### On deciding the confluence of a finite string-rewriting system on a given congruence class

Friedrich Otto
1987 Journal of computer and system sciences (Print)
In general it is undecidable whether or not a given finite string-rewriting system R is confluent on a given congruence class [w]~, even when only length-reducing systems are considered. However, for finite monadic string-rewriting systems this problem becomes decidable in double exponential time. For certain subclasses of monadic string-rewriting systems algorithms of lower complexity are obtained for solving this problem. cl 1987 Academic Press, Inc. String-rewriting systems, also known as
more » ... i-Thue systems, have extensively been studied in computability theory, formal language theory, and combinatorial (semi-) group theory. A string-rewriting system R on alphabet C induces a congruence ++ 2 on the free monoid 27 generated by Z, and hence, the set M, of congruence classes modulo -2 is a monoid. Accordingly, the ordered pair (2'; R) is called a monoidpresentation of M,. The fundamental decision problem associated with R is the word problem: INSTANCE. Two words u, VEX*. Question. Are u and v congruent modulo R? It is well known that this problem is undecidable in general. A string-rewriting system R is called Noetherian, if there is no infinite sequence of reductions modulo R. So if R is Noetherian, then for each word w, an irreducible word wO is reached within a finite number of reduction steps. Here a word is called irreducible if no reduction applies to it. R is called confluent if, for all words w" w2, w3, whenever wi reduces to w2 and to wj, then w2 and w3 can be reduced to a common word w4. This property is equivalent to the Church-Rosser property: for all words w, and w2, if w, and w2 are congruent modulo R, then they can be *