Abstract

MIRANDA-PEREA, Favio E.; OMANA SILVA, Sammantha  and  GONZALEZ HUESCA, Lourdes del Carmen. A Dual-Context Sequent Calculus for S 4 Modal Lambda-Term Synthesis. Comp. y Sist. [online]. 2022, vol.26, n.2, pp.787-799.  Epub Mar 10, 2023. ISSN 2007-9737.  https://doi.org/10.13053/cys-26-2-4238.

In type-based program synthesis, the search of inhabitants in typed calculi can be seen as a process where a specification, given by a type $A$, is considered to be fulfilled if we can construct a $\lambda$-term $M$ such that $M:A$, or more precisely if $\Gamma ⊢M:A$ holds, that is, if under some suitable assumptions $\Gamma$ the term $M$ inhabits the type $A$. In this paper, we tackle this inhabitation/synthesis problem for the case of modal types in the necessity fragment of the constructive logic $S4$. Our approach is human-driven in the sense of the usual reasoning procedures of modern theorem provers. To this purpose we employ a so-called dual-context sequent calculus, where the sequents have two contexts, originally proposed to capture the notions of global and local truths without resorting to any formal semantics. The use of dual-contexts allows us to define a sequent calculus which, in comparison to other related systems for the same modal logic, exhibits simpler typing inference rules for the $\square$ operator. In several cases, the task of searching for a term having subterms with modal types is reduced to the quest for a term containing only subterms typed by non modal propositions.

Keywords : Dual-context sequent calculus; constructive necessity; type inhabitation; modal lambda calculus; program synthesis.

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