Stream Associative Nets and Lambda-mu-calculus

$\Lambda\mu$-calculus has been built as an untyped extension of Parigot's $\lambda\mu$-calculus in order to recover Böhm theorem which was known to fail in $\lambda\mu$-calculus. An essential computational feature of $\Lambda\mu$-calculus for separation to hold is the unrestricted use of abstractions over continuations that provides the calculus with a construction of streams. Based on the Curry-Howard paradigm Laurent has defined a translation of $\Lambda\mu$-calculus in polarized proof-nets. Unfortunately, this translation cannot be immediately extended to $\Lambda\mu$-calculus: the type system on which it is based freezes \Lm-calculus's stream mechanism. We introduce \emph{stream associative nets (SANE)}, a notion of nets which is between Laurent's polarized proof-nets and the usual linear logic proof-nets. SANE have two kinds of \lpar (hence of \ltens), one is linear while the other one allows free structural rules (as in polarized proof-nets). We prove confluence for SANE and give a reduction preserving encoding of $\Lambda\mu$-calculus in SANE, based on a new type system introduced by the second author. It turns out that the stream mechanism at work in $\Lambda\mu$-calculus can be explained by the associativity of the two different kinds of \lpar of SANE. At last, we achieve a Böhm theorem for SANE. This result follows Girard's program to put into the fore the separation as a key property of logic

A Curry-Howard Correspondence for Linear, Reversible Computation

In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching; we show how ensuring syntactical exhaustivity and non-overlapping of clauses is enough to ensure reversibility. The language allows to represent any Primitive Recursive Function. We then give a Curry-Howard correspondence with the logic ?MALL: linear logic extended with least fixed points allowing inductive statements. The critical part of our work is to show how primitive recursion yields circular proofs that satisfy ?MALL validity criterion and how the language simulates the cut-elimination procedure of ?MALL

DRAFT -Do Not Distribute Ludics Programming I: Interactive Proof Search

Abstract Proof theory and Computation are research areas which have very strong relationships: new concepts in logic and proof theory often apply to the theory of programming languages. The use of proofs to model computation led to the modelling of two main programming paradigms which are functional programming and logic programming. While functional programming is based on proof normalization, logic programming is based on proof search. This approach has shown to be very successful by being able to capture many programming primitives logically. Nevertheless, important parts of real logic programming languages are still hardly understood from the logical point of view and it has been found very difficult to give a logical semantics to control primitives. Girard introduced Ludics [12] as a new theory to study interaction. In Ludics, everything is built on interaction or in an interactive way. In this paper, which is the first of a series investigating a new computational model for logic programming based on Ludics, namely computation as interactive proof search, we introduce the interactive proof search procedure and study some of its properties

Phase Semantics for Linear Logic with Least and Greatest Fixed Points

The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (?MALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to ?MALL with explicit (co)induction (i.e. ?MALL^{ind}). We introduce a Tait-style system for ?MALL called ?MALL_? where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property

λμ-calculus and Λμ-calculus: a Capital Difference

Since Parigot designed the λμ-calculus to algorithmically interpret classical natural deduction, several variants of λμ-calculus have been proposed. Some of these variants derived from an alteration of the original syntax due to de Groote, leading in particular to the Λμ-calculus of the second author, a calculus truly different from λμ-calculus since, in the untyped case, it provides a Böhm separation theorem that the original calculus does not satisfy. In addition to a survey of some aspects of the history of λμ-calculus, we study connections between Parigot's calculus and the modified syntax by means of an additional toplevel continuation. This analysis is twofold: first we relate λμ-calculus and Λμ-calculus in the typed case using λμtp-calculus, which contains a toplevel continuation constant tp, and then we use calculi of the family of λμtp-calculi, containing a toplevel continuation variable tp, to study the untyped setting and in particular relate calculi in the modified syntax

λμ-calculus and Λμ-calculus: a Capital Difference

Since Parigot designed the λμ-calculus to algorithmically interpret classical natural deduction, several variants of λμ-calculus have been proposed. Some of these variants derived from an alteration of the original syntax due to de Groote, leading in particular to the Λμ-calculus of the second author, a calculus truly different from λμ-calculus since, in the untyped case, it provides a Böhm separation theorem that the original calculus does not satisfy. In addition to a survey of some aspects of the history of λμ-calculus, we study connections between Parigot's calculus and the modified syntax by means of an additional toplevel continuation. This analysis is twofold: first we relate λμ-calculus and Λμ-calculus in the typed case using λμtp-calculus, which contains a toplevel continuation constant tp, and then we use calculi of the family of λμtp-calculi, containing a toplevel continuation variable tp, to study the untyped setting and in particular relate calculi in the modified syntax

Infinets: The parallel syntax for non-wellfounded proof-theory

International audienceLogics based on the µ-calculus are used to model inductive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. While traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. The present paper investigates a discrepancy found in such proof systems , between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (µMLL ∞) and study their correctness and sequentialization

: De la réduction linéaire de tête à l'évaluation paresseuse

National audienceÀ partir de la réduction linéaire de tête, nous dérivons de manière systématique un calcul en appel par nécessité. L'introduction d'un calcul pour la réduction linéaire de tête, basée sur une analyse fine de la notion de radicaux premiers de Danos et Regnier, nous permet de construire pas à pas un lambda-calcul en appel par nécessité que l'on compare aux calculs présents dans la littérature