Transcript Computational Logic and Cognitive Science – An Overview

Computational Logic
and Cognitive Science:
An Overview
Session 2: Cognitive Challenges
ICCL Summer School 2008
Technical University of Dresden
26th of August, 2008
Helmar Gust & Kai-Uwe Kühnberger
University of Osnabrück
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Overview
 A Bunch of Cognitive Findings / Cognitive
Challenges
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Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
Wason Selection Task
Remarks on Natural Language
Sizes of Cities
Theories of Mind
Creativity
Neuro-Symbolic Integration
Causality
Types of Reasoning
Cognitive Architectures
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Wason Selection Task
 The Wason selection task
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4 cards are given: On one side there is a number and on the
other a letter printed.
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Rule: If there is a vowel at one side, there will be an even
number at the other side.
 The following situation is given:
A
D
4
7
 The task is: Turn as few cards as possible to prove the
rule.
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The correct answer is to turn A and 7.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Wason Selection Task
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The experiment was executed in various versions.
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One showed the following results:
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A and 4:
A:
A and 7:
Others:
46 %
33 %
3%
18 %
Modus Tollens:
 If p, then q. And: not q. Therefore: not p.
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It seems to be the case that humans do not think logically…
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Wason Selection Task
 New rule: Only people over 18 are allowed to drink
alcohol.
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Meaning: If for someone it is allowed to drink alcohol he/she
must be over 18.
 The new situation:
15
Water
Beer
22
 The solution is to turn Beer and 17.
 This version of the Wason selection task seems to be
much easier to solve for humans.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Wason Selection Task
 Some proposals for an explanation of these results:
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Humans do not think logical at all (Gigerenzer).
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Humans think in models not in terms of logical deductions
(Johnson-Laird).
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Humans need to embed their reasoning in concrete
situations. They have problems in reasoning in idealized
situations, i.e. mental models do not reduce the problem
to the idealized (abstracted) situation.
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Humans can solve such problems, if it is placed in a
social context (evolutionary psychology).
 Many theories were proposed to model these data.
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There are logic-based solutions as well as model-based
solutions.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Wason Selection Task
 Another important point to mention is the way to
describe the task in natural language.
 As a matter of fact, many logical connectives in natural
language require a “more complex” interpretation than in
classical logic.
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“Peter is in the living room or in the kitchen.”
“Paul went to the university and gave a speech.” vs.
“Paul gave a speech and went to the university.”
“If Jim works hard for the exam he will pass it.”
 The standard version of the Wason selection task makes
it plausible that a certain number of subjects interpret the
implication as an equivalence.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Natural Language
 Natural language shows many features that cannot be easily modeled with
classical logical approaches. Here are some examples:
 “Many students read different books.”
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“Could you tell me what time is it?”
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Counterfactuals
“The king of France is bald.”
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Temporal aspects require an extension of classical logic.
“If I had been on holidays two weeks ago, I would not have a burnout
now.”
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Implicatures require a non-literal interpretation.
“Yesterday John told me that in 150 years Germany will have a
Mediterranean climate.”
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Generalized quantifiers require an extension of classical logic.
Presuppositions extend the context in a non-trivial way, although there is nothing stated
literally.
“I am here.”
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Indexicals
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
San Diego vs. San Antonio
 An experiment due to Goldstein & Gigerenzer (having to do with
knowledge and rationality in general):
 “Which city has more inhabitants: San Diego or San Antonio?”
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This question was asked American students and German students.
Clearly German students knew little of San Diego, and many had
never heard of San Antonio.
Results:
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62% of the American students answered correctly: San Diego.
100% of the German students answered correctly: San Diego.
Gigerenzer proposes to use heuristics and cues to answer such
questions resulting in a form of bounded rationality.
 In any case, there is a certain tension between bounded rationality
and classical logic and knowledge representation.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Theories of Mind
 Theories of Mind
 Wise men problem (a variation of the famous muddy children
problem).
