA

diversity in the population for a much longer time frame and hence there is a greater chance that

P-EA does not converge at a local optima. This is one of the reasons why P-EA shows meaningful

results and patterns in the experiment.

Table 6.1: Number of generations before the population stagnates,20 experiments

Fn. EA Min Max Mean Std

F1 D-EA 25 54 38 7

P-EA 100 3696 851 1133

F2 D-EA 4 13 7 2

P-EA 82 2091 704 558

F3 D-EA 22 69 37 12

P-EA 361 1099 511 188

F4 D-EA 40 56 46 4

P-EA 335 924 530 156

6.2 Cluster Analysis

In Holland’s schema theorem formulation, disruption analysis is already well known for suggesting

how potentially good schemas can be disrupted early on by the crossover operator (Chen and Smith,

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1999). This has implications for all EA that use crossover operators. We hypothesize that those EA

that partition the population into subpopulations minimize the chances of disruption by crossover.

Peircean-EA first divides the population into clusters and then performs the within-cluster crossover

to generate the next batch of individuals for each cluster, followed by the crossover of the fittest

individuals from across clusters. To monitor the effect of number of clusters on the continual

evolution of good schemas, the following experimental setting was devised. The population size

was fixed to 200 individuals, mutation rate to 0.02 and crossover to 0.7, while the stop-count is fixed

to 500. To make a fair trial, it was necessary to turn off Firstness completely, so that if increasingly

fit individuals keep on getting generated it cannot be because of spontaneity. Instead of uniform

random selection, tournament selection with tournament size of 2 was introduced, so that the only

thing which separates a D-EA and this constricted version of P-EA is that the individuals are

evolved in clusters rather than as one global unit. The only stopping criterion is the stop-Count,

which means that when no better fit individual is found in the last 500 generations, the EA stops

due to this stagnation. To complete the experiment Schwefel’s 30 dimensional function is chosen,

the number of clusters is increased iteratively from 1 to 25 and 10 trials against each number of

clusters are made. The mean number of generations of those 10 trials is plotted against number of

clusters in Figure 6.1.

As is obvious from the y-axis, the increase in clusters does result in prolonged diversity and

exploration up to a point, but beyond that the EA begins to stagnate earlier. Following analysis

explains the behavior observable in the above figure:

1. As the number of clusters is increased from 1, due to less global interactions the retention and

evolution of good schemas becomes more and more probable, as indicated by the prolonged

termination of EA due to stop-count. So for longer duration the better fitted schema keep

on getting explored rather than disrupted.

2. But there is not a linear relation between the number of clusters and the retention/evolution

of schemas, firstly due to a finite and static population size. For instance, against a population

size of 200 and the number of clusters being 25, only 8 individuals on average belong to any

cluster. This is too inexpressive a gene pool to help in evolution, as observable in the latter

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0 5 10 15 20 251950

2000

2050

2100

2150

2200

2250

2300

2350

Number of Clusters

Mea

n N

umbe

r of

Gen

erat

ions

for

Exp

lora

tion

Effect of Number of Clusters on Stagnation

Figure 6.1: Cluster Analysis: Effect of clustering on stagnation using Schwefel’s function

part of the graph.

3. But mainly because as the number of clusters is further increased, inter-cluster crossover

begins to act more disruptively, since intra-cluster would now be a crossover of decreasing

number of individuals closer together in each cluster, while inter-cluster crossover would be

between an increasing number of individuals, and would become characteristic of the global

population of a D-EA.

6.3 Schema Theorem

What is a schema? A schema is John Holland’s formulation of a building block, a string comprising

of either 0s or 1s or * (asterisks). For instance if H = 1****0 is the schema, then it represents all

possible binary individuals in a 6 dimensional search space that begin with a 1 and end at 0. Given

this definition of schema H, there are four more definitions,

1) l, which is the lenght of schema H, here 6;

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2) d(H), which is the defining length of schema H, in other words, the distance between the

outermost defined bits, here 6-1 = 5;

3) o(H), or the order of schema H, which is given by the total number of non-asterisks in the

schema, in this case 2; and

4) m(H, t), which is the number of instances of H present in the GA population at time t.

Now given all of the above, schema thoerem makes a prediction about E(m(H, t + 1)), which

is the expected number of instances in the population belonging to the schema H surviving till the

next generation. Of course, E(m(H, t+ 1)) must be a factor of m(H, t), however, in classical EA,

three operators affect it: selection, crossover, and mutation.

