A Neural-Symbolic Approach to the Contemporary Theory of Metaphor

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
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
The Classical Theory of Metaphor
Dates back to Aristotle
Do not go gentle into that good night.
Dylan Thomas
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
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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]
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
The Contemporary Theory of Metaphor
We’ve hit a dead-end street
We can’t turn back now
Love IS A journey
We‘re driving in the fast lane on the freeway of love
Relationship AS vehicle
Lovers AS passengers
Alternatives AS crossroads
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Source and target domain
T S
x
y
f T ?
f T (x)?
a
b
f S
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Mapping over domains
T S
x
y
m
n
a
b
f S
T S
n(f S (m(x))=y f T (x) = y
x
y
f T
a
b
f S
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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 x1, .., xn of
A, m(fA(x1, .., xn)) = fB(m(x1), .., m(xn))
where fA and fB represent f over A and B respectively.
In our setting, we can’t compute fA, and we wonder what could
fA(x1, .., xn) be. Since m is injective, it can be inverted. Let n be
the inverse function of m. The following transformations hold:
fA(x1, .., xn) ≡ 1 n(m(fA(x1, .., xn))) ≡ 2 n(fB(m(x1), .., m(xn)))
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.
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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).
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Architecture
?
T S
x
y
m
n
n(f S (m(x))=y
a
b
f S
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Source domain
Find other
Reverse Put gas
path
R1
Dead-end
road
R2
Wrong
turn
R3 R4
Low on
gas
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Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Mapping
Dead-end
road
No
promotions
Reverse
Apply
for a job
Wrong
turn
Resign
Find other
path
Volunteer
for
overtime
Put
gas
Low
on gas
Low
salary
11 / 19

Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Target domain
Resign
No
promotions
Apply for
a job
?
Low
salary
Volunteer
for overtime
12 / 19

Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Step one: mapping
Dead-end
road
No
promotions
Reverse
Apply
for a job
Wrong
turn
Resign
Find other
path
Volunteer
for
overtime
Put
gas
Low
on gas
Low
salary
13 / 19

Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Step two: computing
Find other
Reverse Put gas
path
R1
Dead-end
road
R2
Wrong
turn
R3 R4
Low on
gas
14 / 19

Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Step three: mapping back
Dead-end
road
No
promotions
Reverse
Apply
for a job
Wrong
turn
Resign
Find other
path
Volunteer
for
overtime
Put
gas
Low
on gas
Low
salary
15 / 19

Contemporary Theory of Metaphor Sets, Functions and Networks Example Applications and conclusion
Step four: learning
Resign
R1
No
promotions
Apply for
a job
!
Low
salary
Volunteer
for overtime
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
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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.
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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