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“Three wise men know there are three red hats and two blue
hats (and they know that all three know that). The king placed
a hat on each wise man, such that no wise man knows which
color his hat has. Then he asks each wise man in a row
which color his hat has.”
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Assume the first man says: “I don’t know.” and the second
man says “I don’t know.” Why is it possible that the third man
knows the color of his hat?
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Theories of Mind
 BBB is impossible (there are only
P1
P2
P3
two blue hats).
 P1 says: “I don’t know.”
R
R
R
R
R
B
R
B
R
R
B
B
B
R
R
B
R
B
B
B
R
B
B
B
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P2 and P3 infer that P1 sees a red
hat: RBB is impossible.
 P2 says: “I don’t know.”
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P3 infers that P2 sees a read hat:
BRB is impossible.
P3 infers: P2 knows that P1 sees a
red hat. In the remaining models
there is only one where P3 has a
blue hat: RRB. In this case P2
would know that she has a red hat.
 Therefore P3 answers that he has a
red hat.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Theories of Mind
 Reasoning about the knowledge of other agents in a multi-agent
systems seems to be natural to us.
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Maybe this is controversial. Nevertheless, if put into a reasonable
situation, probably we are quite good in solving such puzzles…
 The frameworks proposed for representing and solving such
puzzles are rather complicated.
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Modal logic / epistemic logic
Situation theory
Game theory
 In any case, classical logic needs to be extended in order to model
reasoning about the beliefs of other agents.
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It is quite plausible to assume that humans do not apply game
theory or perform deductions according to a modal logic calculus
in order to solve this problem. They probably solve such problems
differently.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Creativity: Examples
Jan van Eyck: The
Arnolfini Marriage
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Creativity
 Creativity
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It seems to be unquestionable that humans show creative
behavior.
In particular, in problem solving, but also in using language
productively (in particular, semantic productivity), in using
metaphoric expressions, in generating theories, interpreting
visual input, or making sense out of situations, humans show
a remarkable ability of creativity.
There are no really good theories that can describe this kind
of creativity. One candidate may be analogical reasoning.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Neuro-Symbolic Integration
 Symbolic-subsymbolic distinction
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There is an obvious tension between symbolic and subsymbolic
representations.
Symbolic Approaches
Subsymbolic Approaches
Methods
Mainly logical and / or algebraic
Mainly analytic
Strengths
Productivity, recursion, compositionality
Robustness, learning, parsimony,
adaptivity
Weaknesses
Consistency constraints, lower cognitive
abilities
Opaqueness, higher cognitive
abilities
Applications
Reasoning, problem solving, planning etc.
Learning, motor control, vision etc.
Relation to Neurobiology
Not biologically inspired
Biologically inspired
Other Features
Crisp
Fuzzy
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Neuro-Symbolic Integration
 Some interesting facts about the symbolic-subsymbolic distinction
and cognitive science
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Classically natural language is considered to be a domain for
symbolic theories.
Chomsky’s claim was that natural language cannot be learned
without assuming a universal grammar.
 His classical example was auxiliary inversion.
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Nevertheless important insights were provided by Elman who
showed how rather simple recurrent networks (Elman networks)
can learn correctly auxiliary inversion.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
Ecuador is in South America.
 Is Ecuador in South America?
That woman who is walking her dog is Tom’s neighbor.
 *Is that woman who walking her dog is Tom’s neighbor?
 Is that woman who is walking her dog Tom’s neighbor?
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Neuro-Symbolic Integration
 Some further remarks about the symbolic-subsymbolic
distinction and cognitive science
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A further interesting fact is that one of the currently most
influential theories in linguistics was developed by the
neuroscientist Paul Smolensky.
  Optimality theory.
Perhaps linguistics is a good testbed for neural modeling of
complex data structures.
 In total, the integration of symbolic theories (in particular
logic) into neural networks is an ongoing challenge.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Causality
 Causality seems to play an important role in human reasoning.