1) Selection

A standard fitness proportional selection affects the m(H, t) by the following ratio:

E(m(H, t+ 1)) =f(H)

f(p)∗m(H, t) (6.1)

where, f is the fitness function, f(p) is the sum fitness of the entire population, and f(H) is the

sum fitness of those instances belonging to the schema f(H) =∑∀x∈H

f(xi).

2) Crossover

For a single point crossover operator, we can see the more the value of d(H) with respect to

l, higher are its chances of getting disrupted by the crossover. Since the chance of an individual

taking part in the crossover is driven by the crossover probability (pc), hence the chances of the

individual still surviving and being a part of the schema are(

1− pc(d(H)l−1

)), updating equation

6.1 as:

E(m(H, t+ 1)) ≥ f(H)

f(p)∗m(H, t) ∗

(1− pc

(d(H)

l − 1

))(6.2)

3) Mutation

For a mutation rate of pm, the defined bits (o(H)) instances belonging to schema have a survival

chance of (1− pm)o(H), updating equation 6.2 as follows:

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E(m(H, t+ 1)) ≥ f(H)

f(p)∗m(H, t) ∗

(1− pc

(d(H)

l − 1

))∗(

(1− pm)o(H))

(6.3)

Typically pm is kept very low, such that pm � 1 =⇒((1− pm)o(H)

)≈ 1−o(H)pm, simplifying

the equation 6.3 to:

E(m(H, t+ 1)) ≥ f(H)

f(p)∗m(H, t) ∗

(1− pc

(d(H)

l − 1

))∗ (1− o(H)pm) (6.4)

Equation 6.4 captures Holland’s schema theorem for building block hypothesis. What it suggests

is that the number of individuals belonging to the schema that survive the disruption and are still

above average in terms of fitness get increased by a factor of f(H)f(p) every generation!

6.3.1 Clusters as Schema - Effects on Disruption Analysis

Ignoring the effect of mutation in equation 6.4, and denoting the disruption probability caused by

the crossover as dc, we get the following representation:

E(m(H, t+ 1)) ≥ f(H)

f(p)∗m(H, t) ∗ (1− dc) (6.5)

“The schema theorem given in equation 6.4 applies not only to schemas but to any subset of

strings in the search space” (Mitchell, 1998). When we apply it to clusters, meaning, when a cluster

represents a schema, how does that effect the equation for a Peircean EA?

It means that the crossover operator which potentially caused disruption when the entire popula-

tion was classically treated as one big cluster, would not be a problem when it comes to intra-cluster

evolution, since the schema is now the cluster, so any two instances belonging to the cluster, when

crossed-over, would still result within the search space covered by that cluster. Any disruption that

is to come, is to come from the crossover taking place in the inter-cluster evolution. If dc is the

disruption probability for one instance of a schema, this factor gets further reduced by 1m(H,t) ; since

only one instance (fittest) of the cluster can, at max, get disrupted by the inter-cluster evolutionary

crossover now, per generation. This makes the following survival rate when a cluster is a schema:

1− dcm(H,t)

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Within the inter-cluster evolution, the selection factor becomes a ratio of f(Hfittest) and f(C),

where f(C) is the average fitness of all the fittest individuals, one from each cluster. Putting these

two factors in equation 6.5 updates the schema theorem for clusters as follows:

E(m(H, t+ 1)) ≥f(Hfittest)

f(C)∗m(H, t) ∗

(1− dc

m(H, t)

)(6.6)

So, when H is a cluster, and if we have only one cluster, then it means that

f(Hfittest)f(C) = 1, and

(1− dc

m(H,t)

)= 1, since dc = 0 m(H, t+ 1) = m(H, t)

But for any higher number of clusters, equation 6.6 suggests that those clusters H whose best

fitness > average fitness of all clusters’ best instances, keep on surviving, evolving and thriving.

But more importantly, perhaps, those H that are not fit enough relatively speaking, at any time t,

become extinct slowly; it may take m(H, t) generations before they lose all their instances.

