Dynamical Cognition 2010: New Approach to Some Tough Old Problems

Simon D. LevyWashington & Lee University

Lexington, Virginia, USA

Inspiration, 1985-1995

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 onold rotting theories.” − B. MacLennan (1991)

What is Mind?

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)

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)

Pitfalls to Avoid

1. The “Short Circuit” (Localist Connectionist) Approach

• Traditional models of phenomenon X (language) use entities A, B, C, ... (Noun Phrase, Phoneme, ...)

• We wish to model X in a more biologically realistic way.

• Therefore our model of X will have a neuron (pool) for A, one for B, one for C, etc.

QuickTime™ and a decompressor

are needed to see this picture.

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

E.g. Neural Blackboard Model (van der Velde & de

Kamps 2006)

Benefits of Localism (Page 2000)

• Transparent (one node, one concept)

• Supports lateral inhibition / winner-takes all

Lateral Inhibition (WTA)

A B C

L1

L2

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)

The Appeal of Distributed

Representations(Rumelhart &

McClelland 1986)

WALK

WALKED

ROAR

ROARED

SPEAK

SPOKE

GO

WENT

ignores(mary, john)

Mary won’t give John the time of day.

Challenges (Jackendoff 2002)

I. The Binding Problem

+

? ? ? ?

II. The Problem of Two

+

? ? ?

III. The Problem of Variables

ignores(X, Y)

X won’t give Y the time of day.

Vector Symbolic Architectures

(Plate 1991; Kanerva 1994; Gayler 1998)

Tensor Product Binding

(Smolensky 1990)

Binding

Bundling

+ =

Unbinding (query)

Lossy

Lossy

Cleanup

Hebbian / Hopfield /

Attractor Net

Reduction(Holographic;

Plate 2003)

Reduction(Binary;

Kanerva 1994,Gayler 1998)

Composition / Recursion

Variables

X

john

Scaling Up

•With many (> 10K) dimensions, get

• Astronomically large # of mutually orthogonal vectors (symbols)

• Surprising robustness to noise

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.

Banishing the Homunculus

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!

“What is the Dollar of Mexico?”

(Kanerva 2009)• Let X = <country>, Y = <currency>, A = <USA>, B = <Mexico>

• Then A = X*U + Y*D, B = X*M + Y*P

D*A*B =

D*(X*U + Y*D) * (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

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) ?

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

Step II: Distributed “Lateral Inhibition”• Analogical mapping as holistic

graph isomorphsm (Gayler & Levy 2009)

cf. Pelillo (1999)

A

B

C D

P

Q

R S

A

B

C D

P

Q

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?

Replicator EquationsStarting at some initial state (typically just xi = 1/N

corresponding to all xi 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”).

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)

Localist Implementation

Results (Pelillo 1999)

∧

c∧

c

w * cleanup ∑

xt

xt+1πt

VSA “Lateral Inhibition” Circuit

(Levy & Gayler 2009)

tinyurl.com/gidemo

VSA Implementation Results

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

Current / Future Work• Subgraph mapping

• Using Map-Seeking Circuits (Arathorn 2002) to isolate sub-parts

D B

CEA

P Q

SR