dynamical cognition 2010: new approach to some tough old problems
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Dynamical Cognition 2010: New Approach to Some Tough Old Problems. Simon D. Levy Washington & Lee University Lexington, Virginia, USA. Inspiration, 1985-1995. Inspiration, 1995-present. - PowerPoint PPT PresentationTRANSCRIPT
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Dynamical Cognition 2010: New Approach to Some Tough Old Problems
Simon D. LevyWashington & Lee University
Lexington, Virginia, USA
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Inspiration, 1985-1995
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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)
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What is Mind?
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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)
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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)
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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.
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QuickTime™ and a decompressor
are needed to see this picture.
a.k.a. The Reese’s Peanut Butter Cup Model
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E.g. Neural Blackboard Model (van der Velde & de
Kamps 2006)
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Benefits of Localism (Page 2000)
• Transparent (one node, one concept)
• Supports lateral inhibition / winner-takes all
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Lateral Inhibition (WTA)
A B C
L1
L2
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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)
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The Appeal of Distributed
Representations(Rumelhart &
McClelland 1986)
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WALK
WALKED
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ROAR
ROARED
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SPEAK
SPOKE
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GO
WENT
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ignores(mary, john)
Mary won’t give John the time of day.
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Challenges (Jackendoff 2002)
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I. The Binding Problem
+
? ? ? ?
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II. The Problem of Two
+
? ? ?
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III. The Problem of Variables
ignores(X, Y)
X won’t give Y the time of day.
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Vector Symbolic Architectures
(Plate 1991; Kanerva 1994; Gayler 1998)
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Tensor Product Binding
(Smolensky 1990)
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Binding
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Bundling
+ =
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Unbinding (query)
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Lossy
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Lossy
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Cleanup
Hebbian / Hopfield /
Attractor Net
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Reduction(Holographic;
Plate 2003)
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Reduction(Binary;
Kanerva 1994,Gayler 1998)
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Composition / Recursion
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Variables
X
john
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Scaling Up
•With many (> 10K) dimensions, get
• Astronomically large # of mutually orthogonal vectors (symbols)
• Surprising robustness to noise
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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.
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Banishing the Homunculus
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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!
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“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
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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) ?
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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
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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
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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?
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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”).
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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)
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Localist Implementation
Results (Pelillo 1999)
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∧
c∧
c
w * cleanup ∑
xt
xt+1πt
VSA “Lateral Inhibition” Circuit
(Levy & Gayler 2009)
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tinyurl.com/gidemo
VSA Implementation Results
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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
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Current / Future Work• Subgraph mapping
• Using Map-Seeking Circuits (Arathorn 2002) to isolate sub-parts
D B
CEA
P Q
SR