Maximal and Minimal Solutions to Language Equations

Given two languages L1 , L2 7*, we define L1hL2= [uhv | u # L1 , v # L2]. The well-known operations of catenation, right left quotient and shuffle product are examples of such operations. Other examples include the insertion and deletion operations. Recall that (see [3, 4]) given words u, v # 7*, the insertion of v into u is u v= [u1vu2 | u=u1u2] and the deletion of v from u is defined as u v=[w1w2 | u=w1vw2]. Among other binary operations we mention parallel, permuted, controlled insertion, and deletion [4, 3], k-catenation, and k-quotient ([5]). In this paper we study equations of the type LhY=R, XhY=R, XhX=R, RhX=LhY, where h is a binary word (language) operation, R and L are given nonempty languages and X, Y are unknown languages (the variables). In the following, X, Y, Z and their indexed variants will denote the unknowns, while L, R and their indexed variants will denote the given constant languages. The case when h denotes catenation and the languages involved are regular has been considered by Conway in [1]. We consider the existence and uniqueness of solutions. While, when exploring maximal solutions, the results refer to the general case of an abstract binary operation h , when considering the minimal solutions we deal with the particular cases where the operation h is catenation. In Section 2 we deal with equations LhY=R. In the general case, we prove that, if the equation has a solution, it has a unique maximal solution. The fact that all solutions to LY=R have the same set of minimal words aids in showing that if a solution exists, the equation also has a minimal solution. A sufficient condition for the minimal solution to be unique is obtained. The more general equation XhY=R is considered in Section 3. A solution (X, Y) to the equation is called an X-maximal solution (maximal solution) if any other solution (X, Y$) (resp. (X$, Y$)) with Y Y$ (resp. X X$, Y Y$) has the property Y$=Y (resp. X$=X, Y$=Y). If a solution to the equation exists, the equation has a unique Xmaximal solution. The maximal solution, while it always exists, is not necessarily unique. In the case of catenation, we show that the equation (if it has a solution) always has an X-minimal and a minimal solution. The existence of a nontrivial solution to XY=R proves to be decidable if R is a regular language. It remains an open problem whether the problem is decidable or not in case R is a context-free language. Properties of solutions when the constant languages belong to some important classes of languages, for example various types of codes, are also investigated. The concept of a minmax solution is introduced and we show that, if the equation has a solution, it also has a minmax solution. Section 4 deals with equations X =R. If n=2 and the equation has a solution, it also has a maximal solution. In case of catenation, the existence of solutions also implies the existence of a minimal solution, which is not necessarily unique. If n=2, the problem whether the equation X=R has a solution is decidable for given regular languages R (for n>2 the problem remains open). The problem is undecidable for given context-free languages R. In the end of the section, the notion of a square-root language (a language R which can be written as a square X=R) is introduced and its properties are studied. Finally, in Section 5 we deal with equations RhX= LhY. If a solution to such an equation exists, also an X-maximal solution and a maximal solution exists. In the article no. 0082