A Computational Interpretation of Modal Proofs

A Temporal-Logic Approach to Binding-Time Analysis

“…Our natural-deduction formulation uses a judgement annotated with a natural number n, representing the "time" of the conclusion and with each assumption A in Γ also annotated by a time n. These are just like the "levels" in the modal natural deduction systems of Martini and Masini [9], and in fact our system is exactly the same as their rules for modal K, except that because of linearity we do not need any restriction on the introduction rule for . Our rules for the non-temporal fragment are relatively standard for natural deduction for pure classical logic, which will later allow us to depend on the equivalence between the axiomatic and natural-deduction systems for pure classical logic.…”

“…In the following section, we start with L , an axiomatic formulation due to Stirling [13] for a small classical linear-time temporal logic including . We then formulate a natural-deduction system in a similar style to the modal systems of Martini and Masini [9], and prove that it has the same theorems as the axiomatic formulation. This allows us to directly apply the Curry-Howard isomorphism to the natural-deduction system, yielding the typed λ -calculus with the operator in the types.…”

The basic intuitionistic logic of proofs

“…[4]). Furthermore, by adding also the choice operation "+", we will gain a capacity to naturally capture the intuitionistic version of the modal logic S4 and hence the modal λ-calculus [11,26,27]. Note, that in iLP every admissible rule of HA will be represented by a proof term.…”

“…The intuitionistic logic of proofs provides a more expressive version of the modal λ-calculus [11,26,27] which has interesting applications.…”

Call-by-Name and Call-by-Value in Normal Modal Logic

“…In Section 2, the calculus for IK is defined as a refinement of Bellin et al's calculus [4] rather than Martini and Masini's [22]. Our calculus is sound and complete for the categorical semantics given in [4].…”