Global and local invariants in transition systems

Abstract

Given a transition system and a cover P of the set of its states, a set of local invariants with respect to P is defined as a set of predicates in bijection with the blocks of P and in such a way that a local invariant be true every time the system is in a state belonging to the corresponding block of the cover.This definition is proved to be sufficiently general in the sense that any proof made by using global invariants can be also made by using sets of local invariants. The same result is proved for two more restrictive definitions of the notion of local invariant by using well-known properties of connections between lattices.Finally, it is shown how the theory of connections can provide a general frame for tackling the problem of decomposing a global assertion into a logically equivalent set of local assertions.