6.4 Summary

The previous chapter was focused on guaging the performance of the P-EA, especially in contrast

with the D-EA. This chapter focused itself on why the performance of P-EA came out as a more

robust EA of the lot. The hunch that the factors of Firstness (continual introduction of individ-

uals to the population) and Thirdness (cluster based population dynamics,) besides the standard

selection and variation operators in EA, has been evaluated in this chapter. The three investigative

questions analyzed for P-EA were: 1) how long before the diversity of the population is lost; 2) what

is the effect of number of clusters on diversity; and, 3) the effect of clusters on Holland’s schema

theorem. The finding of the first investigation is that both P-EA as well as D-EA stagnate, but

P-EA stagnation is delayed by quite a margin comparitively speaking. The finding of the second

investigation is that turning every factor off in P-EA that departs from our implementation of a

standard D-EA, apart from clustering in P-EA, still makes a difference in terms of it’s exploratory

potential. Two less a number of clusters, and too many clusters both result in a deteriorated ex-

ploration and early stagnation. Finally, the theoretical finding in the last investigation suggests

that clusters with good fitness have a greater potential for growing, exploring and exploiting, while

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those with weaker fitnesses do not die out quickly, rather they also get a fair chance of explorative

survival before getting extinct.

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Chapter 7

Future of EC & our Conclusions

This chapter lists the future directions as hypothesized by the pioneering figures from EC - De Jong

and Schwefel. It also then maps the Peircean framework on to those future directions. Towards

the end some conclusions are presented.

7.1 Hans-Paul Schwefel’s Future Directions for EC

Schwefel lays out a comprehensive plan of how to engage with the future directions in EC (Schwefel,

1997). Schwefel identifies that “there is a bulk of as yet unincorporated phenomena and mechanisms

of organic evolution underpinning the hope for further breakthroughs in devising ever more useful

evolutionary algorithms.” He suggests that by comparing what is deficient in current EAs with

what is empirically and observably present in organic evolution and nature, further improvements

and directions in EC development can be defined:

“Current evolutionary algorithms are certainly better models of organic evolution. Nev-




within the whole universe. We must be more modest in order to understand at least a

little of what really happens – as always within natural sciences.”

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In general we term this as Schwefel’s hypothesis, but additionally he has also pointed out

the specific aspects of organic evolution which should be modelled by an EA. We shall represent

Schwefel’s future directions point by point, using his own words:

1) Evolvability: Organic evolution certainly does not only aim at finding static optima just once

and with ultimate precision. Organic evolution happens within an ever-changing environment,

where evolvability is more important than precision.

2) Dynamic interaction of agents: The environment is not only intrinsically dynamic; it is

changed by the mutual actions of all participants in the evolutionary game. From the per-

spective of one species, the search for meliorization (a term preferred to optimization by the

author) takes place on a trampoline, deformed by (re-)actions of other species, as well.

3) Multiple selection criteria: Organic evolution always deals with a situation of multiple se-

lection criteria.

4) Spatial distribution of population: Larger populations in the real world are spatially dis-

tributed (small populations are always prone to extinction). Neither mating nor predation

thus takes place in a way that includes all individuals with equal chance. There is no global

selection in either case. Consequently modeling selection as an asynchronous and spatially

distributed process with several predators at a time among the prey, which is the normal case

in ecological systems, has not yet been done but should be considered.

5) Cooperation as important as competition: Many attacks against EAs as optimization, or,

better, meliorization, algorithms have been mounted by those who emphasize cooperative be-

havior within evolutionary processes. This controversy is not necessary. Cooperative problem

solving by sharing resources or dividing the problem into subproblems to conquer may lead

to novel approaches not yet taken into account thoroughly. At least, theoretical results are

missing.

6) Self-adaptive parameter rates: ES with self-adaptive individual mutation rates: Very low

rates as observed in temporary optimal or near-equilibrium situations may not be good for

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starting conditions. Autoadaptation of strategy parameters, be they discrete or real valued,

can and has been handled in two different ways: these parameters vary either from individual

to individual or from subgroup to subgroup. The possibilities of on-line adaptation of a

variable epigenetic apparatus are by far not exhausted in all EAs today.

7) Social learning and epigenetic factors: Haeckel component of an evolutionary algorithm

can be taken as any kind of individual adaptation to the local – temporal environmental

conditions; but the result of this cannot be transferred genetically to the next generation.

Completely different from this mechanism is the life-long individual learning and all kinds of

social transmission of the knowledge (and prejudice) gained this way. EA individuals so far

have no kind of brain of their own but this might be added in the future.

8) Gender: [EA] do not yet handle two (or more) sexes really ... [for] a purposeful employment

in optimization.

9) Variable genome length: Genome length variation should be an important ingredient in all

cases of so-called structure optimization tasks, where the number of variables is not known

in advance, but is a variable itself.

