The Sphynx's new riddle: How to relate the canonical formula of myth to quantum interaction

The Sphynx’s new riddle: How to relate thecanonical formula of myth to quantum

interaction

Daranyi Sandor1, PETER WITTEK1, and Kirsty Kitto2

1University of Boras

2Queensland University of Technology

July 25, 2013

Motivation Mythology The Canonical Formula Quantum Interaction Conclusions

Outline

1 Motivation

2 Mythology

3 The Canonical Formula

4 Quantum Interaction

5 Conclusions

Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle


Argumentation

Advanced access to DL begs to focus on text genres otherthan scientific articles, with complexities of meaning beingan obstacle to process semantic contentOne reason to include belief-based narratives in DL is toadd documents of and about the collective unconscious,relevant for cognitive studiesWith folk narratives, a major implication is that theirprocessing encourages methodology outside of linguistics,such as biology and physicsWe look at cases where bag-of-words methods do not helpand probabilistic approaches have not been tested this far.



Outline

Work in progress with basic examples of a new researchidea linking the structural study of myth with group theoryand Bloch vectorsBriefly discussing:

Narrative processingFolklore, mythology and text variationFormulaity (with examples)The structural study of mythInsights combined with QIExperiment and results



Outline

1 Motivation

2 Mythology



5 Conclusions



Narrative processing

Emerging field of interest in text analytics, with focus onstoryline analysis and generation

Typical genres to be analyzed are folktales and mythsThese present special challenges to computer analysis:

Formalization, formulaic structureBuilding block identificationGenre identification based on available metadata

Broad field: digital humanities, own forum: CMNWhence the need:

Special problems but related to S & T document indexing,classification, retrieval and visualization

E.g. sentence-based indexing by tensor product,Holographic Reduced Representations (Plate 1994), circularconvolution, etc.

Information filtering according to predefined semanticcriteria for semantic markupRecurrent semantic pattern identification



Basic concepts

Folklore:Term first used by English antiquarian William Thoms in aletter published in the London journal The Athenaeum in1846Consists of legends, [music], oral history, proverbs, jokes,popular beliefs, fairy tales, stories, tall tales, and customsthat are the traditions of a culture, subculture, or groupThe above genres are also called folk narratives



Basic concepts

Mythology:A body of myths, as that of a particular people or thatrelating to a particular person, e.g. Greek mythology.Myths collectivelyThe science or study of mythsA set of stories, traditions, or beliefs associated with aparticular group or the history of an event, arising naturallyor deliberately fostered, e.g. the Fascist mythology of theinterwar years



Basic concepts

In folklore text analytics, a myth is a sacred narrativeusually explaining how the world or humankind came to bein its present form, although, in a very broad sense, theword can refer to any traditional story of e.g. originsText variation: in folklore/anthropology/ethnology, artifactssuch as texts, songs, objects etc. exist in variants ratherthan canonical (archetypical) single examples, leading toclassification (conceptualization) problems (i.e. which oneis “the” original?)



Two more categories

The fertility myth:Fertility-related deities are a global phenomenonOnce widespread in the Mediterranean and the AncientNear East, this myth is a symbolic prescription of how toregulate individual and community welfareBriefly, proper moral conduct being the key, disaster strikesdue to ill behavior or violation of social norms, whereas therole of the regulator (a deity or a human, a male or a femalesuch as a sacred king or queen) is to remedy the insult tothe supernatural, and thereby bring back fertility, anindicator to signal if things are on the right track



Two more categories

The dying god:Now debated, a type proposed by Frazer (The goldenbough, 1890)In comparative mythology, the motif refers to a deity whodeparts and returns, e.g. is resurrected or reborn, in eithera literal or symbolic senseOften related to the vegetation cycle, examples include:

Ancient Mesopotamia: during the journey of Inanna or Ishtar to the underworld, the earthbecomes sterile, and neither humans nor animals are able to procreate. After confrontingher sister Ereshkigal, the ruler of the underworld, Inanna is killed, but an emissary from thegods administers potions to restore her to life. She is allowed to return to the upper worldonly if someone else will take her place. Her husband, the vegetation god Dumuzi, agreesto spend half the year in the underworld, during which time vegetation dies off. His returnbrings regrowth.Ancient Egypt: the cultural achievements of Osiris among the peoples of the earthprovokes the envy of his brother Set, who kills and dismembers him. Osiris’s wife Isisjourneys to gather his fourteen scattered body parts. In some versions, she buries eachpart where she finds it, causing the desert to put forth vegetation. In other versions, shereassembles his body and resurrects him, and he then becomes the ruler of the afterlife.



