Multiplicative Binding, Representation Operators & Analogical Inference Ross Gayler & Roger Wales Psychology Department, University of Melbourne [email protected] [email protected] 1-Feb-2000

5th Australasian Cognitive Science Conference

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Overview • • • •

Aim: connectionist implementation of analogy Construed task as structural pattern completion Strategy: extend recurrent autoassociator (BSB) Developed distributed representational system – Represent complex structures in fixed dimension – Naturally supports primitive operations of analogy

• Proposal for extension of BSB memory architecture based on representational system 1-Feb-2000

5th Australasian Cognitive Science Conference

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Analogy as Mapping & Substitution • Mapping = identify best correspondences

• map( (x (y z z) x) , (a (b c c) a (a b c) ) )

 {xa, yb, zc}

• Substitution = apply mappings • sub( {xa, yb, zc} , (a (b c c) a (a b c)) )

 (x (y z z) x (x y z))

• analogy( p , q) = sub( map( p , q) , q ) 1-Feb-2000

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Analogy as Retrieval • BSB autoassociators for pattern completion

– Currently: completion of literal, flat patterns cue = (x y z z x ? ? ?) trace = (x y z z x x y z) + ...  (x y z z x x y z) – Objective: completion of structural patterns cue = (x (y z z) x) trace = (a (b c c) a (a b c)) + ...  (x (y z z) x (x y z))

• Analogy as a failure mode of literal retrieval • Analogy is conjectured to be memory retrieval with mapping and substitution 1-Feb-2000

5th Australasian Cognitive Science Conference

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Convergent Substitution Notional Architecture (a (b c c) a)

AutoAssociator (literal pattern completion from AA trace)

(a (b c c) a (a b c))

(x (y z z) x)

sub( )

Cue {xa, yb, zc}

Generate mappings

? Retrieved

sub( )

(x (y z z) x (x y z))

Lines indicate flows of connectionist distributed representations of structures 1-Feb-2000

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Representation of Structures • “MAP” binding of high-dimensional vectors – Multiply  associate or bind – Add  superpose or insert in multiset – Permute  quote

• Desirable properties for analogy – – – – –

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MAP

Represent structures of arbitrary complexity  Fixed dimension vectors (implies distributed)  One-step construction and manipulation  Able to represent mappings  Implicit interaction between mappings  5th Australasian Cognitive Science Conference

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Representations & Operations • Any structure can be represented as a sum of products of permutations of vectors • Represent frame as sum of products (naïve) nameTom + speciescat + colourblue

• Nested frames give higher-order bindings • Design of these structures is tricky – Commutative operations lose ordering – Vectors are self-inverse for binding – Permutation helps with both problems

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Literal Retrieval & Clean-up • Retrieve by binding cue (a) with trace (ab) retrieve( a , ab ) = (a )  (ab) = aa-1b = b

• Retrieval generates noise terms

(a’)  (ab) = (a + )  (ab) = b + ab (a)  (ab + x + …) = b + ax + ... = b + 

• Noise terms are suppressed by a clean-up memory populated with known vectors • Clean-up = Literal pattern completion 1-Feb-2000

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Substitution by Retrieval • Represent mappings by bindings { ax , by , cz } = ax + by + cz

• Retrieval with a mapping cue is substitution

sub( { bluered } , colourblue )  colourred retrieve( bluered , colourblue ) = (bluered)  (colourblue) = colourred

• Parallel substitution in one operation

(ab)  (ax + ay + ...) = bx + by + … (ab + cd )  (ax + cy + ...) = bx + dy + …

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Retrieval Generates Mappings • Failed retrieval gives only noise (a)  (b + c + ...) = ab + ac + ...

• Noise terms are representations of mappings ab = { ab }

• Failed retrieval generates all mappings between the cue and every trace term (a)  (b + c + ...) = ab + ac + ...

• Binding gives retrieval, substitution, or mappings/noise (depends on cue and trace) 1-Feb-2000

5th Australasian Cognitive Science Conference

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Conjectured Analogical Retrieval 1 Try to retrieve from trace using literal cue Noise contains all possible cue mappings

2 Filter the mappings to amplify the best 3 Apply best mappings to cue as substitution 4 Use substituted cue to retrieve from AA trace 5 AA trace provides literal pattern completion 6 Reverse the substitution from the results 7 Iterate to convergence (over map & retrieved) 1-Feb-2000

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Convergent Substitution Proposed Architecture transformed

cue

mapping literal pattern completion

literal component clean-up

retrieved literal component clean-up

mapping transformed 1-Feb-2000

retrieved 5th Australasian Cognitive Science Conference

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BSB with Substitution transformed

cue

mapping literal pattern completion

literal component clean-up

retrieved literal component clean-up

mapping transformed 1-Feb-2000

retrieved 5th Australasian Cognitive Science Conference

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Estimation of Mapping transformed

literal pattern completion

cue

mapping literal component clean-up

retrieved literal component clean-up

mapping transformed 1-Feb-2000

retrieved 5th Australasian Cognitive Science Conference

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Two Types of Clean-up transformed

cue

mapping literal pattern completion

literal component clean-up

retrieved literal component clean-up

mapping transformed 1-Feb-2000

retrieved 5th Australasian Cognitive Science Conference

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Clean-up by Decomposition • Literal pattern completion = amplify known structures and suppress all others • Analogy generates novel literal structures • How to clean-up novel literal structures? • Take advantage of compositional structure • Clean-up as decomposition into known components that recombine to the original 1-Feb-2000

5th Australasian Cognitive Science Conference

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Literal Component Clean-up a

literal pattern completion

a

ab

a'b

b

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literal pattern completion

b

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Summary

• Conjectured analogy to be an extension of BSB memory with mapping and substitution • Developed a distributed representational system with properties useful to analogy – – – – –

Represent structures of arbitrary complexity Fixed dimension vectors One-step construction and manipulation Able to represent mappings Substitution is the same process as retrieval

• Proposed analogical memory architecture 1-Feb-2000

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