The Sphynx’s new riddle: How to relate the canonical formula of myth to quantum interaction 1 , PETER WITTEK1 , and Kirsty Kitto2 ´ ´ Daranyi Sandor 1 University 2 Queensland ˚ of Boras University of Technology July 25, 2013 Motivation Mythology The Canonical Formula Quantum Interaction 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 Conclusions Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Argumentation Advanced access to DL begs to focus on text genres other than scientific articles, with complexities of meaning being an obstacle to process semantic content One reason to include belief-based narratives in DL is to add documents of and about the collective unconscious, relevant for cognitive studies With folk narratives, a major implication is that their processing encourages methodology outside of linguistics, such as biology and physics We look at cases where bag-of-words methods do not help and probabilistic approaches have not been tested this far. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Outline Work in progress with basic examples of a new research idea linking the structural study of myth with group theory and Bloch vectors Briefly discussing: Narrative processing Folklore, mythology and text variation Formulaity (with examples) The structural study of myth Insights combined with QI Experiment and results ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction 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 Conclusions Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Narrative processing Emerging field of interest in text analytics, with focus on storyline analysis and generation Typical genres to be analyzed are folktales and myths These present special challenges to computer analysis: Formalization, formulaic structure Building block identification Genre identification based on available metadata Broad field: digital humanities, own forum: CMN Whence 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), circular convolution, etc. Information filtering according to predefined semantic criteria for semantic markup Recurrent semantic pattern identification ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Basic concepts Folklore: Term first used by English antiquarian William Thoms in a letter published in the London journal The Athenaeum in 1846 Consists of legends, [music], oral history, proverbs, jokes, popular beliefs, fairy tales, stories, tall tales, and customs that are the traditions of a culture, subculture, or group The above genres are also called folk narratives ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Basic concepts Mythology: A body of myths, as that of a particular people or that relating to a particular person, e.g. Greek mythology. Myths collectively The science or study of myths A set of stories, traditions, or beliefs associated with a particular group or the history of an event, arising naturally or deliberately fostered, e.g. the Fascist mythology of the interwar years ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Basic concepts In folklore text analytics, a myth is a sacred narrative usually explaining how the world or humankind came to be in its present form, although, in a very broad sense, the word can refer to any traditional story of e.g. origins Text variation: in folklore/anthropology/ethnology, artifacts such as texts, songs, objects etc. exist in variants rather than canonical (archetypical) single examples, leading to classification (conceptualization) problems (i.e. which one is “the” original?) ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Two more categories The fertility myth: Fertility-related deities are a global phenomenon Once widespread in the Mediterranean and the Ancient Near East, this myth is a symbolic prescription of how to regulate individual and community welfare Briefly, proper moral conduct being the key, disaster strikes due to ill behavior or violation of social norms, whereas the role of the regulator (a deity or a human, a male or a female such as a sacred king or queen) is to remedy the insult to the supernatural, and thereby bring back fertility, an indicator to signal if things are on the right track ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Two more categories The dying god: Now debated, a type proposed by Frazer (The golden bough, 1890) In comparative mythology, the