Computational Logic and Cognitive Science: An Overview

1. 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

2. 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 Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

3. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

4. Wason Selection Task • 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… Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

5. Wason Selection Task • 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

6. Wason Selection Task • 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

7. 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. • “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

8. 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 Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

9. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

10. 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? Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

11. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

12. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

13. 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

14. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

15. Neuro-Symbolic Integration • Symbolic-subsymbolic distinction • There is an obvious tension between symbolic and subsymbolic representations. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

16. Neuro-Symbolic Integration • 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

17. Neuro-Symbolic Integration • 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

18. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

19. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

20. 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) Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

21. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

22. Non-Monotonicity monotonic extension Theorems Theorems without p Axioms + p Axioms new theorems because of + p Theorems without p non-monotonic extension Axioms Theorems incl. p + p Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

23. 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

24. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

25. 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. Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008

26. 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

27. 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

28. Thank you very much!! Helmar Gust & Kai-Uwe Kühnberger Universität Osnabrück ICCL Summer School 2008 Technical University of Dresden, August 25th – August 29th, 2008