Intentional Set Constraints for Formal Speci

Sets are widely used as data structures in programs or speciications. Many powerful speciica-tion languages such as Z 8] and B 1] derived from VDM 3] are based on set theory. In this context, to validate a speciication one needs tools to manipulate logical formulas on sets. Be-havioral properties of modules speciied in this way can be naturally described using intentional sets and constraints on the global state variables ; the same is also true for studying labelled state machines, either nite or innnite. This is the motivation for the work of this paper, where we propose a language to form intentional set expressions, and to express relational constraints between them. A sequent calculus is developed for proving logical formulas involving set expressions. A method for checking the satissability of a system of intentional set constraints and for constructing a solution in terms of a logic program is proposed. In a second part of the work, we give a method for synthesizing a B-abstract machine, also in terms of a logic program, if its speciication does not use sets of sets in its data structures. Techniques from the domain of logic programming are therefore mingled with our development, since a natural way of specifying intentional sets is in terms of logic programs. Our approach is thus applicable for the formal validation of set based speciications, where the operations can be deened using implicit theories on some of the state variables.