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A Coinductive Approach to Proof Search

Abstract : We propose to study proof search from a coinductive point of view. In this paper, we consider intuitionistic logic and a focused system based on Herbelin's LJT for the implicational fragment. We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the "solution spaces" (called Böhm forests), which are a representation of all (not necessarily well-founded but still locally well-formed) proofs of a given formula (more generally: of a given sequent).

As main result we obtain, for each given formula, the reduction of a coinductive definition of the solution space to a effective coinductive description in a finitary term calculus with a formal greatest fixed-point operator. This reduction works in a quite direct manner for the case of Horn formulas. For the general case, the naive extension would not even be true. We need to study "co-contraction" of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point variables are typed with sequents. Co-contraction enters the interpretation of the formal greatest fixed points - curiously in the semantic interpretation of fixed-point variables and not of the fixed-point operator.

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  • HAL Id : hal-01226485, version 1
  • OATAO : 12636

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José Espírito Santo, Ralph Matthes, Luís Pinto. A Coinductive Approach to Proof Search. Fixed Points in Computer Science (FICS 2013), Sep 2013, Turin, Italy. pp. 28-43. ⟨hal-01226485⟩

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