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Although the concept of causality is complicated and not very well
understood, humans tend to structure the dynamics of the world
by causes and effects.
 Reduction of causality to logical relations:
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Mackie: Causality can be explained by insufficient and nonredundant parts of unnecessary but sufficient causes (INUS
condition).
 Example
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Short circuit is the cause of the house burning down (plus side
conditions): together these events are unnecessary but sufficient
for the destruction; the short circuit is insufficient but nonredundant.
 Nevertheless there are many different proposals for a logical
reduction of causality, e.g. counterfactuals.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Reasoning Aspects
 Manifold of reasoning abilities:
Deductions, inductions, abductions, analogical reasoning,
associations, non-monotonic reasoning etc.
 An integration of these reasoning abilities is desirable.
 From a pure logical approach this does not seem to be a
straightforward task.
 Even worse reasoning abilities are highly context dependent:
 Humans have the ability to jump easily from one context to
another context, finding re-interpretations of a given input, and
applying different types of reasoning types.
 Classical logical theories have their problems in modeling
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such situations.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Context Dependencies
Suppose you are in a forest and
you want to heat some water. You
do not have a container of any
kind. You can cut a vessel of wood,
but it would burn in the fire. How
can you heat the water in this
wooden vessel?
Kokinov & Petrov (2001)
“I am here.”
Davies & Goel (2001)
“Oh, it’s raining.”
“Every student answered
every question.”
Indurkhya (1992)
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Non-Monotonicity
Axioms
Axioms
Birds can usually fly.
Penguins are birds.
Tweety is a Penguin.
Birds can usually fly.
Penguins are birds.
Tweety is a Penguin.
Penguins can’t fly.
Theorem
Theorem
Tweety can fly.
Tweety cannot fly.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Non-Monotonicity
Theorems
monotonic
extension
Axioms
+p
Axioms
non-monotonic
extension
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
new theorems
because of + p
Theorems
without p
Axioms
+p
Theorems
without p
Theorems
incl. p
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Analogical Reasoning
“Electrons are the planets of the
atom.”
“Current is the water in an electric
circuit.”
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Analogical Reasoning
 Some statements about analogical reasoning right at the
beginning:
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Analogy making is in general not case-based reasoning.
Most interesting cases of analogies are cross-domain
analogies.
Analogical reasoning can be modeled with logical means.
Analogical reasoning requires but cannot be reduced to
deductions, inductions, and abductions.
Analogical reasoning is the core of human creativity.
 A logical framework modeling analogical reasoning
requires some non-standard techniques.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Cognitive Architectures
 The attempt to model cognitive behavior currently results in an
inflationary number of different cognitive architectures.
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Examples are: ACT-R (Anderson), SOAR (Laird), AMBR (Kokinov),
Clarion (Sun), NARS (Wang), Icarus (Langely), PSI (Dörner, Bach)
etc.
 Some features of several (not of all) of these architectures:
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Integration of different reasoning types.
“Non-rational” behaviors (associations, emotions etc.).
Hybrid (neuro-symbolic) representations.
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Remark: not in the sense of neuro-symbolic integration, but
more in the sense of “semantic networks + activation
potentials”.
Integration of various cognitive abilities.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
What Do We Have so Far?
 Wason selection task
 Remarks on Natural Language
 San Diego vs. San Antonio
 Theories of mind
 Creativity
 Symbolic-subsymbolic distinction
 Causality
 Reasoning
 Context, non-monotonicity, analogy
 Cognitive architectures
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Conclusion
 The mentioned cognitive capacities (or
deficiencies) are relatively hard to model with
standard logic techniques.
 The aim is to build intelligent systems that can
come up with solutions of such problems.
 This requires non-classical forms of reasoning,
extensions of classical logic into various
directions, and the integration of different
reasoning mechanisms.
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008
Thank you very much!!
Helmar Gust & Kai-Uwe Kühnberger
Universität Osnabrück
ICCL Summer School 2008
Technical University of Dresden, August 25th – August 29th, 2008