7.1.1 Equivalence between Peircean Framework and Schwefel’s Future Direc-

tions

“The invention of broadly efficient [evolutionary] algorithms is a design challenge as stiff as the

most difficult that have been faced this century.” Here (Goldberg, 1993) effectively says that ad-

hoc design “hacks” often do not work in EA. It is not the case that we can take Schwefel’s guidelines

point by point and build a piece-meal yet efficient algorithm. This section highlights how most of

Schwefel’s demands for the future of EA are met naturally by Peircean evolutionary framework.

Their relationship is also captured in Table 7.1.

1) Evolvability: Thirdness of Peirce is a true representation of what Schwefel means by evolvabil-

ity. It is this process of evolutionary growth through generalizing tendency that converts a

possibility of getting to the global optima into an actuality. Otherwise, stressing on efficiency

95

and precision through selection has been the primary cause of premature convergence Her-

rera and Lozano (2000). In P-EA the focus on selectionism is balanced due to the dynamics

introduced by Thirdness, such as parameter tuning and clustering. This makes it possible to

stress more on the function of evolvability, as all the clusters separately evolve.

2) Dynamic interaction of agents: Through the inter-cluster operations, the fittest clusters

affect other clusters and individuals. Eventually the clusters get pulled by the fittest towards

the best explored optima. This action and reaction of different clusters addresses Schwefel’s

call for meliorization.

3) Multiple selection criteria: Unlike a global flat selection operator, P-EA has multiple pos-

sibilities:

• There are multiple levels, and hence selection can take place at individual as well as

cluster and global levels

• Due to clusters, there is a natural possibility of having multiple selection operators now

• We have experimented by introducing a cooperative selection operator as well, but this

idea has unexplored potential for future work

4) Spatial distribution of population: By evolving the populations in clusters, P-EA directly

meets this demand of Schwefel.

5) Cooperation as important as competition: There are three sides to this party. On the

one hand is the group which reduces every phenomenon to competition, and consequently all

empirical behaviors of cooperation are also explained away as epi-phenomenon, only keeping

selfish competition as real. The group at the other extreme suggests the opposite. They read

everything from a cooperative angle (Todes, 1987). Peirce on the other hand is the third.

He acknowledges reality in its multiplicity. His understanding of evolution has real place for

competition as well as cooperation, given certain conditions. This fact translates into P-EA,

as there is clustering and cooperation as well as selection of the fittest through competition.

96

6) Self-adaptive parameter rates: Peircean system does not take anything as a given, as Peirce

applies his evolutionary principles at the cosmological levels as well. For him even the physical

laws are at different evolutionary stages. The presence of exceptions against every law is a

sign for that. Any EA that also evolves the parameters would be more Peircean in spirit. We

have only emulated this self-adaption of cluster radii and mutation rate so far, as described

in chapters above.

7) Social learning and epigenetic factors: P-EA does employ the social learning especially

when the inter-cluster fittest individuals change within their lifetime, and when during the

addition of Firstness individuals to the population, some are locally affected by their cluster

representatives. At the moment our P-EA is handicapped by the fact that there is nothing

much beyond the genetic makeup, and that is also primarily due to the type of mathematical

benchmark we have used. Perhaps for another type of problem set, a brain or extra-genetic

elements could be added to the genotype and then epigenetic operators could be involved

more meaningfully.

Table 7.1: Relation between Peircean framework and Schwefel’s future challenges

Schwefel’s Research Guide-lines

Peircean EA

1 Evolvability Thirdness guarantees evolv-ability

2 Dynamic interaction of agents Inter-Cluster evolution

3 Multiple selection criteria Cooperative selection besidesnatural selection

4 Spatial distribution of popula-tion

Clustering

5 Cooperation as important ascompetition

P-EA treats cooperation asreal just like competition

6 Self-adaptive parameter rates Cluster radius, mutation rate

7 Social learning and epigeneticfactors

Firstness, inter-cluster infor-mation exchange

8 Gender N/A

9 Variable genome length N/A

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7.2 Kenneth De Jong’s Agenda for the 21stCentury and Peirce

While presenting a history of the field of GA, Kenneth De Jong concludes with an extended “agenda

for the 21stcentury” (De Jong, 2005). This section briefly outlines them and then draws parallels

with Peircean framework, wherever possible.

