Abstract
The Brouwer-Heyting-Kolmogorov interpretation of intuitionistic 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 finite canonical model of q contains a finite 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 finite Kripke structure for propositional intuitionistic 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 finite Kripke structures K and K′ are bisimilar if and only if they have the same set of valid formulas.
We then generalize these results to a variety of infinite structures; finite principal filter structures and saturated structures.
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Patterson, A. (1997). Bisimulation and propositional intuitionistic logic. In: Mazurkiewicz, A., Winkowski, J. (eds) CONCUR '97: Concurrency Theory. CONCUR 1997. Lecture Notes in Computer Science, vol 1243. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63141-0_24
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DOI: https://doi.org/10.1007/3-540-63141-0_24
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