Undecidability of ground reducibility for word rewriting systems with variables

Abstract

Given a nite alphabet A and an alphabet of variables X we consider words over A X called words with variables or patterns and words over A called simply words or strings A nite set R of rewrite rules w v where w v are words with variables is called a word rewriting system with variables WRSV From the viewpoint of rewriting systems theory WRSV s are rewriting systems over a signature consisting of one binary associative function concatenation and nitely many constant symbols letters If the rules of R do not contain variables we obtain word rewriting systems string rewriting systems semi Thue systems that have been investigated for a long time In this work we are interested in the reductive power of WRSV s For this reason we simply identify a WRSV with the set of its left hand sides Also we sometimes call patterns of R rules without adding the pre x the left hand side of Let Inst R denote the set of instances of patterns from R that is the set of p for all p R and all substitutions of variables by non empty words di erent occurrences of the same variable are substituted by the same word A pattern p applies to or reduces a word v if v A Inst fpg A We are interested in the properties of the set Red R A Inst R A that is the set of all words reducible by R In this paper we study the following ground reducibility problem for a given word with variables w does Inst fwg Red R hold i e are all the instances of w reducible by R If the inclusion holds w is said to be ground reducible by R The ground reducibility problem has been proved decidable for ordinary term rewriting systems However in the presence of associative functions the problem is more complex It was shown undecidable in for term rewriting systems over a signature containing among other functions an associative function This result however cannot be generalized to WRSV s as the auxiliary non associative functions are essential in the proof The ground reducibility problem can be also stated in the setting of pattern lan guages or non linear pattern matching Here pattern matching should be under stood in the extended sense a pattern w matches a word v i v has a factor which is