Computational types from a logical perspective

Abstract

Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the Curry–Howard correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove strong normalisation and confluence results.

Categorical and Kripke Semantics for Constructive S4 Modal Logic

“…These models were in fact the original semantics for Moggi's computational lambda-calculus and PLL can be seen as reverse engineering from that [BBdP98]. Hence we refrain from stating categorical soundness and completeness for this system, but of course they hold as expected [Kob97].…”

“…Still another use of the 2 modality, to model the quote mechanism of Lisp, is proposed by Goubault-Larrecq [GL96]. A 3-style modality has been extensively used to distinguish a computation from its result in the λ-calculus: Moggi's [Mog91] influential work on computational monads describes the computational λ-calculus, which corresponds to an intuitionistic modal type theory with a 3-like modality (see [BBdP98]). Fairtlough and Mendler [Men93,FMW97,Men00] use the same modality, which they call , in their work on lax logic for constraints and hardware verification.…”

“…This system is discussed under different names and in slightly differing but equivalent axiomatic presentations, such as Computational Logic [BBdP98] or Propositional Lax Logic (PLL) [FM97]. Henceforth we shall call it PLL.…”

“…In this paper we develop a suitable categorical semantics and associated λ-calculus for CS4 and PLL. It should be mentioned that the results for PLL are not new (see [FM97] for the Kripke and [BBdP98] for categorical semantics for PLL). Our contribution here is to show how PLL is related to CS4 and how these known results for PLL can be derived from those from CS4, or, to put it the other way round, how the known constructions for PLL may be generalised to CS4.…”

Logic in Access Control (Tutorial Notes)

“…CDD is a formalism related to lax logic and the computational lambda calculus [14,24,52]. CDD can be seen both as a type system and as a logic, via the Curry-Howard isomorphism.…”