Propositional Lax Logic Propositional Lax Logic

We investigate a peculiar intuitionistic modal logic, called Propositional Lax Logic (PLL), which has promising applications to the formal veri cation of computer hardware. The logic has emerged from an attempt to express correctness `up to' behavioural constraints | a central notion in hardware veri cation | as a logical modality. As a modal logic it is special since it features a single modal operator that has a avour both of possibility and of necessity. In the paper we provide the motivation for PLL and present several technical results. We investigate some of its proof-theoretic properties, presenting a cutelimination theorem for a standard Gentzen-style sequent presentation of the logic. We go on to de ne a new class of fallible two-frame Kripke models for PLL. These models are unusual since they feature worlds with inconsistent information; furthermore, the only frame condition imposed is that the -frame be a subrelation of the -frame. We give a natural translation of these models into Goldblatt's J -space models of PLL. Our completeness theorem for these models yields a G odel-style embedding of PLL into a classical bimodal theory of type (S4, S4) and underpins a simple proof of the nite model property. We proceed to prove soundness and completeness of several theories for specialized classes of models. We conclude with a brief exploration of two concrete and rather natural types of model from hardware veri cation for which the modality models correctness up to timing constraints. We obtain decidability of -free fragment of the logic of the rst type of model, which coincides with the stable form of Maksimova's intermediate logic L . 3