José Espírito Santo (Centro de Matemática, Universidade do Minho, Braga, Portugal) |
Ralph Matthes (Institut de Recherche en Informatique de Toulouse (IRIT), C.N.R.S. and University of Toulouse, France) |
Luís Pinto (Centro de Matemática, Universidade do Minho, Braga, Portugal) |

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. |

Published: 28th August 2013.

ArXived at: http://dx.doi.org/10.4204/EPTCS.126.3 | bibtex | |

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