### When is an extension of a specification consistent? Decidable and undecidable cases

Friedrich Otto
1991 Journal of symbolic computation
The problem of consistency is the following decision problem: Instance: Two finite specifications (E, E) and (.Xl, El), where the signature X is contained in the signature X~. Question: Is (E l , El) a consistent extension of (:Z, E)? For the special ease of string-rewriting systems we present decidable and undeeidable eases of this problem. As it turns out in order to obtain decidability rather severe restrictions are necessary either on the syntactic form of'the equations in E and E1 or on
more » ... algebraic structure of the algebras (i.e. monoids) that are presented by (E. E) and (El, El) , respectively. of the form So> s~ > s2>' ' '. If So> s~, we say that s~ is simpler than So. Now the given set E of defining equations can be oriented into a replacement system R by taking R := {l--, r[l> r, and (l, r)e E or (r, l)e E}, provided that for each equation (l, r) of E, the two objects 1 and r are comparable under >. If the ordering > is compatible with t This work was performed while the author was visiting at F. Otto the structure of the elements of S, then this choice of R implies that each sequence of applications of rules of R terminates after a finite number of steps. Thus, the system R is Noetherian. If R is also confluent, then, for each object s ~ S, there exists a unique object t ~ S such that s reduces to t (mod R), and no rule of R applies to t, i.e. t is irreducible (mod R). The replacement system R is called canonical if it is both Noetherian and confluent. One can verify that, if R is canonical, then two objects s, t ~ S are congruent if and only if they reduce to the same irreducible object. Thus, in this situation the word problem for (5, ~-) is decidable, if the process of reduction is an effective one. If R is a canonical replacement system, then the process of computing the irreducible descendant of a given object is called normalization. Hence, normalization is the one fundamental algorithm when dealing with canonical replacement systems. The second fundamental algorithm is the so-called Knuth-Bendix completion procedure. In general, the replacement system R obtained by orienting the equations of E is not confluent. Given the system R (or E) and the ordering > as input, the Knuth-Bendix completion procedure tries to derive a system R~ such that R and R~ are equivalent, i.e. they both define the congruence ~-on S, and R1 is canonical. In its simplest form the Knuth-Bendix completion procedure does this by introducing certain additional rules. Obviously, this procedure will not always succeed, since in general the word problem for (S, ~-) will be undecidable. In fact, this procedure may fail although the word problem for (S, ~) is decidable. For example, let S = T({f, g}, X), i.e. S is the set of first-order terms built from the variables in X and the unary function symbols f and g, and let -~