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

We study theπ-calculus, where the family of names is enriched with pairing, and define a notion of type assignment that uses th type constructor→. We encode the circuits of the calculus X into this variant ofπ, and show that reduction and type assignment are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen’s calculusLK , this implies that all proofs inLK have an encoding inπ. 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, calle d nets, for derivable sequents. Nets in X have multiple named inputs and multiple named outputs, that are collectively called connectors . Reduction inX is expressed via a set of rewrite rules that representcut-elimination, eventually leading to renaming of connectors, and gives computational meaning to classical (sequen t) proof reduction. It is well known that cut-elimination inLK is not confluent, and, sinceX is Curry-Howard forLK , neither is reduction in X . These two features –non-confluence and reduction as connec tion of nets via the exchange of names– lead us to consider the πcalculus as an alternative computational model for cut-elimination and proofs inLK . 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 foun d [20]), which should be developed further, widening further the roa d to practical application of classical logic in computation. T he aim of this paper is to focus on linking LK andπ viaX ; the main achievements of this paper are: [copyright notice will appear here] • an encoding ofX 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 show ing that all proofs inLK have an encoding in π, thereby representing classical logic directly in a process calculus; • to achieve the necessary non-standard notion of types (type s do not contain channel information), the target calculus is en riched with pairing [2], and a notion of type assignment is defined which encompasses implication; • in addition to [8], we treat the full classical logic, includ ing the connectives→, ¬, &, and∨, not only forX , but also forπ. Classical sequents Thesequent 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 connec tiv s in the logical formulae. Natural deduction derives statement s wi h a single conclusion, whereas LK allows for multiple conclusions, deriving sequents of the formA1, . , An ⊢ B1, . , Bm. Implicative LK has four rules:axiom, left introductionof the arrow,right introduction, andcut. (Ax) : Γ, A ⊢ A,∆ (⇒L) : Γ ⊢ A,∆ Γ, B ⊢∆