Semantical Evaluations as Monadic Second-Order Compatible Structure Transformations

Abstract

Abstract.A transformation of structures τ is monadic second-order compatible (MS-compatible) if every monadic second-order property P can be effectively rewritten into a monadic second-order property Q such that, for every structure S, if T is the transformed structure τ (S), then P (T ) holds iff Q(S) holds. We will review Monadic Second-order definable transductions (MS-transductions): they are MS-compatible transformations of a particular form, i.e., defined by monadic second-order (MS) formulas. The unfolding of a directed graph into a tree is an MS-compatible transformation that is not an MS-transduction. The MS-compatibility of various transformations of semantical interest follows. We will present three main cases and discuss applications and open problems. Overview of the lectureOur working logical language is Monadic Second-Order Logic, i.e., the extension of First-Order Logic with variables denoting sets of elements of the considered structures. It enjoys a number of interesting properties regarding decidability and construction of polynomial algorithms [4].We consider certain semantical evaluations that can be formalized as transformations of discrete structures like graphs or trees (and not as mappings from terms to values belonging to semantical domains as this is usual in denotational semantics).Our main concern will be to identify transformations such that the verification of an MS property P, of a structure T = τ (S) reduces to the verification of an MS property Q of the input structure S, where Q depends only on τ and P and, of course, of the fixed relational signatures of S and T .In such a case, if the MS theory of an infinite structure S is decidable (which means that there exists an algorithm that decides whether a monadic secondorder formula is true or not in S), then so is that of T = τ (S). We say that τ is Monadic Second-order compatible (MS-compatible).