1) Developing a more general EC/EA framework: Pitching the variety of design choices in

classical EAs (GA, ES, EP) and other variant EAs, there is a better chance of understanding

the implications of those design choices when a more general framework of EC is developed,

and the variant EAs are seen as instances of that general framework. Abstractly speaking,

the basic components of that general EC framework are:

• A population of individuals

• A notion of fitness

• A notion of population dynamics biased by fitness

• A notion of inheritance of properties from parent to child

Peircean framework would only add two more building blocks to complete De Jong’s proposal

- which mainly covers Secondness:

• A notion of introducing freshness (Firstness) at various levels

• A notion of Habit/Generalization (Thirdness)

2) Decentralized and Speciation models: EAs, no doubt, are naturally parallelizable. Some

of the evolutionary effects that can be covered by this parallelism include speciation, niching,

and punctuated equilibria, with subpopulations developing in equilibrium according to the

underlying parallel architectural topology, punctuated by infrequent migrations from neigh-

boring subpopulations.

While De Jong admits that the theories for traditional EAs do not help understand the

effects of design choices for these parallel implementations. We believe that the Peircean

98

formulation, both in its form of evolutionary theory, as well as coupled by its EC theory, can

be strengthened further to help the EC community.

3) Self adapting and coevolutionary systems: Natural reality of course shows more complex

phenomena, such as self-adaptive mutation, selection and other parameteric rates. It also

shows individuals and species co-evolving each other in more modes than just through com-

petition. The fact is that the traditional simple EA does not have explicit support for these

complex behaviours. The challenge is to come up with a theoretical framework for EC that

also incorporates self-adaptation and co-evolution.

The previous section already explains well how Peircean EA naturally has a place for both

these complex phenomena. It will make for a very constructive future work to come up with

a formal theoretic formulation that helps in coordination of the self-adaptive parameters.

4) Incorporating more biology into EA, especially Lamarckian ideas: Since Holland’s clas-

sical GA was inspired from Modern Synthesis’ point of view, it is not surprising to see Lamar-

ckian factors being factord out from the GA consequently. There is enough evidence that the

genetic viewpoint is a limited account of evolution, and that the epi-genetic factors are heri-

table and sustain through generations (Noble, 2013).

Although Peircean framework has scope for other-than-genetic learning, it will make for a

very interesting future work to construct intelligent agents that learn over their lifetime and

also transmit this learning to their next generation.

5) Using EA to further our evolutionary understanding: It is always easy to see how an

algorithm can and will be viewed as only a degenerated form of the actual evolutionary

processes at work in natural reality. But one positive strand within EC is to look at EAs as

tools that further our understanding of the natural evolutionary reality. This sentiment is

more vividly elaborated by (Mitchell and Taylor, 1999) and we quote:

Evolutionary computation can sometimes serve as a useful model for biological evo-

lution. It allows dissection and repetition in ways that biological evolution does not.

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Computational evolution can be a useful tool for education and is beginning to pro-

vide new ways to view patterns in evolution, such as power laws and descriptions of

non-equilibrium systems. Evolutionary theory, as developed by biologists, typically

tries to linearize systems, for ease of analysis with differential equations, or to treat

units in isolation, as in single-locus selection. While evolutionary computation is

not inconsistent with such theory, it tends to be outside it, in that real difference

in capacity and complexity are often observed, and are not really describable by

stable equilibria or simple changes in gene frequencies, at least in ways that are

interesting. There is reason to believe that theories of evolutionary computation

might extend the language of biological evolutionary theory and contribute to new

kinds of generalizations and analyses that have not been available up to now.

We feel that with this experimental verification of Charles Peirce’s ideas, we have not only

tried to put forth a viable EC framework, but have also brought new interpretive and scientific

meaning for the Charles Peirce scholarship and community (Akhtar et al., 2013). We also

hope that this work triggers appropriate discussions within the community of evolutionary

biologists as well.

7.3 Conclusions

Science works by rigorous experimental testing of the theories that make claims or predictions about

the working principles of the natural world. In this article we have presented an interpretation of

Peirce’s theory of evolution, modeled it algorithmically, and tested the model scientifically through

lab experiments. We understand Peirce’s pragmatism to include the idea that in order to attain

more clarity with regard to a concept, one must first of all translate the concept into a set of

testable abductive hypotheses, and then test them under laboratory conditions for their practical

considerations. We feel that, remaining true to computer science, we have formulated testable

hypotheses out of Peirce’s theory and set up experiments that could test their validity.

