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Quantum Representation Theory for Nonlinear Dynamical Automata

Quantum Representation Theory for Nonlinear Dynamical Automata. 8th Szklarska Poreba Workshop. Overview. nonlinear dynamical automata representation theory quantum linguistics. controversy. Motivation. Fodor. Smolensky. among others…. beim Graben (2004).

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Quantum Representation Theory for Nonlinear Dynamical Automata

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  1. Quantum Representation Theory for Nonlinear Dynamical Automata 8th Szklarska Poreba Workshop

  2. Overview • nonlinear dynamical automata • representation theory • quantum linguistics

  3. controversy Motivation Fodor Smolensky among others… beim Graben (2004)

  4. A Toy-Model of Natural Language Processing the speakerAMB has the advisor sought

  5. Context-Free Grammar (1) CP die Rednerin + C’ (2) C’ hat + IP (3) IP t + I1’ (4) I1’  VP + I0 (5) VP den Berater + V’ (6) V’ t + gesucht

  6. Gödel Encoding of Variables die Rednerin = 0 den Berater = 1 der Berater = 2 hat = 3 gesucht = 4 I0 = 5 t = 6 CP = 7 C' = 8 IP = 9 I1' = A (10) I2' = B (11) VP = C (12) V' = D (13) terminals, b = 7 non-terminals, n = 14 “tetradecimal”

  7. arithmetic functions Gödel Encoding of Rules (1) 70 8 (2) 83 9 (3) 96 A (4) AC 5 (5) C1 D (6) D6 4

  8. 0 0 most significant digit Gödel Encoding of Strings well-formed input string Gödel( ) = 0. 0 3 6 1 6 4 57 0 3 6 1 6 4 5 = 07-1 + 37-2 + 67-3 + 17-4 + 67-5 + 47-6 + 57-7 = 0.079510

  9. Top-Down Recognizer time stack input operation 1. 70361645predict (1) 2. 080361645attach 3. 8361645predict(2) 4. 39361645attach 5. 961645predict(3) 6. 6A61645attach 7. A1645predict(4) 8. C51645 predict(5) 9. 1D51645attach 10. D5 645predict(6) 11. 645645attach 12. 4545attach 13. 55attach 14.   accept

  10. quantitative dynamics: piecewise affine-linear map (beim Graben et al. 2004) Nonlinear Dynamical Automaton time = 2: stack x input = [08] x [0361645] [0,1]2 symbolic dynamics: generalized shifts (Moore, 1990) attach time = 3: stack x input = [8] x [361645] [0,1]2

  11. (5B, 24) (4, 1) Phase Space

  12. Phase Space Dynamics • initialization: prepare set of initial conditions • evolve according the nonlinear map: • predict: squeeze and shift horizontally • attach: expand • accept:state 13 covers whole unit square

  13. Disadvantages • dynamics unfolds the syntactic structure of the whole string as encoded in the initial conditions deterministically: • unnatural with respect to continuous stream of speech • unnatural with respect to word-by-word presentation in psycholinguistic experiments • cannot explain surprising events such as garden-path effects

  14. restrict input tape to finite “working memory”, e.g. to the first two digits: • after each attachment, read next input symbol A into working memory: Solution • regard A as a perturbation in the NDAs phase space

  15. electron in state . Observing position mediates state transition . Observing momentum mediates transition . Observations don’t commute: . • non-commutative algebra: . • representation of algebra elements by state space maps: • representations are function-functions Quantum Representation Theory

  16. automaton’s microstate Gödel code msd WM base state transition Representation Theory for NDAs remainder

  17. iterate 1. 2. Word Semi-Group word semi-group homomorphism

  18. der Berater gesucht Quantum Linguistics

  19. gesucht der Berater Quantum Linguistics non-commutativity

  20. Input-Perturbed Phase Space Dynamics • initialization: prepare set of initial conditions • evolve according the nonlinear map: • predict: squeeze and shift horizontally • attach: expand • scan: squeeze and shift vertically • accept:state 18 covers whole unit square

  21. Summary • nonlinear dynamical automata bridge the cleft between symbolic processing and deterministic dynamics • encoding sentences to be processed by complete initial conditions is psycholinguistically unsatisfactory • modeling finite working memory as input tape • regarding uncertain inputs as perturbations upon the system’s state space • representations of the “phrase space” in the “phase space”

  22. Thank you! and Leticia Pablos Robles and Doug Saddy (UoR)

  23. After Eight

  24. 03 000... ... 03 666... Cylinder sets of languages die Rednerin hatblah blah blah…  cylinder set [03] = [03000...,03666...] = [0.0612, 0.0816] is interval in [0, 1]

  25. predict : if there is a rule : • attach : if : • do not accept : if : Domains of dependence State descriptions provide a partition of the unit square, the domains of dependence (DoD).

  26. Phase space DoD images

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