motif refers to a deity who departs and returns, e.g. is resurrected or reborn, in either a literal or symbolic sense Often related to the vegetation cycle, examples include: Ancient Mesopotamia: during the journey of Inanna or Ishtar to the underworld, the earth becomes sterile, and neither humans nor animals are able to procreate. After confronting her sister Ereshkigal, the ruler of the underworld, Inanna is killed, but an emissary from the gods administers potions to restore her to life. She is allowed to return to the upper world only if someone else will take her place. Her husband, the vegetation god Dumuzi, agrees to spend half the year in the underworld, during which time vegetation dies off. His return brings regrowth. Ancient Egypt: the cultural achievements of Osiris among the peoples of the earth provokes the envy of his brother Set, who kills and dismembers him. Osiris’s wife Isis journeys to gather his fourteen scattered body parts. In some versions, she buries each part where she finds it, causing the desert to put forth vegetation. In other versions, she reassembles his body and resurrects him, and he then becomes the ruler of the afterlife. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Formulatity, formulaic The theory of oral-formulaic composition originated in the scholarly study of epic poetry, being developed in the 2nd quarter of the 20th century. It seeks to explain two related issues: The mechanism whereby some oral poets are able to improvise poetry, and Why orally improvised poetry has the characteristics it has The key idea of the theory is that poets have a store of formulae and that by linking these in conventionalized ways, they can rapidly compose verse A formula being “an expression which is regularly used, under the same metrical conditions, to express a particular essential idea” ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Formulatity, formulaic Milman Parry (1902-1935), Albert Lord (1912-1991): their approach transformed the study of ancient and medieval poetry, and oral poetry in general, with an impact on narratology Major finding: standard sequences of content elements (formulae) pertain to documents and document parts ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions The structure of Greek flower myths Example: structural similarities of Greek myths about the origins of plants (flowers and trees) The structure demonstrates erosion of content Typical narrative elements at typical locations in the plot are in canonical relations with one another ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Examples of formulaity (= structure) in narrative research Propp (1929): Russian fairy tales have 7 actors (dramatis personae), 31 functions (types of actions) and 150 narrative elements Thompson (1932-37): Folktales can be indexed by their structure. Motif index system to catalog individual motifs. ´ Levi-Strauss (1954): both narrative segments in myths (called mythemes), and myth variants, manifest canonical content transformations ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Conclusions Motivation Mythology The Canonical Formula Quantum Interaction ´ Levi-Strauss’ reading of the Oidipus (1955) Paradigmatic reading Syntagmatic reading Cadmos seeks his sister Europa, ravished by Zeus Cadmos kills the dragon The Spartoi kill one another Oedipus kills his father, Laios Oedipus kills the Sphinx Labdacos (Laois' father) = lame (?) Laios (Oedipus father) = lef-sided (?) Oedipus = swollen-foot (?) Oedipus marries his mother, Jocasta Eteocles kills his brother, Polynices Antigone buries her brother, Polynices, despite prohibition Overestimating blood relations Underestimating blood relations ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto Denial of matriarchal order ("born from one") Affirmation of matriarchal order ("born from one") The Sphynx’s new riddle Conclusions Motivation Mythology The Canonical Formula Quantum Interaction 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 Conclusions Motivation Mythology The Canonical Formula Quantum Interaction Conclusions The canonical formula of myth: formulaic expression of the above (by Andre´ Weil) “Weak” form vs. “strong”, canonical form: The four components stand for two oppositions of four paradigms From two “weak” formulae, two “strong” (= canonical) versions by symmetry breaking as interaction between two Klein groups: They manifest the orbit of a Klein group ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Research problem The fundamental difficulty with myths is conceptual contamination, also called eclecticism or syncretism, i.e. different concepts belonging to the same category (e.g. the dying deity) can appear in the same plot so that nobody can tell them apart The other is the fundamental insecurity of not knowing what factor may be important and how much of its manifestations can be out there. So a probabilistic tool, should one exist or could one be designed, would be a significant step forward E.g. 