Use of logical models for proving infeasibility in term rewriting

Abstract

Given a (Conditional) Rewrite System R and terms s and t, we consider the following problem: is there a substitution σ instantiating the variables in s and t such that the reachability test σ(s) → * R σ(t) succeeds? If such a substitution does not exist, we say that the problem is infeasible; otherwise, we call it feasible. Similarly, we can consider reducibility, involving a single rewriting step. In term rewriting, a number of important problems involve such infeasibility tests (e.g., confluence and termination analysis). We show how to recast infeasibility tests into the problem of finding a model of a set of (first-order) sentences representing the operational semantics of R together with some additional sentences representing the considered property which is formulated as an infeasibility test.

CoCo 2019: report on the eighth confluence competition

“…In this case, the critical pair is said to be infeasible [31, Definition 7.1.8]. Sufficient conditions for infeasibility of conditional critical pairs are reported in [19,42].…”

“…The new tool nonreach 24 participated in the INF category. Among others [23] it implements decomposition techniques 19 http://cl-informatik.uibk.ac.at/software/FORT/. 20 http://zenon.dsic.upv.es/infChecker/.…”

Reachability Analysis for Termination and Confluence of Rewriting

“…Our work is inspired by the work of Lucas and Gutiérrez [11]. Their feasibility sequences serve the same purpose as our reachability constraints, but are limited to atoms and conjunctions.…”

“…Reachability analysis for term rewriting [6] is concerned with the problem of, given a rewrite system R, a source term s and a target term t, deciding whether the source reduces to the target by rewriting, which is usually written s − → * R t. A useful generalization of this problem is the (un)satisfiability of the following reachability problem: given terms s and t containing variables, decide whether there is a substitution σ such that sσ − → * R tσ or not. This problem, also called (in)feasibility by Lucas and Guitiérrez [11], has various applications in termination and confluence analysis for plain and conditional rewriting.…”

Confluence Competition 2019

“…In this case the critical pair is said to be infeasible [14,Definition 7.1.8]. Sufficient conditions for infeasibility of conditional critical pairs are reported in [15,16].…”