From X to π Representing the Classical Sequent Calculus in π - calculus

We study the π-calculus, where the family of names is enriched with pairing, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that reduction and type assignment are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have an encoding in π. We then enrich the logic with the connectors &,∨ and ¬, and show that also these can be represented in π. Introduction In this paper we present an encoding of proofs of Gentzen’s LK [15] into the π-calculus, that respects cut-elimination as well as assignable types. The encoding of classical logic into π-calculus is attained by using the intuition of the calculus X , which gives a computational meaning to LK (a first version of this calculus was proposed in [30, 29, 28]; the implicative fragment of X was studied in [8]). X enjoys the Curry-Howard property for LK; it achieves the isomorphism by constructing witnesses, called nets, for derivable sequents. Nets in X have multiple named inputs and multiple named outputs, that are collectively called connectors. Reduction in X is expressed via a set of rewrite rules that represent cut-elimination, eventually leading to renaming of connectors, and gives computational meaning to classical (sequent) proof reduction. It is well known that cut-elimination in LK is not confluent, and, since X is Curry-Howard for LK, neither is reduction in X . These two features –non-confluence and reduction as connection of nets via the exchange of names– lead us to consider the πcalculus as an alternative computational model for cut-elimination and proofs in LK. The relation between process calculi and classical logic is an interesting and very promising area of research (a very similar attempt, in a natural deduction context, can be found [20]), which should be developed further, widening further the road to practical application of classical logic in computation. The aim of this paper is to focus on linking LK and π via X ; the main achievements of this paper are: [copyright notice will appear here] • an encoding of X into π is defined, that preserves the operational semantics; the non-confluent nature of reduction in X /LK is neatly reflected by the non-determinism of π; • the encoding preserves assignable types, effectively showing that all proofs in LK have an encoding in π, thereby representing classical logic directly in a process calculus; • to achieve the necessary non-standard notion of types (types do not contain channel information), the target calculus is enriched with pairing [2], and a notion of type assignment is defined which encompasses implication; • in addition to [8], we treat the full classical logic, including the connectives →, ¬, &, and ∨, not only for X , but also for π. Classical sequents The sequent calculus, LK introduced by Gentzen in [15], is a logical system in which the rules only introduce connectives (but on either side of a sequent), in contrast to natural deduction (also introduced in [15]) which uses rules that introduce or eliminate connectives in the logical formulae. Natural deduction derives statements with a single conclusion, whereas LK allows for multiple conclusions, deriving sequents of the form A1, . . . , An ⊢ B1, . . . , Bm. Implicative LK has four rules: axiom, left introduction of the arrow, right introduction, and cut. (Ax) : Γ, A ⊢ A,∆ (⇒L) : Γ ⊢ A,∆ Γ, B ⊢∆