Formulatity, formulaic

The theory of oral-formulaic composition originated in thescholarly study of epic poetry, being developed in the 2ndquarter of the 20th century. It seeks to explain two relatedissues:

The mechanism whereby some oral poets are able toimprovise poetry, andWhy orally improvised poetry has the characteristics it has

The key idea of the theory is that poets have a store offormulae and that by linking these in conventionalizedways, they can rapidly compose verse

A formula being “an expression which is regularly used,under the same metrical conditions, to express a particularessential idea”



Formulatity, formulaic

Milman Parry (1902-1935), Albert Lord (1912-1991): theirapproach transformed the study of ancient and medievalpoetry, and oral poetry in general, with an impact onnarratologyMajor finding: standard sequences of content elements(formulae) pertain to documents and document parts



The structure of Greek flower myths

Example: structuralsimilarities of Greekmyths about theorigins of plants(flowers and trees)The structuredemonstrates erosionof contentTypical narrativeelements at typicallocations in the plotare in canonicalrelations with oneanother



Examples of formulaity (= structure) in narrativeresearch

Propp (1929): Russian fairy taleshave 7 actors (dramatis personae),31 functions (types of actions) and150 narrative elementsThompson (1932-37): Folktales canbe indexed by their structure. Motifindex system to catalog individualmotifs.Levi-Strauss (1954): both narrativesegments in myths (calledmythemes), and myth variants,manifest canonical contenttransformations



Levi-Strauss’ reading of the Oidipus (1955)

Syntagmatic readingParadigmatic

reading

Cadmos seeks his sister Europa, ravished by Zeus

Oedipus marries his mother, Jocasta

Antigone buries her brother, Polynices, despite prohibition

Overestimating blood relations

The Spartoi kill one another

Oedipus kills his father, Laios

Eteocles kills his brother, Polynices

Underestimating blood relations

Cadmos kills the dragon

Oedipus kills the Sphinx

Denial of matriarchal order ("born from one")

Labdacos (Laois' father) = lame (?)Laios (Oedipus father) = lef-sided (?)

Oedipus = swollen-foot (?)

Affirmation of matriarchal order ("born from one")



Outline

1 Motivation

2 Mythology



5 Conclusions



The canonical formula of myth: formulaic expressionof the above (by Andre Weil)

“Weak” form vs. “strong”,canonical form:

The four componentsstand for two oppositionsof four paradigmsThey manifest the orbit of aKlein group

From two “weak” formulae, two“strong” (= canonical) versionsby symmetry breaking asinteraction between two Kleingroups:



Research problem

The fundamental difficulty with myths is conceptualcontamination, also called eclecticism or syncretism, i.e.different concepts belonging to the same category (e.g. thedying deity) can appear in the same plot so that nobodycan tell them apartThe other is the fundamental insecurity of not knowingwhat factor may be important and how much of itsmanifestations can be out there. So a probabilistic tool,should one exist or could one be designed, would be asignificant step forward

E.g. what is the probability that a text fragment is in statefx(a), or a whole text as a mix of fx(a) : fy (b) :: fx(b) : fa−1(y)has a given outcome for fa−1(y)?This is a weighted superposition

For a probabilistic tool, enter QI



Recent insights

Lost in translation for 50 years: Weil used group theory toformalize the CFThe CF encodes a plotMorava (2005) demonstrated that the CF is a(non-commutative) quaternion group of order eightQuaternions correspond to Pauli matrices and can bedisplayed by Bloch spheres, so that a set of story variantsinfluence the behavior of the state vector in the space ofthe CF, i.e. the Bloch sphereNot one but 32 CF, and possibly many more

In other words the CF as a narrative generation toolperforms the same transformations on the plot but underrotation of its group, leading to new actors and actions innew situations, i.e. plot variantsAll CFs are pure state vectors in a Bloch sphere

The CF is candidate for information filteringDaranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle


