The Calculus of Re nements , a

Programming in the large require the use of formal speci cation languages for describing program requirements and a method to test (automatically) such requirements. These methods can also be applied in other areas like complex system modeling. In this thesis we study the theoretical kernel of a formal speci cation language, named Calculus of Re nements (COR), based on the use of monotonic inclusion relations. These relations are more general than equality relations, therefore inclusion speci cations can be considered as a generalization of equational speci cations. Moreover, we propose the substitution of the typing relation \:" by an inclusion relation, therefore, the Calculus of Re nements can also be considered as a new typing discipline. The theoretical study of the Calculus of Re nements consists of the de nition of a denotational semantics and of an operational semantics for it. They are described on the two rst parts of the thesis. In the third part we approach the speci cation of nondeterministic programs by means of inclusions. In the rst part of the thesis we describe the Calculus of re nements as a logic, giving its syntax, a set of inference rules and de ning a class of models based on the class of environment models of the -calculus. We also study a concrete model where expressions are interpreted as order ideals. Such ideal domains have been used to give semantics to polymorphic types. On it we base the view of the Calculus of Re nements as a typing discipline. In the second part we give an operational semantics based on rewrite techniques. We de ne a pair of rewriting systems, namely a bi-rewriting system, which implement the deduction on inclusion theories. The main idea is using one of the relations to rewrite terms into smaller terms, and the other one to rewrite terms into bigger terms. Using a bi-rewriting system is possible to implement an algorithm to test if an inclusion a b is deducible in a theory. We rewrite a into bigger terms, and b into smaller terms till we obtain a common term. We have studied such technique for rst-order theories and linear secondorder theories (where bindings bind one and only one variable occurrence). xiii In the third part, we propose the use of bi-rewriting systems for the veri cation of nondeterministic program speci cations. We model nondeterministic computation by means of a relation satisfying, among others, the inclusion axioms. Therefore, the rewriting technique is sound (although not necessarily complete). We prove that adding more axioms to the speci cation such technique is also complete. xiv Chapter