Computational Logic and Cognitive Science: An Overview Session 2: Cognitive Challenges ICCL Summer...

Computational Logic and Cognitive Science: An Overview

Session 2: Cognitive Challenges

ICCL Summer School 2008Technical University of Dresden26th 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 Wason Selection Task Remarks on Natural Language Sizes of Cities Theories of Mind Creativity Neuro-Symbolic Integration Causality Types of Reasoning Cognitive Architectures





Wason Selection Task

The Wason selection task 4 cards are given: On one side there is a number and on the

other a letter printed. 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. The correct answer is to turn A and 7.






The experiment was executed in various versions.

One showed the following results: A and 4: 46 % A: 33 % A and 7: 3 % Others: 18 %

Modus Tollens: If p, then q. And: not q. Therefore: not p.

It seems to be the case that humans do not think logically…






New rule: Only people over 18 are allowed to drink alcohol. 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.






Some proposals for an explanation of these results: Humans do not think logical at all (Gigerenzer). Humans think in models not in terms of logical deductions

(Johnson-Laird). 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.

Humans can solve such problems, if it is placed in a social context (evolutionary psychology).

Many theories were proposed to model these data. There are logic-based solutions as well as model-based

solutions.






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. “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.





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.”

Generalized quantifiers require an extension of classical logic. “Could you tell me what time is it?”

Implicatures require a non-literal interpretation. “Yesterday John told me that in 150 years Germany will have a

Mediterranean climate.” Temporal aspects require an extension of classical logic.

“If I had been on holidays two weeks ago, I would not have a burnout now.”

Counterfactuals “The king of France is bald.”

Presuppositions extend the context in a non-trivial way, although there is nothing stated literally.

“I am here.” Indexicals





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?”

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:

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.





Theories of Mind

Theories of Mind Wise men problem (a variation of the famous muddy children

problem).

“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.”

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?





Theories of Mind

BBB is impossible (there are only two blue hats).

P1 says: “I don’t know.” P2 and P3 infer that P1 sees a red

hat: RBB is impossible. P2 says: “I don’t know.”

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.

P1 P2 P3

R R R

R R B

R B R

R B B

B R R

B R B

B B R

B B B





Theories of Mind

Reasoning about the knowledge of other agents in a multi-agent systems seems to be natural to us. 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. 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. 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.





Creativity: Examples





Jan van Eyck: The Arnolfini Marriage

Creativity

Creativity 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.





Neuro-Symbolic Integration

Symbolic-subsymbolic distinction 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






Some interesting facts about the symbolic-subsymbolic distinction and cognitive science 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.

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?

Nevertheless important insights were provided by Elman who showed how rather simple recurrent networks (Elman networks) can learn correctly auxiliary inversion.






Some further remarks about the symbolic-subsymbolic distinction and cognitive science 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.





Causality

Causality seems to play an important role in human reasoning. 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: Mackie: Causality can be explained by insufficient and non-

redundant parts of unnecessary but sufficient causes (INUS condition).

Example 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 non-redundant.

Nevertheless there are many different proposals for a logical reduction of causality, e.g. counterfactuals.





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 such situations.





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)

Davies & Goel (2001) “I am here.”

“Oh, it’s raining.”

“Every student answered every question.”





Indurkhya (1992)

Non-Monotonicity

Axioms

Birds can usually fly.Penguins are birds.Tweety is a Penguin.

Theorem

Tweety can fly.

Axioms

Birds can usually fly.Penguins are birds.Tweety is a Penguin.Penguins can’t fly.

Theorem

Tweety cannot fly.





Non-Monotonicity

Axioms

TheoremsAxioms

+ p

new theoremsbecause of + p

Axioms

Theoremswithout p

Theoremsincl. p

monotonic extension

non-monotonic extension

Theoremswithout p

+ p





Analogical Reasoning

“Electrons are the planets of the atom.”

“Current is the water in an electric circuit.”





Analogical Reasoning

Some statements about analogical reasoning right at the beginning: 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.





Cognitive Architectures

The attempt to model cognitive behavior currently results in an inflationary number of different cognitive architectures. 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: Integration of different reasoning types. “Non-rational” behaviors (associations, emotions etc.). Hybrid (neuro-symbolic) representations.

Remark: not in the sense of neuro-symbolic integration, but more in the sense of “semantic networks + activation potentials”.

Integration of various cognitive abilities.





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





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.





Thank you very much!!





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Computational Logic and Cognitive Science: An Overview Session 2: Cognitive Challenges ICCL Summer...

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