Insight 1: Not 1 but at least 32 “weak” vs. “strong”versions of the CF exist

“Weak” forms “Strong” formsOctet A (term +, function +) Octet E (term +, function +)NF1 = x(a) : y(b) :: x(b) : y(a) CF1 = x(a) : y(b) :: x(b) : a−1(y)NF2 = x−1(a) : y−1(b) :: x−1(b) : y−1(a) CF2 = x−1(a) : y−1(b) :: x−1(b) : a−1(y−1)NF3 = x(a−1) : y(b−1) :: x(b−1) : y(a−1) CF3 = x(a−1) : y(b−1) :: x(b−1) : a(y)NF4 = x−1(a−1) : y−1(b−1) :: x−1(b−1) : y−1(a−1) CF4 = x−1(a−1) : y−1(b−1) :: x−1(b−1) : a(y−1)NF5 = a(x) : b(y) :: b(x) : a(y) CF5 = a(x) : b(y) :: b(x) : y−1(a)NF6 = a(x−1) : b(y−1) :: b(x−1) : a(y−1) CF6 = a(x−1) : b(y−1) :: b(x−1) : y(a)NF7 = a−1(x) : b−1(y) :: b−1(x) : a−1(y) CF7 = a−1(x) : b−1(y) :: b−1(x) : y−1(a−1)NF8 = a−1(x−1) : b−1(y−1) :: b−1(x−1) : a−1(y−1) CF8 = a−1(x−1) : b−1(y−1) :: b−1(x−1) : y(a−1)Octet B (term -, function +) Octet F (term -, function +)NF9 = x(−a) : y(−b) :: x(−b) : y(−a) CF9 = x(−a) : y(−b) :: x(−b) :−a−1(y)NF10 = x−1(−a) : y−1(−b) :: x−1(−b) : y−1(−a) CF10 = x−1(−a) : y−1(−b) :: x−1(−b) :−a−1(y−1)NF11 = x(−a−1) : y(−b−1) :: x(−b−1) : y(−a−1) CF11 = x(−a−1) : y(−b−1) :: x(−b−1) :−a(y)NF12 = x−1(−a−1) : y−1(−b−1) :: x−1(−b−1) : y−1(−a−1) CF12 = x−1(−a−1) : y−1(−b−1) :: x−1(−b−1) :−a(y−1)NF13 =−a(x) :−b(y) ::−b(x) :−a(y) CF13 =−a(x) :−b(y) ::−b(x) : y−1(−a)NF14 =−a(x−1) :−b(y−1) ::−b(x−1) :−a(y−1) CF14 =−a(x−1) :−b(y−1) ::−b(x−1) : y(−a)NF15 =−a−1(x) :−b−1(y) ::−b−1(x) :−a−1(y) CF15 =−a−1(x) :−b−1(y) ::−b−1(x) : y−1(−a−1)NF16 =−a−1(x−1) :−b−1(y−1) ::−b−1(x−1) :−a−1(y−1) CF16 =−a−1(x−1) :−b−1(y−1) ::−b−1(x−1) : y(−a−1)Octet C (term +, function -) Octet G (term +, function -)NF17 =−x(a) :−y(b) ::−x(b) :−y(a) CF17 =−x(a) :−y(b) ::−x(b) : a−1(−y)NF18 =−x−1(a) :−y−1(b) ::−x−1(b) : (−y−1(a) CF18 =−x−1(a) :−y−1(b) ::−x−1(b) : a−1(−y−1)NF19 =−x(a−1) :−y(b−1) ::−x(b−1) : a(−y) CF19 =−x(a−1) :−y(b−1) ::−x(b−1) : a(−y)NF20 =−x−1(a−1) :−y−1(b−1) ::−x−1(b−1) :−y(a−1 CF20 =−x−1(a−1) :−y−1(b−1) ::−x−1(b−1) : a(−y−1)NF21 = a(−x) : b(−y) :: b(−x) : a(−y) CF21 = a(−x) : b(−y) :: b(−x) :−y−1(a)NF22 = a(−x−1) : b(−y−1) :: b(−x−1) : a(−y−1) CF22 = a(−x−1) : b(−y−1) :: b(−x−1) :−y(a)NF23 = a−1(−x) : b−1(−y) :: b−1(−x) : a−1(−y) CF23 = a−1(−x) : b−1(−y) :: b−1(−x) :−y−1(a−1)NF24 = a−1(−x−1) : b−1(−y−1) :: b−1(−x−1) : a−1(−y−1) CF24 = a−1(−x−1) : b−1(−y−1) :: b−1(−x−1) :−y(a−1)Octet D (term -, function -) Octet H (term -, function -)NF25 =−x(−a) :−y(−b) ::−x(−b) :−y(−a) CF25 =−x(−a) :−y(−b) ::−x(−b) :−a−1(−y)NF26 =−x−1(−a) :−y−1(−b) ::−x−1(−b) :−y−1(−a) CF26 =−x−1(−a) :−y−1(−b) ::−x−1(−b) :−a−1(−y−1)NF27 =−x(−a−1) :−y(−b−1) ::−x(−b−1) :−y(−a−1) CF27 =−x(−a−1) :−y(−b−1) ::−x(−b−1) :−a(−y)NF28 =−x−1(−a−1) :−y−1(−b−1) ::−x−1(−b−1) :−y−1(−a−1) CF28 =−x−1(−a−1) :−y−1(−b−1) ::−x−1(−b−1) :−a(−y−1)NF29 =−a(−x) :−b(−y) ::−b(−x) :−a(−y) CF29 =−a(−x) :−b(−y) ::−b(−x) :−y−1(−a)NF30 =−a(−x−1) :−b(−y−1) ::−b(−x−1) :−y(−a) CF30 =−a(−x−1) :−b(−y−1) ::−b(−x−1) :−y(−a)NF31 =−a−1(−x) :−b−1(−y) ::−b−1(−x) :−a−1(−y) CF31 =−a−1(−x) :−b−1(−y) ::−b−1(−x) :−y−1(−a−1)NF32 =−a−1(−x−1) :−b−1(−y−1) ::−b−1(−x−1) :−b−1(−y−1) CF32 =−a−1(−x−1) :−b−1(−y−1) ::−b−1(−x−1) :−y(−a−1)

