Propositional Linear Temporal Logic and Language Homomorphisms

When verifying speci cations of reactive systems|which are at least nonterminating computer programs|the complexity of the veri cation algorithm is a major problem of practical relevance. Speci cations of practical interest are usually too large to be veri ed by standard veri cation methods in an acceptable amount of time. So what can we do? We can try to get rid of information in the speci cation that is of no use due to the properties that are to be veri ed. Such an abstraction of a system's speci cation based on homomorphic mappings on !-languages can lead to much smaller descriptions of the system's interesting behaviour. We can now verify properties of the small speci cation and using these properties calculate properties of the large speci cation. In this paper we will develop a mapping R from formulae of the Propositional Linear Temporal Logic (abbr.: PLTL) to PLTL-formulae in such a way that if L is a nitary language describing a reactive system, h is an alphabetic language homomorphism that maps sequences of system states to some other state sequences and is a PLTL-formula then lim(L) j= R( ) holds if lim(h(L)) j= does hold; i.e. it su ces to verify that holds for lim(h(L)) to know that R( ) holds for lim(L). The relation \j=" has to be read as \satis es".