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forthcoming)). Applying lakatosstyle reasoning to ai problems
 Thinking Machines and the philosophy of computer science: Concepts and principles. IGI Global
, 2010
"... One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning ..."
Abstract

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One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning styles collaborate to develop mathematical ideas. Lakatos argued that mathematics is a quasiempirical, flexible, fallible, human endeavour, involving negotiations, mistakes, vague concept definitions and disagreements, and he outlined a heuristic approach towards the subject. In this chapter we apply these heuristics to the AI domains of evolving requirement specifications, planning and constraint satisfaction problems. In drawing analogies between Lakatos’s theory and these three domains we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos’s philosophy can inform AI problem solving. Thus, we show how we might begin to produce a philosophicallyinspired AI theory of combined reasoning. 1
Normalization of intuitionistic set theories
"... Abstract. IZF is a wellinvestigated impredicative constructive counterpart of ZermeloFraenkel set theory. We define a weaklynormalizing lambda calculus λZ corresponding to proofs in an intensional version of IZF with Replacement according to the CurryHoward isomorphism principle. By adapting a c ..."
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Abstract. IZF is a wellinvestigated impredicative constructive counterpart of ZermeloFraenkel set theory. We define a weaklynormalizing lambda calculus λZ corresponding to proofs in an intensional version of IZF with Replacement according to the CurryHoward isomorphism principle. By adapting a counterexample invented by M. Crabbé, we show that λZ does not strongly normalize. Moreover, we prove that the calculus corresponding to a nonwellfounded IZF does not even weakly normalize. Thus λZ and IZF are positioned on the fine line between weak, strong and lack of normalization. 1
California, US
"... We present an action theory with the power to represent recursive plans and the capability to reason about and synthesize recursive workflow control structures. In contrast with the software verification setting, reasoning does not take place solely over predefined data structures, and neither is th ..."
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We present an action theory with the power to represent recursive plans and the capability to reason about and synthesize recursive workflow control structures. In contrast with the software verification setting, reasoning does not take place solely over predefined data structures, and neither is there a process specification available in recursive form. Rather, specification takes the form of goals, and domain structure takes the form of a physical application setting containing objects. For this reason, wellfounded induction is employed for its suitability for practical action domains where recursive structures must be described or inferred. Under this method, termination of the synthesized recursive workflow is a property that follows automatically. We show how a general workflow recursive construct is added to an action language that is then augmented with induction. This formalism is then transformed in a way amenable to automated reasoning. We demonstrate the method with a particular example specified in the theory, and then extracted from a proof constructed by the SNARK firstorder theorem prover.
NORMALIZATION OF IZF WITH REPLACEMENT
, 2007
"... Abstract. IZF is a well investigated impredicative constructive version of ZermeloFraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF − R. We define a typed lambda calculus λZ corresponding to proofs in IZF − R acc ..."
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Abstract. IZF is a well investigated impredicative constructive version of ZermeloFraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF − R. We define a typed lambda calculus λZ corresponding to proofs in IZF − R according to the CurryHoward isomorphism principle. Using realizability for IZF − R, we show weak normalization of λZ. We use normalization to prove the disjunction, numerical existence and term existence properties. An inner extensional model is used to show these properties, along with the set existence property, for full, extensional IZFR. 1.