By interaction between their respective fourth arguments, 32 CF can be generated from 32 NF.



Insight 2: For the fertility myth, a linguistic rendering ofthe CF translates group components into text variants

fx(a) : fy(b ):: fx(b) : fa-1(y)

„divine adult male creates [other]:mortal adult male destroys [other] ::mortal adult male creates [other] :divine adolescent male destroys[himself]”



Outline

1 Motivation

2 Mythology



5 Conclusions



The quaternion group of order eight

Q = {±1,±i ,±j ,±k}The noncommutative product operation defined asij = k = −ji , jk = i = −kj , ki = j = −ik , ii = jj = kk = −1,and (−1)2 = 1.

The canonical formula: Fx(a) : Fy (b) 7→ Fx(b) : Fa−1(y).x 7→ 1, a 7→ i , y 7→ j , and b 7→ k

This automorphism reproduces the canonical formula.



The Pauli matrices

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

).

With the identity matrix I, they form a basis for the realHilbert space of 2× 2 complex Hermitian matrices.The real linear span of {I, iσx , iσy , iσz} is isomorphic to thereal algebra of quaternions H.

1 7→ I, i 7→ −iσx , j 7→ −iσy , k 7→ −iσz .

A density matrix can be written as ρ = 12(I + sσ)

s is called the Bloch vector



Mapping to the Bloch sphere

(a) A pure state. (b) A mixed state.

Figure : A pure state corresponds to a point on the surface of theBloch sphere, whereas a mixed state is inside the Bloch sphere.



Applying the probabilistic description

“Belief contamination”Different concepts belonging to the same category (e.g. thedying deity) can appear in the same plot

Insecurity of not knowing what factor may be important andhow much of its manifestations can be out there

What is the probability that a text fragment is in state Fx(a),or a whole text as a mix of Fx(a) : Fy (b) 7→ Fx(b) : Fa−1(y)has a given outcome for Fa−1(y)?



Case study

Linguistic rendering of the CF exemplified on the myths ofAdonis and Attis (minor deities from Asia Minor in theHellenistic period, 323-31 BC)

Typical plot: a youth invites harm by being too beautiful,losing his virility from which a specific plant springs upAttis’ story (13 variants) relates the loss to directself-mutilation (8); to mutual castration with partner (indirectself-mutilation, 1); to being born as an eunuch or killed byspear through an unspecified wound (indirect not-selfmutilation, 2); or the goddess mutilating him as punishmentfor his infidelity (direct not-self mutilation, 2)Direct self (DS) : not-direct self (DNS) :: not-direct not-self(NDNS) : direct not-self (DNS) constitute two oppositions offour paradigms as per the CF

The axes Fy (b) = x , Fx(b) = z, and Fa−1(y) = y , wherethe latter can have four outcomes as above, the result is amixed state vector weighted by the outcome probabilities.Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle


Outline

1 Motivation

2 Mythology



5 Conclusions



Summary

Bridging the gapAnalytical studies in need of processing methodologyProcessing methodology development in need of rawmaterial

A concrete example how a topical set of myth variantscorrespond via their syntagmatic transcripts to narrativeformulae

Families of narrative formulae, some with double invertedvalues in their arguments, some without, all share the samegroup structure with a certain quaternion group of ordereight.

A quantum probabilistic framework further generalises theformulae.


The Sphynx's new riddle: How to relate the canonical formula of myth to quantum interaction

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