Dynamic Cognitive Modeling IV

Transcription

1 Dynamic Cognitive Modeling IV CLS Computational Linguistics Summer Events University of Zadar Department of German Language and Linguistics Humboldt Universität zu Berlin

2 Overview 1. Introduction to linear algebra and calculus 2. Dynamical systems and neural networks 3. Dynamic automata: dynamic recognizers, fractal automata, nonlinear dynamical automata, and quantum automata 4. Language processing with neural networks: context-free and minimalist grammars 5. Dynamic field theory: functional representations, logics, and brain dynamics Literature: beim Graben, P. & Potthast, R. (2009). Inverse problems in dynamic cognitive modeling. Chaos: An Interdisciplinary Journal of Nonlinear Science, 19, ; and references therein:

3 Dynamic Cognitive Modeling

4 Dynamic Cognitive Modeling

5 Recursive Filler/Role Binding

6 Filler/Role Binding labeled trees left subtree: right subtree: complex fillers whole tree:

7 Dynamic Cognitive Modeling

8 Tensor Product Representations Let be a filler/role binding for some symbolic data structures with (simple) fillers and roles. A mapping to a vector space is a tensor product representation if 1. is a subspace of. 2. is a subspace of. 3.

9 Tensor Product Representations Let be a filler/role binding for some symbolic data structures with (simple) fillers and roles and a tensor product representation for the filler/role binding. The concatenation product is then the tensor product representation for the symbolic structures.

10 Fock Space As a consequence, is the Fock space used in quantum field theory

11 Computations and Processes Let be a set of symbolic data structures. Symbolic computations are partial functions. Let be a data structure. If the computations process can be concatenated to a Then, where P denotes the set of processes, is a semigroup.

12 Filler/Role Unbinding Let be a filler/role binding for some symbolic data structures with (simple) fillers and roles and a tensor product representation for the filler/role binding. A mapping for some is called unbinding if

13 Filler/Role Unbinding Unbinding can be achieved, e.g. by means of linear forms. Let, and be the dual space of the role subspace. Then, the linear form implements an unbinding via where denotes the identity map at filler subspace.

14 Realizing Computations Let be symbolic computations on. Two linear maps are called realizations of the computations in Fock space, if there is a tensor product representation such that mediates between semigroup homomorphisms

15 Example: String Processing strings: symbol alphabet: fillers: roles:

16 Example: String Processing tensor product representation fillers: roles:

17 Example: String Processing computations like in symbolic dynamics

18 Example: String Processing realization: Proof:

19 labeled trees Example: Tree Processing symbol alphabet: fillers: roles:

20 Example: Tree Processing tensor product representation fillers: roles:

21 computations Example: Tree Processing

22 Example: Tree Processing Passivization á la Smolensky (2006): passive sentence logical form

23 realization: Example: Tree Processing

24 Continuous Time Symbolic processes take place in discrete time: Brain dynamics takes place in continuous time! Language-related brain potentials for Die Rednerin hat der Berater gesucht the speaker has sought the advisor...

25 Order Parameter continuoustime dynamics in Fock space amplitude of k-th representation, between 0 and 1 tensor product representation of symbolic process in p time steps

26 Amplitude Dynamics initial state decayed state

27 Amplitude Dynamics excitation

28 Amplitude Dynamics

29 Fock Space Dynamics

30 Fock Space Dynamics repulsion saddle point attraction

31 Stable Heteroclinic Sequences SHS

32 Particular Representations Fillers and roles can be chosen from different vector spaces: number fields: Gödel representations arithmetic spaces: finite-dimensional dynamical systems combinations: fractal representations function spaces: dynamic fields

33 Dotted Sequences Turing machine state description: 1 a dotted sequences :

34 Symbolic Dynamics phase space dynamics symbolic dynamics

35 Cylinder Sets s k s 1 s 1 s k s a i1 a i2... a 2 in s 2 s n s n

36 Generalized Shifts current time domain of dependence: equivalent to Turing machine

37 Dotted Sequences obviously: filler roles however:

38 Dotted Sequences Decompose into left- and right substrings: tensor product representation:

39 Gödel Representations filler: Gödel numbers roles

40 Symbologram

41 Domains of Dependence cylinder set: domain of dependence: domain of dependence

42 Nonlinear Dynamical Automata The symbologram representation of a generalized shift is called nonlinear dynamical automaton.

43 Example: Parsing Process sentence: the dog chased the cat. context-free grammar (1) Gödel code (2)

44 Algorithm: Top-Down Recognizer time stack input operation 1 S NP V NP predict by rule (1) 2 NP VP NP V NP attach 3 VP V NP predict by rule (2) 4 V NP V NP attach 5 NP NP attach 6 ε ε accept

45 Domains of Dependence State descriptions provide a partition of the unit square, the domains of dependence. predict : if there is a rule : attach : if : do not accept : if : analogous to the Bernoulli map. piecewise affine-linear maps:

46 Symbologram domains of dependence images

47 Microstate Dynamics phase space initial condition

48 Microstate Dynamics

49 Ensemble Dynamics cloud of initial conditions ( ensemble )

50 Ensemble Dynamics

51 Contextual Partition

52 Dynamical Parsing Preparation: randomly distributed initial conditions. Predict: squeeze and shift horizontally. Attach: expand to unit square. Accept: unit square