All other things being equal (experimentally), evolutionary algorithms based on an interpre-

100

tation of Peirce’s theory maintain diversity far longer than those based on Darwin’s theory. This

issue of diversity is significant for the EC community, and any solution to this problem warrants

serious consideration. We have not introduced any ad hoc mechanism in our proposed algorithm

that seeks to redress this problem artificially. If there is an apparent improvement in the main-

tenance of diversity in the Peircean EA (as indicated by the results) we would like to attribute

it to Peirce’s theory of evolution (or our understanding of it). The experimental evidence for the

predictive accuracy of Peircean theory presented in this paper has been consistent so far, at least

in the domain of evolutionary computation.

More importantly, Schwefel’s and Peirce’s abductive hypotheses have passed our inductive tests.

Schwefel’s hypothesis-that the more an evolutionary algorithm models natural evolution, the better

it will perform-is supported by our experimental results. Peirce’s hypothesis-that in the beginning

was chaos and over long periods of time, habitually that chaos is transformed into regularity-has also

been affirmed. Consequently, it also helped explain the fact of stagnation, which has been a problem

for the EC community. Darwin’s theory predicts diversity to result from simple beginnings by using

blind chance and mechanistic laws alone. However, models of Darwin’s theory in computer science,

namely evolutionary algorithms, lead to the opposite of diversity: stagnation at a very early stage.

This unpredicted behavior leads, in optimization terms, to premature convergence, and this has

been amply documented in the literature. Darwin’s theory does not say a word about stagnation,

as it only focuses on diversity, so the issue of stagnation could not get any support from within

the Darwinian paradigm. On the other hand, Peirce’s theory anticipates stagnation (complete

regularity) in the long run. This was confirmed by the experiments as well. We feel that we have a

strong case of understanding Peirce’s theory pragmatically here, given the constraints of computer

science. Of course the real pragmatic tests would need to come from evolutionary biology.

We have also indicated how new findings in cellular biology, epigenetics and systems biology are

already unknowingly heading in the direction of Peirce and knowingly heading away from Darwinian

notions. It is concluded that our interpretation of Peirce’s work is significant enough to help solve

some of the contemporary problems in evolutionary algorithms as well. We think that the passing

of this pragmatic test, plus the movement within contemporary physics as well as biology, are

101

convergent signs that point towards the truth of Peirce’s evolutionary framework.

102

Appendix A

Cooperation vs. Competition in

Evolution

A.1 Cooperation

A recent publication conducting a 43 years survey of the theoretical and empirical literature on

cooperation by (West et al., 2007) suggests:

Explaining cooperation remains one of the greatest challenges for evolutionary biology,

irrespective of whether it is altruistic or mutually beneficial.

Darwin was aware of the problem in his lifetime, and flirted with some ideas of group selection

to work around it. In Darwin’s own words (Darwin, 1859):

I will not here enter on these several cases, but will confine myself to one special difficulty,

which at first appeared to me insuperable, and actually fatal to my whole theory. I

allude to the neuters or sterile females in insect-communities;.. This difficulty, though

appearing insuperable, is lessened, or, as I believe, disappears, when it is remembered

that selection may be applied to the family, as well as to the individual, and may thus

gain the desired end. (Chapter 7, pages 236-237)

103

Natural selection cannot possibly produce any modification in any one species exclu-

sively for the good of another species; though throughout nature one species incessantly

takes advantage of, and profits by, the structure of another. But natural selection can

and does often produce structures for the direct injury of other species, as we see in

the fang of the adder, and in the ovipositor of the ichneumon, by which its eggs are

deposited in the living bodies of other insects. If it could be proved that any part of the

structure of any one species had been formed for the exclusive good of another species,

it would annihilate my theory, for such could not have been produced through natural

selection. (Chapter 6, page 200)

Natural selection will modify the structure of the young in relation to the parent, and

of the parent in relation to the young. In social animals it will adapt the structure of

each individual for the benefit of the community; if each in consequence profits by the

selected change. What natural selection cannot do, is to modify the structure of one

species, without giving it any advantage, for the good of another species. (Chapter 4,

page 86)

Professor Stephen Stearns, an accomplished neo-Darwinist evolutionary biologist at Yale’s Ecol-

ogy department (Stearns, 2009b), has summarized Darwin’s words in this fashion: If ever it could

be shown that individuals repeatedly and reliably sacrificed their own fitness to in-

crease the fitness of others, the theory of natural selection would be refuted. The

post-Darwinian work which has been referred to as neo-Darwinism has persistently tried to ex-

plain cooperation, altruism, and symbiosis by first and foremost reducing it to various forms of

selfishness. This is one hypothesis. Another equally valid hypothesis is that cooperation is an

independent agent in the evolutionary process - at least as much a factor as mutation (chance) and

selfish struggle (competition). Cooperation is understood as being irreducible to chance mutation or

selfishness or competition - it is actually the opposite of both. Cooperation means that selfish

replicators forgo some of their reproductive potential to help one another (Taylor and

Nowak, 2009). What neo-darwinists miss is that cooperation could be understood as a deliberate,

purposive collaboration or association: a long term process rather than a cost-benefit happenstance.

