Dynamical Cognition 2010: New Approaches to Some Tough Old Problems. Simon D. Levy Washington & Lee University Lexington, Virginia, USA

Transcription

1 Dynamical Cognition 2010: New Approaches to Some Tough Old Problems Simon D. Levy Washington & Lee University Lexington, Virginia, USA

2 Inspiration,

3 Inspiration, 1995-present [I]t turns out that we don t think the way we think we think!... The scientific evidence coming in all around us is clear: Symbolic conscious reasoning, which is extracted through protocol analysis from serial verbal introspection, is a myth. J. Pollack (2005) [W]hat kinds of things suggested by the architecture of the brain, if we modeled them mathematically, could give some properties that we associate with mind? P. Kanerva (2009)... a fresh coat of paint on old rotting theories. B. MacLennan (1991)

4 What is Mind?

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8 The Need for New Representational Principles Ecological affordances (Gibson 1979); exploiting the environment (Clark 1998) Distributed/Connectionist Representations (PDP 1986) Holographic Representations (Gabor 1971; Plate 2003) Fractals / Attractors / Dynamical Systems (Tabor 2000; Levy & Pollack 2001)

9 The Need for New Representational Principles Ecological affordances (Gibson 1979); exploiting the environment (Clark 1998) Distributed/Connectionist Representations (PDP 1986) Holographic Representations (Gabor 1971; Plate 2003) Fractals / Attractors / Dynamical Systems (Tabor 2000; Levy & Pollack 2001)

10 Pitfalls to Avoid 1. The Short Circuit (Localist Connectionist) Approach i) Traditional models of phenomenon X (language) use entities A, B, C,... (Noun Phrase, Phoneme,...) ii) We wish to model X in a more biologically realistic way. iii) Therefore our model of X will have a neuron (pool) for A, one for B, one for C, etc.

11 a.k.a. The Reese s Peanut Butter Cup Model

12 E.g. Neural Blackboard Model (van der Velde & de Kamps 2006)

13 Benefits of Localism (Page 2000) Transparent (one node, one concept) Supports lateral inhibition / winner-takes all

14 Lateral Inhibition (WTA) L 2 A B C L 1

15 Problems with Localism Philosophical problem: fresh coat of paint on old rotting theories (MacLennan 1991): what new insights does neuro-x provide? Engineering problem: need to recruit new hardware for each new concept/ combination leads to combinatorial explosion (Stewart & Eliasmith 2008)

16 The Appeal of Distributed Representations (Rumelhart & McClelland 1986)

17 WALKED WALK

18 ROARED ROAR

19 SPOKE SPEAK

20 WENT GO

21 Mary won t give John the time of day. ignores(mary, john)

22 Challenges (Jackendoff 2002)

23 I. The Binding Problem +????

24 II. The Problem of Two +???

25 III. The Problem of Variables ignores(x, Y) X won t give Y the time of day.

26 Vector Symbolic Architectures (Plate 1991; Kanerva 1994; Gayler 1998)

27 Tensor Product Binding (Smolensky 1990)

28 Binding

29 Bundling + =

30 Unbinding (query)

31 Lossy

32 Lossy

33 Cleanup Hebbian / Hopfield / Attractor Net

34 Reduction (Holographic; Plate 2003)

35 Reduction (Binary; Kanerva 1994, Gayler 1998)

36 Composition / Recursion

37 Variables john X

38 Scaling Up With many (> 10K) dimensions, get Astronomically large # of mutually orthogonal vectors (symbols) Surprising robustness to noise

39 Pitfalls to Avoid 2. The Homunculus problem, a.k.a. Ghost in the Machine (Ryle 1949) In cognitive modeling, the homunculus is the researcher: supervises learning, hand-builds representations, etc.

40 Banishing the Homunculus

41 Step I: Automatic Variable Substitution If A is a vector over {+1,-1}, then A*A = vector of 1 s (multiplicative identity) Supports substitution of anything for anything: everything (names, individuals, structures, propositions) can be a variable!

42 What is the Dollar of Mexico? (Kanerva 2009) Let X = <country>, Y = <currency>, A = <USA>, B = <Mexico> Then A = X*U + Y*D, D*A*B = D*(X*U + Y*D) * (X*M + Y*P) = B = X*M + Y*P (D*X*U + D*Y*D) * (X*M + Y*P) = (D*X*U + Y) * (X*M + Y*P) = D*X*U*X*M + D*X*U*Y*P + Y*X*M + Y*Y*P = P + noise

43 Learning Grammatical Constructions from a Single Example (Levy 2010) Given Meaning: KISS(MARY, JOHN) Form: Mary kissed John Lexicon: KISS/kiss, MARY/Mary,... What is the form for HIT(BILL, FRED)?

44 Learning Grammatical Constructions from a Single Example (Levy 2010) (ACTION*KISS + AGENT*MARY + PATIENT*JOHN) * (P1*Mary + P2*kissed + P3*John) * (KISS*kissed + MAY*Mary + JOHN*John + BILL*Bill + FRED*Fred + HIT*hit) * (ACTION*HIT + AGENT*BILL + PATIENT*FRED) =... = (P1*Bill + P2*hit + P3*Fred) + noise

45 Step II: Distributed Lateral Inhibition Analogical mapping as holistic graph isomorphsm (Gayler & Levy 2009) A P B Q C D R S cf. Pelillo (1999)

46 A P B Q C D R S Possibilities x: A*P + A*Q + A*R + A*S D*S Evidence w: A*B*P*Q + A*B*P*R B*C*Q*R C*D*R*S X*W = A*Q + B*R A*P D*S What kind of program could work with these data structures to yield a single consistent mapping?

47 Replicator Equations Starting at some initial state (typically just xi = 1/N corresponding to all x i being equally supported as part of the solution), x can be obtained through iterative application of the following equation: where and w is a linear function of the adjacency matrix of the association graph ( evidence matrix ).

48 Replicator Equations Origins in Evolutionary Game Theory (Maynard Smith 1982) xi is a strategy (belief in a strategy) πi is the overall payoff from that strategy wij is the utility of playing strategy i against strategy j Can be interpreted as a continuous inference equation whose discrete-time version has a formal similarity to Bayesian inference (Harper 2009)

49 Localist Implementation Results (Pelillo 1999)

50 VSA Lateral Inhibition Circuit (Levy & Gayler 2009) xt w * cleanup / πt xt+1 c c

51 VSA Implementation Results tinyurl.com/gidemo

52 Conclusions Vector Symbolic Architectures: A new kind of distributed representation for cognitive computing robust to noise rapid (one-shot) learning everything is a variable solves complicated problems in parallel Replicator equations: Dynamical system from evolutionary game theory, adapted to solve graph problems (analogies); can be made more plausible by using VSA instead of localist representation

53 Current / Future Work Subgraph mapping D E B C A P R Q S Using Map- Seeking Circuits (Arathorn 2002) to isolate sub-parts