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Finding small counterexamples for abstract rewriting properties

Published online by Cambridge University Press:  25 July 2018

Department of Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Email: Institute for Computing and Information Sciences, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands


Rewriting notions like termination, normal forms and confluence can be described in an abstract way referring to rewriting only as a binary relation. Several theorems on rewriting, like Newman's lemma, can be proved in this abstract setting. For investigating possible generalizations of such theorems, it is fruitful to have counterexamples showing that particular generalizations do not hold. In this paper, we develop a technique to find such counterexamples fully automatically, and we describe our tool Carpa that follows this technique. The basic idea is to fix the number of objects of the abstract rewrite system, and to express the conditions and the negation of the conclusion in a satisfiability (SAT) formula, and then call a current SAT solver. In case the formula turns out to be satisfiable, the resulting satisfying assignment yields a counterexample to the encoded property. We give several examples of finite abstract rewrite systems having remarkable properties that are found in this way fully automatically.

Copyright © Cambridge University Press 2018 

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