A FRAMEWORK FOR REPRESENTING AND PROCESSINGARBITRARY MATHEMATICS

Arnold Neumaier and Peter SchodlFakultat fur Mathematik, University of Vienna,Nordbergstr. 15, A-1090 Wien, Austria

Keywords: Knowledge representation, Human-machine coorperation.

Abstract: While mathematicians already benefit from the computer as regards numerical problems, visualization, sym-bolic manipulation, typesetting, etc., there is no common facility to store and process information, and math-ematicians usually have to communicate the same mathematical content multiple times to the computer. Weare in the process of creating and implementing a framework that is capable of representing and interfacingoptimization problems, and we argue that this framework can be used to represent arbitrary mathematics andcontribute towards a universal mathematical database.

1 INTRODUCTION

Mathematicians nowadays rely heavily on comput-ers. They use them to communicate with colleagues,search the web for information, create documentsthey want to publish, perform numerical and symboliccomputations, check their proofs, store the work theyhave done, etc. However, since mathematicians ad-dress very diverse parties with their writing, a mathe-matician usually has to formulate the same idea (e.g.,a proof, a numerical problem, a conjunction) multi-ple times, depending on the recipient: for a studentin great detail, for a foreign colleague working in thesame field in less detail but a common language, for apublication in a document markup language, for a nu-merical solver in an algebraic modeling language, fora proof checker in a special language and at a tremen-dous level of detail.Our vision is that, when represented in an adequateway, the same mathematical content only needs to becommunicated to the computer once, and the machinecan then extract the information in different formats,depending on the addressee. We have achieved somepromising results representing and reformulating op-timization problems into different formal and (con-trolled) natural languages, and we envision that ourframework used for optimization problems is generalenough for representing and communicating arbitrarymathematics.If a general representation with rich possibilities tointerface the information proves feasible, it may alsocontribute to a huge electronic database containing

essential amounts of the known mathematics. Thisvision is not new, it dates back at least to the QEDproject (Boyer, 1994). Its goal was to represent all im-portant mathematical knowledge, conforming to thehighest standards of mathematical rigor. Another vi-sion in this direction was the universal automated in-formation system for all sciences (Andrews, 2003).This is even more ambitious than a universal math-ematical database, but the prominent role of mathe-matics in such a system, also discussed by Andrews,would make a mathematical database probably a cor-ner stone of such a system and could be a startingpoint.

2 THE SEMANTIC MEMORY

The semantic memory is the data structure that is,in our opinion, most adequate to represent arbitrarymathematics. It is not a fixed file format like an Excel-sheet or an XML-file, but rather an abstract conceptof a data structure, like a binary search tree or a heap.The semantic entities we want to refer to are calledobjects. The set of objects is not fixed, but frequentlyused mathematical notions like set, function, deriva-tive, manifold, R, π etc. may be objects. All the in-formation is then stored in a way that is akin to thesemantic web, namely we store relations of the form

object1.object2 = object3 (1)

with the only restriction that

476Neumaier A. and Schodl P. (2010).A FRAMEWORK FOR REPRESENTING AND PROCESSING ARBITRARY MATHEMATICS.In Proceedings of the International Conference on Knowledge Engineering and Ontology Development, pages 476-479DOI: 10.5220/0003119104760479Copyright c© SciTePress

object1.object2 = object3 andobject1.object2 = object4 impliesobject3 = object4.

One possible way to store such relations is in the formof a matrix, where the relation (1) is represented asan entry object3 in the row with name object1 andcolumn with name object2 . We call this matrix thesemantic matrix.Such relations can also be stored and visualized ina labeled, directed graph, where the relation (1) isrepresented as an arc from vertex object1 to ver-tex object3 , labeled with object2 . Such a directedgraph will be called a semantic graph.For example, the information 7+5 = 12 may be rep-resented by the following set of relations:equation1.RHS = 12equation1.LHS = term_lhsequation1.OP = EQUALterm_lhs.1 = 7term_lhs.2 = 5term_lhs.OP = PLUS

This corresponds to the semantic matrix given in Fig-ure 1 and to the semantic graph in Figure 2.

LHS RHS OP 1 2equation1 term lhs 12 EQUALterm lhs PLUS 7 5

Figure 1: 7+5 = 12 represented in a semantic matrix.

equation1

term_lhs

LHS

EQUALOP

12

RHS

7

1

5

2

PLUSOP

Figure 2: 7+5 = 12 represented in a semantic graph.

A semantic graph is a special case of a concept map,a graphical tool to organize and represent knowledge,but that is not defined rigorously (Novak and Canas,2006).

3 THINGS THAT ALREADYEXIST

We list some of the facilities that already providevaluable aid for mathematicians. Interfaces to thesesystems would be desirable.

Semantic Markup Languages for Mathematics.This is a group of languages that represent mathemat-ics as annotated text. They are used to communicateformulas between machines, display formulas on theweb, etc. Examples are OMDoc (Kohlhase, 2001) andOpenMath (Abbott et al., 1996).

Computer Algebra Systems. These are programsthat are able to perform symbolic manipulations onterms, solve equations, plot graphs, etc. Examplesfor widely used CAS’s are Mathematica, Maple andMaxima.

Algebraic Modeling Languages. Numerical opti-mization problems can be expressed quite comfort-able in algebraic modeling languages. The machinereads the description of the problem and the numeri-cal data and is able to interface a variety of solvers.Examples are AMPL, GAMS, GLPK and NOP-2(Kallrath, 2004).

