Automatic synthesis of optimal invariant assertions: Mathematical foundations

The problem of discovering invariant assertions of programs is explored in light of the fixpoint approach in the static analysis of programs, Cousot [1977a], Cousot[1977b]. In section 2 we establish the lattice theoric foundations upon which the synthesis of invariant assertions is based. We study the resolution of a fixpoint system of equations by Jacobi's successive approximations method. Under continuity hypothesis we show that any chaotic iterative method converges to the optimal solution. In section 3 we study the deductive semantics of programs. We show that a system of logical forward equations can be associated with a program using the predicate transformer rules which define the semantics of elementary instructions. The resolution of this system of semantic equations by chaotic iterations leads to the optimal invariants which exactly define the semantics of this program. Therefore these optimal invariants can be used for total correctness proofs (section 4). Next we show that usually a system of inequations is used as a substitute for the system of equations. Hence the solutions to this system of inequations are approximate invariants which can only be used for proofs of partial correctness (section 5). In section 6 we show that symbolic execution of programs consists in fact in solving the semantic equations associated with this program. The construction of the symbolic execution tree corresponds to the chaotic successive approximations method. Therefore symbolic execution permits optimal invariant assertions to be discovered provided that one can pass to the limit, that is consider infinite paths in the symbolic execution tree. Induction nrinciDles can be used for that purpose. In section 7 we show how difference equations can be utilized to discover the general term of the sequence of successive approximations so that optimal invariants are obtained by a mere passage to the limit. In section 8 we show that an approximation of the optimal solution to a fixpoint system of equations can be obtained by strengthening the term of a chaotic iteration sequence. This formalizes the synthesis of approximate invariants by heuristic methods. Various examples provide a helpful intuitive support to the technical sections.