104

From the perspective of our thesis the most important difference between the Darwinian theory

of evolution and the Peircean alternative is that the latter identifies various forms and levels of

cooperation as genuine factor in the process of evolution whereas the former sees cooperation as

an epiphenomenon which in reality is nothing other than selfish cost-benefit calculations. We shall

now entertain both these hypotheses one by one.

A.1.1 Hypothesis 1: Only selfishness is real, cooperation is not

The plentiful forms of selection employed by neo-Darwinism are simply variants of the first hypothe-

sis - only selfishness is real and any appearance of cooperation or altruism is an epiphenomenon that

can ultimately be explained in terms of selfishness. All recent neo-Darwinist attempts to reduce

cooperation to selfishness are themselves the result of an evolutionary process within Darwinism.

The first stage was the abstract theory of group selection, an idea explored by Darwin himself as ev-

ident from some of his work quoted above. While it solved some cases, it had a major shortcoming:

group selected altruism results in evolutionarily unstable strategy, as it favors the selfish mutant

invader over the otherwise cooperating group of un-selfish individuals. This was followed by the

theory of kin selection, which avoided the selfish invader problem through cost-benefit calculations

proposed by Bill Hamilton’s equations. Kin selection operates at the genetic level and it involves

maximizing the inclusive fitness of kin even if the individual has to bear the cost of sacrificing its

own phenotype under certain scenarios. This theory did help explain some of the altruistic and

cooperative behavior in nature, but factually not all the individuals involved in cooperation in the

natural world are kin. Furthermore, any theory using inclusive fitness makes certain mathematical

assumptions (Nowak et al., 2010):

First, for inclusive fitness theory all interactions must be additive and pairwise. This

limitation excludes most evolutionary games that have synergistic effects or where more

than two players are involved. Many tasks in an insect colony, for example, require the

simultaneous cooperation of more than two individuals, and synergistic effects are easily

demonstrated.

Second, inclusive fitness theory can only deal with very special population structures.

105

It can describe either static structures or dynamic ones, but in the latter case there

must be global updating and binary interactions. Global updating means that any two

individuals compete uniformly for reproduction regardless of their (spatial) distance.

Binary interaction means that any two individuals either interact or they do not, but

there cannot be continuously varying intensities of interaction.

These particular mathematical assumptions, which are easily violated in nature, are

needed for the formulation of inclusive fitness theory. If these assumptions do not hold,

then inclusive fitness either cannot be defined or does not give the right criterion for

what is favoured by natural selection.

The final stage was the theory of reciprocal altruism or mutualism, which obviously avoided the

problems of kin selection by the incorporation of game theoretic aspects - especially the repeated

trials of prisoner’s dilemma. The most famous and simplest win-win strategy for players seems

to be tit-for-tat so far (Axelrod, 1984) which helps retain long term self-interests of individuals,

presuming they start with cooperation. It has many variations, but for reciprocal altruism to work,

it has a very stringent set of pre-requisites that need to be met. The repeatedly interactive species

need to have good memory systems, good recognition system for distinction between individuals,

and disincentive to cheat, to begin with (Stearns, 2009a). Even though its simplistic mappings

onto natural systems have been questioned (Clutton-Brock, 2009), this reciprocity does seem to fit

in some scenarios very well, especially in some higher animals, and it also gives good results in tit-

for-tat game theoretic strategies, but these higher animals are only a negligibly recent phenomenon

when seen on the geological scale.

Evolutionary transformations such as single cells to multi-cellular specialized colonies strictly

require cooperative coordination mechanisms for interaction between different cells (Moser et al.,

2009). Not surprisingly, most of these lower and basic level phenomena do not have any of the

strong pre-requisites for reciprocal altruism to take place, and yet cooperation is ascribed to them

in one game-theoretic form or another, frequently using gene-centric view.