Proof Assistants. A proof assistant is software thatchecks the validity of a proof, expressed in a special,highly detailed and annotated language. Since trans-lating a proof into such a language is a lot of work,proof checkers are not widely used among mathemati-cians.

4 NATURAL LANGUAGE IN- ANDOUTPUT

For being attractive for a working mathematician, theability to interface existing systems is one key fea-ture, another one is communication in an almost nat-ural language.There is a consensus among mathematicians and lin-guists that the communication of mathematics to acomputer is much easier than the communication ofarbitrary content. This has several reasons:

• Mathematical discourse has a well-defined do-main, is highly structured, and has relatively smallset of discourse relations. The reasoning patternsapplied in mathematics are widely studied and un-derstood (Zinn, 2004). Building an ontology for,say, number theory, is much easier than for a nat-ural domain, because mathematicians define con-cepts before they use them. It was even claimedthat “[. . . ] if we fail to construct an understanderfor mathematical discourse, then we will also failto write one for other (non-trivial) domains”, seep. 8 in (Zinn, 2004).

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• Due to the fact that mathematicians want to com-municate unambiguously, they tend to use a rela-tively small set of phrases to express their ideas,and there is a standard interpretation for thesephrases. About 700 phrases suffice for the es-sential part of mathematics (definitions, theorems,proofs, etc.) but this does not include the more in-formal motivational part (Trzeciak, 1995).

• Mathematicians use words and phrases in a veryrigid way. The language of mathematics is sim-ple: very few variety in time, person, etc. (Gane-salingam, 2009).

• Another reason why mathematics is apt to berepresented by a machine is that in mathematicswe are in the (probably unique) position that ev-ery meaningful rigorous statement can, at leastin principle, be translated into a formal language.Therefore, it is possible for a machine to faithfullyrepresent the complete content of an arbitrary (butmeaningful) mathematical statement.

However, we do not intend to allow general naturallanguage as input, even though we expect only rela-tively simple sentences, but we intend to exploit thefact that mathematical language is simple by defininga controlled natural language (CNL) that is expressiveenough to fulfill the needs of mathematicians, whilestill sounding like natural language.For formulas, since LATEX has been de facto-standardin the mathematical community for decades, we en-vision a reasonable subset of LATEX as the main inputformat.

5 THINGS ALREADYIMPLEMENTED

• Optimization problems can be represented in thesemantic memory, and a description of the prob-lem can then be automatically generated in the al-gebraic modeling language AMPL and in almostnatural language. Below is an example of a simpleoptimization problem, in Figure 3 formulated as amathematician would do, and then in in Figure 4in the AMPL-format. Both texts have been gener-ated automatically from a common representationin the semantic memory comprising about 550 re-lations of the form of (1).

• For the semantic memory we have two implemen-tations, one written in Matlab where the seman-tic memory is a sparse matrix and the objects arenatural numbers, and another one in Soprano, aframework for RDF data.

Multi-dimensional knapsack.Let integer N be number of contract , let integer Mbe number of budget, let c j be contract volume ofproject j for j = 1, . . . ,N, let Ai, j be estimated costof budget i for project j for i = 1, . . . ,M and j =1, . . . ,N, let Bi be available amount of budget i fori = 1, . . . ,M and let x j = 1 if project j is selected,and let x j = 0 otherwise for j = 1, . . . ,N.Problem : Given integer N, integer M , vector c,matrix A and vector B find binary vector x such that

N

∑j=1

c j x j

is maximal under the constraint ∑Nj=1 Ai, j x j ≤ Bi

for i = 1, . . . ,M.

Figure 3: The knapsack-problem in (almost) natural mathe-matical language.

param N ;param M ;param c {j in 1..N } ;param A {i in 1..M , j in 1..N } ;param B {i in 1..M } ;var x {j in 1..N } binary ;

maximize target : sum {j in 1..N }(c[j]* x[j]);subject to constraint 3014 {i in 1..M }: sum {j in 1..N }(A[i , j] * x[j]) <=B[i];

Figure 4: The knapsack-problem in AMPL.

• We created an interface to the controlled natu-ral language of the Naproche project (Kuhlweinet al., 2009), a project carried out at the univer-sity of Bonn that enables proof checking of proofswritten in a controlled natural language.

• Creation of LATEX-output of simple general mathe-matical text represented in the semantic memory:basic forms of definitions, assumptions, interfer-ences, etc.

• Grammatically correct text-output is generatedvia an interface to the Grammatical Framework(Ranta, 2004), a programming language and soft-ware package for multilingual grammar applica-tions.

• We implemented a parser for problem files ofthe TPTP (Thousands of Problems for TheoremProvers, available at http://www.tptp.org/ ),and parsed and represented large parts in the se-mantic memory, adding up to several thousand

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formulas. These have been represented in the se-mantic memory and written into LATEX-files.

6 CONCLUSIONS

We gave our arguments why we think that our frame-work to represent optimization problems can be astarting point for a more general system to repre-sent and process arbitrary mathematics. We think thatone of the key features of a universal mathematicaldatabase, that will actually be used, is the retrievalof information in many different forms, together withthe ability to communicate with the user in a way nat-ural to the human. There is already a huge amountof mathematical knowledge on the web, e.g., in on-line encyclopedias, in archives for scientific papers,libraries for proof assistants etc., but as long as theyonly serve single purposes, they will stay separated.

ACKNOWLEDGEMENTS

Support by the Austrian Science Fund (FWF) undercontract number P20631 is gratefully acknowledged.

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