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) :: fx (b) : fa−1 (y ) has a given outcome for fa−1 (y)? This is a weighted superposition For a probabilistic tool, enter QI ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Recent insights Lost in translation for 50 years: Weil used group theory to formalize the CF The CF encodes a plot Morava (2005) demonstrated that the CF is a (non-commutative) quaternion group of order eight Quaternions correspond to Pauli matrices and can be displayed by Bloch spheres, so that a set of story variants influence the behavior of the state vector in the space of the CF, i.e. the Bloch sphere Not one but 32 CF, and possibly many more In other words the CF as a narrative generation tool performs the same transformations on the plot but under rotation of its group, leading to new actors and actions in new situations, i.e. plot variants All CFs are pure state vectors in a Bloch sphere The CF is candidate for information filtering ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Insight 1: Not 1 but at least 32 “weak” vs. “strong” versions of the CF exist “Weak” forms Octet A (term +, function +) NF 1 = x(a) : y(b) :: x(b) : y(a) NF 2 = x− 1 (a) : y− 1 (b) :: x− 1 (b) : y− 1 (a) NF 3 = x(a− 1 ) : y(b− 1 ) :: x(b− 1 ) : y(a− 1 ) NF 4 = x− 1 (a− 1 ) : y− 1 (b− 1 ) :: x− 1 (b− 1 ) : y− 1 (a− 1 ) NF 5 = a(x) : b(y) :: b(x) : a(y) NF 6 = a(x− 1 ) : b(y− 1 ) :: b(x− 1 ) : a(y− 1 ) NF 7 = a− 1 (x) : b− 1 (y) :: b− 1 (x) : a− 1 (y) NF 8 = a− 1 (x− 1 ) : b− 1 (y− 1 ) :: b− 1 (x− 1 ) : a− 1 (y− 1 ) Octet B (term -, function +) NF 9 = x(−a) : y(−b) :: x(−b) : y(−a) NF 10 = x− 1 (−a) : y− 1 (−b) :: x− 1 (−b) : y− 1 (−a) NF 11 = x(−a− 1 ) : y(−b− 1 ) :: x(−b− 1 ) : y(−a− 1 ) NF 12 = x− 1 (−a− 1 ) : y− 1 (−b− 1 ) :: x− 1 (−b− 1 ) : y− 1 (−a− 1 ) NF 13 = −a(x) : −b(y) :: −b(x) : −a(y) NF 14 = −a(x− 1 ) : −b(y− 1 ) :: −b(x− 1 ) : −a(y− 1 ) NF 15 = −a− 1 (x) : −b− 1 (y) :: −b− 1 (x) : −a− 1 (y) NF 16 = −a− 1 (x− 1 ) : −b− 1 (y− 1 ) :: −b− 1 (x− 1 ) : −a− 1 (y− 1 ) Octet C (term +, function -) NF 17 = −x(a) : −y(b) :: −x(b) : −y(a) NF 18 = −x− 1 (a) : −y− 1 (b) :: −x− 1 (b) : (−y− 1 (a) NF 19 = −x(a− 1 ) : −y(b− 1 ) :: −x(b− 1 ) : a(−y) NF 20 = −x− 1 (a− 1 ) : −y− 1 (b− 1 ) :: −x− 1 (b− 1 ) : −y(a− 1 NF 21 = a(−x) : b(−y) :: b(−x) : a(−y) NF 22 = a(−x− 1 ) : b(−y− 1 ) :: b(−x− 1 ) : a(−y− 1 ) NF 23 = a− 1 (−x) : b− 1 (−y) :: b− 1 (−x) : a− 1 (−y) NF 24 = a− 1 (−x− 1 ) : b− 1 (−y− 1 ) :: b− 1 (−x− 1 ) : a− 1 (−y− 1 ) Octet D (term -, function -) NF 25 = −x(−a) : −y(−b) :: −x(−b) : −y(−a) NF 26 = −x− 1 (−a) : −y− 1 (−b) :: −x− 1 (−b) : −y− 1 (−a) NF 27 = −x(−a− 1 ) : −y(−b− 1 ) :: −x(−b− 1 ) : −y(−a− 1 ) NF 28 = −x− 1 (−a− 1 ) : −y− 1 (−b− 1 ) :: −x− 1 (−b− 1 ) : −y− 1 (−a− 1 ) NF 29 = −a(−x) : −b(−y) :: −b(−x) : −a(−y) NF 30 = −a(−x− 1 ) : −b(−y− 1 ) :: −b(−x− 1 ) : −y(−a) NF 31 = −a− 1 (−x) : −b− 1 (−y) :: −b− 1 (−x) : −a− 1 (−y) NF 32 = −a− 1 (−x− 1 ) : −b− 1 (−y− 1 ) :: −b− 1 (−x− 1 ) : −b− 1 (−y− 1 ) “Strong” forms Octet E (term +, function +) CF 1 = x(a) : y(b) :: x(b) : a− 1 (y) CF 2 = x− 1 (a) : y− 1 (b) :: x− 1 (b) : a− 1 (y− 1 ) CF 3 = x(a− 1 ) : y(b− 1 ) :: x(b− 1 ) : a(y) CF 4 = x− 1 (a− 1 ) : y− 1 (b− 1 ) :: x− 1 (b− 1 ) : a(y− 1 ) CF 5 = a(x) : b(y) :: b(x) : y− 1 (a) CF 6 = a(x− 1 ) : b(y− 1 ) :: b(x− 1 ) : y(a) CF 7 = a− 1 (x) : b− 1 (y) :: b− 1 (x) : y− 1 (a− 1 ) CF 8 = a− 1 (x− 1 ) : b− 1 (y− 1 ) :: b− 1 (x− 1 ) : y(a− 1 ) Octet F (term -, function +) CF 9 = x(−a) : y(−b) :: x(−b) : −a− 1 (y) CF 10 = x− 1 (−a) : y− 1 (−b) :: x− 1 (−b) : −a− 1 (y− 1 ) CF 11 = x(−a− 1 ) : y(−b− 1 ) :: x(−b− 1 ) : −a(y) CF 12 = x− 1 (−a− 1 ) : y− 1 (−b− 1 ) :: x− 1 (−b− 1 ) : −a(y− 1 ) CF 13 = −a(x) : −b(y) :: −b(x) : y− 1 (−a) CF 14 = −a(x− 1 ) : −b(y− 