106

A.1.2 Hypothesis 2: Cooperation is as real as Selfishness

Microbiologists such as Lynn Margulis pursue the second hypothesis: that cooperation is as real a

phenomenon as selfishness; and not only do living organisms possess both characteristics but it is

historically necessary that, under certain circumstances, the cooperative aspect overshadowed the

selfish part for life to have developed to the extent it has till today. Through her work she has

come to the conclusion that (non-game-theoretic) symbiosis is so important that it is responsible

for the creation of new species in certain scenarios. This conception of cooperation involves two

organisms coming into a long term relationship and in the process a new entity bigger than the

two participating individuals comes into being. Some examples from (Margulis and Sagan, 2002)

(page 91) include: coral reef animals and their algae, photosynthetic clams, sulfide-oxidizing, two-

meter-long tube worms, urbanized termites; and grass munching cows. Now an established term,

symbiogenesis is considered as an explanation for the evolution of eukaryotic cells from the primitive

prokaryotic cells (Margulis, 1981; Lake, 2009). Molecular biology has come to the conclusion that

life as we see it today, could not have evolved unless there were high levels of cooperative associations

amongst the basic elements of life at all key stages of evolutionary history (Taylor and Nowak, 2009;

Smith and Szathmary, 1997):

Evolution occurs in populations of reproducing individuals. Mutation, selection, and

cooperation can be seen as the three fundamental principles of evolution. Cooperation

is needed for evolution to construct new levels of organization. The origin of life, the

emergence of the first cell, the arrival of eucarya, the rise of multi-cellular organisms,

and the advent of human language are all based on cooperation. A higher level of orga-

nization emerges, whenever the competing units on the lower level begin to cooperate.

On another front, Systems biology and epigenetics hold responsible not the genes but the

interaction of information between the intra-cellular elements and the extra-cellular environment

as an equally important factor in the evolutionary process. (Noble, 2006, 2010), (Woese, 2004), and

(Shapiro, 1997) through their work prove that reducing the entire natural system to selfish gene-eye-

view does not work anymore. In fact, they say that there is no single level that explains everything.

107

Rather their biology is going towards probabilistic and dynamically interactive systems and this

is another blow to the explicitly genetic level kin selection, mutualism and other explanations of

cooperation in neo-Darwinian literature.

It seems reasonable at this point to emphasize that most of the jargon used in evolutionary

biology to explain the natural phenomenon of symbiotic relationships is an example of misplaced

concreteness in scientific language. The following terminologies become instantly meaningless for

species not belonging to the negligible class of “higher animals”: altruism, benefit, game payoff

matrix, selfish gene, inclusive fitness, mutualism, reciprocal altruism, etc. The relation of symbiotic

entities is one of association not “benefits” or “costs” or “cooperation” or “competition” in the

game-theoretic terms (Margulis and Sagan, 2002) (pages 15-18). It is evident that this ignorance

of metabolic and physical associations in the contemporary evolutionary analyses is a legacy of

Darwin’s own writings. (Caldwell, 1999) has done a histogram analysis of frequent words used in

The Origin of Species, and while we find hundreds of repetitions for words such as destruction, kill,

exterminate, individual, race, perfect, selection, species, by contrast the following terms are missing:

association, affiliation, cooperation, community, and symbiosis. It is time that these more natural

metaphors are developed for a revision of otherwise stagnant evolutionary analyses on cooperation

(Goldenfeld and Woese, 2007). It is also time that nature be accepted along with its multifaceted

mechanisms rather than enforcing upon it simplistic conveniences.

We conclude this section by highlighting another important difference between Darwin and

Peirce. Whereas Darwin’s theory requires a negation of an important part of Peirce’s theory (re-

ducing Peircean cooperation to selfishness), there is nothing in Peircean theory that requires a

negation of Darwinian principles. Darwin pursues the first hypothesis, in which only selfishness is

real, and cooperation emerged because individuals sought to exploit the benefits offered by simu-

lations of cooperation. On the other hand, Peirce pursues the second hypothesis, where relations

such as cooperation are taken to be as real as competition in the empirical reality. It is noticed

that Peirce does not present an anti-thesis of Darwin. He does not say that only cooperation is

real and competition emerged from it and is explainable and reducible to cooperative elements.

In the Peircean framework cooperation is an important element represented by his third agency.

108

In short, Darwin’s dyadic theory of evolution requires a rejection of Peirce’s theory. But Peirce’s

triadic theory embraces and corrects Darwin’s theory.

109

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Evolutionary Algorithms based on non-Darwinian Theories of Evolution: The Peircean Perspective

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