Logic of computational semi-effects and categorical gluing for equivariant functors Yuichi Nishiwaki University of Tokyo, Japan
[email protected] Toshiya Asai University of Tokyo, Japan
[email protected] Abstract In this paper, we revisit Moggi’s celebrated calculus of computational effects from the perspective of logic of monoidal action (actegory). Our development takes the following steps. Firstly, we perform proof-theoretic reconstruction of Moggi’s computational metalanguage and obtain a type theory with a modal type B as a refinement. Through the proposition-as-type paradigm, its logic can be seen as a decomposition of lax logic via Benton’s adjoint calculus. This calculus models as a programming language a weaker version of effects, which we call semi-effects. Secondly, we give its semantics using actegories and equivariant functors. Compared to previous studies of effects and actegories, our approach is more general in that models are directly given by equivariant functors, which include Freyd categories (hence strong monads) as a special case. Thirdly, we show that categorical gluing along equivariant functors is possible and derive logical predicates for B-modality. We also show that this gluing, under a natural assumption, gives rise to logical predicates that coincide with those derived by Katsumata’s categorical >>-lifting for Moggi’s metalanguage. 2012 ACM Subject Classification Theory of computation → Categorical semantics Keywords and phrases computational effects, actegories, logical relations, categorical gluing, proof theoretic semantics, modal logic, Curry-Howard correspondence, fibrations 1 Introduction It has passed about three decades since the deep connection between the notion of computation and monads in category theory was revealed by Moggi [29, 30].