Completion is an instance of abstract canonical system inference

Guillaume Burel

Claude Kirchner

pp. 497-520

in: Kokichi Futatsugi, Jean-Pierre Jouannaud, José Meseguer (eds), Algebra, meaning, and computation, Berlin, Springer, 2006

Abstract

Abstract canonical systems and inference (ACSI) were introduced to formalize the intuitive notions of good proof and good inference appearing typically in first-order logic or in Knuth-Bendix like completion procedures.Since this abstract framework is intended to be generic, it is of fundamental interest to show its adequacy to represent the main systems of interest. This has been done for ground completion (where all equational axioms are ground) but was still an open question for the general completion process.By showing that the standard completion is an instance of the ACSI framework we close the question. For this purpose, two proof representations, proof terms and proofs by replacement, are compared to built a proof ordering that provides an instantiation adapted to the abstract canonical system framework.