Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Philosophy and Machine Learning - Workshop (@ Neural Information Processing Systems 2011) A Neural-Symbolic Approach to the Contemporary Theory of Metaphor Guido Boella - University of Turin, Italy Artur d’Avila Garcez - City University, London Alan Perotti - University of Turin, Italy 1 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion The Classical Theory of Metaphor Do not go gentle into that good night. Dylan Thomas Dates back to Aristotle Metaphors: Instances of novel poetic language in which words (like go and night ) are not used in their normal everyday sense. Defines metaphor as a matter of language Describes metaphorical expression as mutually exclusive with the realm of ordinary language 2 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion The Contemporary Theory of Metaphor ”The generalizations governing poetic metaphorical expressions are not in language, but in thought: they are general mappings across conceptual domains. Moreover, these general principles which take the form of conceptual mappings, apply not just to novel poetic expressions, but to much of ordinary everyday language. In short, the locus of metaphor is not in language at all, but in the way we conceptualize one mental domain in terms of another. The general theory of metaphor is given by characterizing such cross-domain mappings. And in the process, everyday abstract concepts like time, states, change, causation, and purpose also turn out to be metaphorical.” [G. Lakoff, The Contemporary Theory of Metaphor] 3 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion The Contemporary Theory of Metaphor Love IS A journey We’ve hit a dead-end street We can’t turn back now We‘re driving in the fast lane on the freeway of love Relationship AS vehicle Lovers AS passengers Alternatives AS crossroads 4 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Source and target domain T S x a fT? fS y b fT(x)? 5 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Mapping over domains T S T S m x a x a fT fS fS y b y b n n(fS(m(x))=y fT(x) = y 6 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion The Mapping Given two algebraic structures A and B , a function m is a monomorphism iff: m is injective ∀ n-ary operation f over the structures, ∀ n-tuple x 1 , .., x n of A , m ( f A ( x 1 , .., x n )) = f B ( m ( x 1 ) , .., m ( x n )) where f A and f B represent f over A and B respectively. In our setting, we can’t compute f A , and we wonder what could f A ( x 1 , .., x n ) be. Since m is injective, it can be inverted. Let n be the inverse function of m . The following transformations hold: f A ( x 1 , .., x n ) ≡ 1 n ( m ( f A ( x 1 , .., x n ))) ≡ 2 n ( f B ( m ( x 1 ) , .., m ( x n ))) Where ( ≡ 1 ) is justified because m and n are inverse functions (and therefore n ( m ( x )) ≡ x ) and ( ≡ 2 ) follows from the definition of monomorphism. 7 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Knowledge representation and Learning The Neural-Symbolic paradigm We model the source and target domains as connectionist inductive learning and logic programming (CILP) system The CILP system ([1]) is a neural-symbolic system showing a one-to-one correspondence between logic programming and neural networks that are trainable by backpropagation. We model the mapping functions m and n as a single restricted Boltzmann machine (RBM). A RBM defines a probability distribution P(V=v,H=h) over pairs of vectors v and h encoded in these layers, where v encodes the input data in binary or real values and h encodes the posterior probability P(H | v). 8 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Architecture T S m x a ? fS y b n n(fS(m(x))=y 9 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Source domain Find other Reverse Put gas path R1 R2 R3 R4 Dead-end Low on Wrong road gas turn 10 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Mapping Dead-end Wrong Find other Put Low Reverse road turn path gas on gas Volunteer No Apply Low Resign for promotions for a job salary overtime 11 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Target domain Apply for Volunteer Resign a job for overtime ? No Low promotions salary 12 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Step one: mapping Dead-end Wrong Find other Put Low Reverse road turn path gas on gas Volunteer No Apply Low Resign for promotions for a job salary overtime 13 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Step two: computing Find other Reverse Put gas path R1 R2 R3 R4 Dead-end Low on Wrong road gas turn 14 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Step three: mapping back Dead-end Wrong Find other Put Low Reverse road path turn gas on gas Volunteer No Apply Low Resign for promotions for a job salary overtime 15 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Step four: learning Apply for Volunteer Resign a job for overtime ! R1 No Low promotions salary 16 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Applications Knowledge (and reasoning patterns) recycling Software reuse and encapsulation Blackbox use via interfaces Commitment-based multiagent interaction 17 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Conclusions In this work, we model the cognitive theory of metaphor, as defined by Lakoff, as a monomorphism. With this approach we are able to prove that local computation can be performed over a more familiar domain. We propose a framework that relies on the CILP system and RBMs and allows to perform learning and reasoning over unknown domains. 18 / 19
Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion Thank you. References: 1 A. d’Avila Garcez, K. B. Broda, and D. M. Gabbay. Neural-Symbolic Learning Systems. Per- spectives in Neural Computing. Springer, 2002. 2 L. de Penning, A. S. d’Avila Garcez, L. C. Lamb, and J.-J. C. Meyer. A neural-symbolic cogni- tive agent for online learning and reasoning. In IJCAI, pages 1653–1658, 2011. 3 G. E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Com- put., 14:1771–1800, August 2002. 4 G. Lakoff. The Neural Theory of Metaphor and Thought, page 17–39. Cambridge University Press, Cambridge, 2008. 5 G. Lakoff and M. Johnson. Metaphors we Live by. University of Chicago Press, Chicago, 1980. 6 P. Smolensky. Information processing in dynamical systems: foundations of harmony theory, pages 194–281. MIT Press, Cambridge, MA, USA, 1986. 19 / 19
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