Temporal Refinements for Guarded Recursive Types (full version)

Guilhem Jaber (1, 2) , Colin Riba (3)

1 UN - Université de Nantes
2 LS2N - équipe GALLINETTE - Gallinette : vers une nouvelle génération d'assistant à la preuve
3 PLUME - Preuves et Langages

We propose a logic to reason on temporal properties of higher-order programs that handle infinite objects like streams or infinite trees, represented via coinductive types. Specifications of programs are defined using safety and liveness properties. A given program can then be proven to satisfy its specification, in a compositional way, our logic being based on a type system. The logic is presented as a refinement type system over the guarded lambda-calculus, a λ-calculus with guarded recursive types. The refinements are formulae of a modal μ-calculus which embeds usual temporal modal logics such as LTL and CTL. The semantics of our system is given within a rich structure, the topos of trees, in which we build a realizability model of the temporal refinement type system. We use in a crucial way the connection with set-theoretic semantics to handle liveness properties.

Mots clés: Guarded recursive types; µ-calculus; refinement types; coinductive types; topos of trees; guarded recursive types

Domaines: Logique en informatique [cs.LO]

Guilhem Jaber, Colin Riba. Temporal Refinements for Guarded Recursive Types (full version). 2021. ⟨hal-02512655v5⟩