A Coinductive Approach to Proof Search

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.

In David Baelde and Arnaud Carayol: Proceedings Workshop on Fixed Points in Computer Science (FICS 2013), Turino, Italy, September 1st, 2013, Electronic Proceedings in Theoretical Computer Science 126, pp. 28–43.
Published: 28th August 2013.

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