Mona: Decidable Arithmetic in Practice

In this note, we describe how a fragment of arithmetic can be decided in practice. We follow essentially the ideas of 8], which we have embedded in the Mona tool. Mona is an implementation of Monadic Second-order Logic on nite strings (and trees). The previous semantics used in Mona is the one provided in current literature 7, 9], where the meaning of rst-order terms is restricted to being a position in the string over which the formula is interpreted. In this note, we describe our new semantics, where terms are interpreted relative to all natural numbers. With this semantics Mona becomes a decision procedure for the calculus called WS1S, the Weak Second-order theory of 1 Successor. We also show how the Mona representation of automata subsumes a recent proposal for representing queues. We exploit the natural semantics to carry out automated reasoning about queue operations. In practice, the fundamental concept of regularity ((nite-state acceptance of strings) is often exceedingly hard to express. For example, regular expressions (as used in text editors and UNIX shell programming) are elegant when expressing simple patterns, but often unreadable for patterns of even modest complexity. The aim of the FIDO/Mona project pursued at the University of Aarhus is to devise new practical means of describing nite-state systems in formal logics that naturally capture informal requirements. Mona is a tool that translates formulas to nite-state automata. The formulas may express search patterns, temporal properties of reactive systems, or parse tree constraints. Mona is based on Monadic Second-order Logic on nite strings. (FIDO is a high-level language, incorporating logic and many usual programming language concepts like recursive data types. A FIDO program is translated into Mona, which in turn is translated into an automaton.) In this paper, we discuss a new semantics for Mona that we have recently implemented. We also show that a recently proposed data structure for representing queues 2] is but a special case of the BDD-represented automata of Mona. Some Mona applications The previous Mona implementationfor nite strings is described in 3]. We have applied Mona to hardware veriication 1], veri-cation of complicated behavioral descriptions of distributed systems 5], and