Bisimulation and Propositional Intuitionistic Logic

The Brouwer-Heyting-Kolmogorov interpretation of intuition-istic logic suggests that p q can be interpreted as a computation that given a proof of p constructs a proof of q. Dually, we show that every nite canonical model of q contains a nite canonical model of p. If q and p are interderivable, their canonical models contain each other. Using this insight, we are able to characterize validity in a Kripke structure in terms of bisimilarity. Theorem 1 Let K be a nite Kripke structure for propositional intu-itionistic logic, then two worlds in K are bisimilar if and only if they satisfy the same set of formulas. This theorem lifts to structures in the following manner. Theorem 2 Two nite Kripke structures K and K 0 are bisimilar if and only if they have the same set of valid formulas. We then generalize these results to a variety of innnite structures; nite principal lter structures and saturated structures. 1 Background The standard model theory for intuitionistic logic (Heyting 1966) was introduced by Kripke (1965). This model theory was originally introduced as a characterization of modal logics (Kripke 1963). A natural question that arises for any model theory is characterizing when two models satisfy the same set of formulas. A partial answer to this for modal logics was given by van Benthem, who showed that for well behaved modal structures the notion of bisimulation was appropriate. We now apply the same techniques to intuitionistic logic. Although intuition-istic logic is not a traditional modal logic we can use the relationship between S4 and topology to provide a bridge. The algebra of S4 (Lewis and Langford 1932) is a topological Boolean algebra (McKinsey and Tarski 1944) where the modality 2 can be taken to be an interior operation and the modality 3 is taken to be a closure operation (McKinsey 1941, Rasiowa and Sikorski 1963). It is well known (Birkhoo 1948) that the open subsets of a topological Boolean algebra form a relatively pseudo-complemented lattice. However, a relatively pseudo-complemented lattice with 1