1 ) :: −b(x− 1 ) : y(−a) CF 15 = −a− 1 (x) : −b− 1 (y) :: −b− 1 (x) : y− 1 (−a− 1 ) CF 16 = −a− 1 (x− 1 ) : −b− 1 (y− 1 ) :: −b− 1 (x− 1 ) : y(−a− 1 ) Octet G (term +, function -) CF 17 = −x(a) : −y(b) :: −x(b) : a− 1 (−y) CF 18 = −x− 1 (a) : −y− 1 (b) :: −x− 1 (b) : a− 1 (−y− 1 ) CF 19 = −x(a− 1 ) : −y(b− 1 ) :: −x(b− 1 ) : a(−y) CF 20 = −x− 1 (a− 1 ) : −y− 1 (b− 1 ) :: −x− 1 (b− 1 ) : a(−y− 1 ) CF 21 = a(−x) : b(−y) :: b(−x) : −y− 1 (a) CF 22 = a(−x− 1 ) : b(−y− 1 ) :: b(−x− 1 ) : −y(a) CF 23 = a− 1 (−x) : b− 1 (−y) :: b− 1 (−x) : −y− 1 (a− 1 ) CF 24 = a− 1 (−x− 1 ) : b− 1 (−y− 1 ) :: b− 1 (−x− 1 ) : −y(a− 1 ) Octet H (term -, function -) CF 25 = −x(−a) : −y(−b) :: −x(−b) : −a− 1 (−y) CF 26 = −x− 1 (−a) : −y− 1 (−b) :: −x− 1 (−b) : −a− 1 (−y− 1 ) CF 27 = −x(−a− 1 ) : −y(−b− 1 ) :: −x(−b− 1 ) : −a(−y) CF 28 = −x− 1 (−a− 1 ) : −y− 1 (−b− 1 ) :: −x− 1 (−b− 1 ) : −a(−y− 1 ) CF 29 = −a(−x) : −b(−y) :: −b(−x) : −y− 1 (−a) CF 30 = −a(−x− 1 ) : −b(−y− 1 ) :: −b(−x− 1 ) : −y(−a) CF 31 = −a− 1 (−x) : −b− 1 (−y) :: −b− 1 (−x) : −y− 1 (−a− 1 ) CF 32 = −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. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Insight 2: For the fertility myth, a linguistic rendering of the 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]” ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction 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 Conclusions Motivation Mythology The Canonical Formula Quantum Interaction Conclusions The quaternion group of order eight Q = {±1, ±i, ±j, ±k} The noncommutative product operation defined as ij = 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. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions The Pauli matrices σx = 0 1 , 1 0 σy = 0 −i , i 0 σz = 1 0 . 0 −1 With the identity matrix I, they form a basis for the real Hilbert space of 2 × 2 complex Hermitian matrices. The real linear span of {I, iσx , iσy , iσz } is isomorphic to the real 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 ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction 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 the Bloch sphere, whereas a mixed state is inside the Bloch sphere. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Conclusions Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Applying the probabilistic description “Belief contamination” Different concepts belonging to the same category (e.g. the dying deity) can appear in the same plot Insecurity of not knowing what factor may be important and how 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 )? ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Case study Linguistic rendering of the CF exemplified on the myths of Adonis and Attis (minor deities from Asia Minor in the Hellenistic period, 323-31 BC) Typical plot: a youth invites harm by being too beautiful, losing his virility from which a specific plant springs up Attis’ story (13 variants) relates the loss to direct self-mutilation (8); to mutual castration with partner (indirect self-mutilation, 1); to being born as an eunuch or killed by spear through an unspecified wound (indirect not-self mutilation, 2); or the goddess mutilating him as punishment for 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 of four paradigms as per the CF The axes Fy (b) = xˆ , Fx (b) = zˆ , and Fa−1 (y ) = yˆ , where the latter can have four outcomes as above, the result is a mixed state vector weighted by the outcome probabilities. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle Motivation Mythology The Canonical Formula Quantum Interaction 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 Conclusions Motivation Mythology The Canonical Formula Quantum Interaction Conclusions Summary Bridging the gap Analytical studies in need of processing methodology Processing methodology development in need of raw material A concrete example how a topical set of myth variants correspond via their syntagmatic transcripts to narrative formulae Families of narrative formulae, some with double inverted values in their arguments, some without, all share the same group structure with a certain quaternion group of order eight. A quantum probabilistic framework further generalises the formulae. ´ ´ Daranyi Sandor, Peter Wittek, and Kirsty Kitto The Sphynx’s new riddle

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