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Monad as modality. (English) Zbl 0895.03011
Summary: In 1989, Eugenio Moggi proposed a categorical framework for program semantics based on the notion of a strong monad. He showed that various kinds of computation can be modeled in his framework. On the other hand, strong monads are not suited for the categorical semantics of traditional modal logics. According to these observations, Moggi thought that the Curry-Howard correspondence would not hold between programs and constructive proofs in modal logics. However, contrary to his view, we can show that proofs in a certain kind of modal logics are actually considered as programs. In this paper, first we shall introduce the notion of an $${\mathcal L}$$-strong monad which is a generalization of strong monads. Using this new notion, we can generalize Moggi’s semantics-preserving soundness and completeness with respect to his equational logic. Next we shall show that $${\mathcal L}$$-strong monads give a sound and complete semantics of a constructive version of S4 modal logic. Finally, we present a method to extract a monad-based imperative functional program from a proof in the modal logic. Interestingly, this method can also be understood in terms of $${\mathcal L}$$-strong monads.

##### MSC:
 03B70 Logic in computer science 68Q55 Semantics in the theory of computing 03B45 Modal logic (including the logic of norms) 03G30 Categorical logic, topoi 18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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