Alexis Saurin

From proofs to focused proofs: a modular proof of focalization in linear logic

Probably the most significant result concerning cut-free sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications -- e.g. in game semantics, Ludics, and proof search -- and more computer science applications -- e.g. logic programming, call-by-name/value evaluation. Andreoli proved this theorem for first-order linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cut-free proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreolis original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic.

Canonical Sequent Proofs via Multi-Focusing

The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative canonicity directly in the cut-free sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multi-focused proofs. We validate this definition by proving a bijection to the well-known proof-nets for the unit-free multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments.

Separation with Streams in the ?µ-calculus

The /spl lambda//spl mu/-calculus is an extension of the /spl lambda/-calculus introduced in 1992 by Parigot (M. Parigot, 1992) in order to generalize the Curry-Howard isomorphism to classical logic. Two versions of the calculus are usually considered in the literature: Parigots original syntax and an alternative syntax introduced by de Groote. In 2001, David and Py (R. David, 2001) proved that the Separation Property (also referred to as Bohm theorem) fails for Parigots /spl lambda//spl mu/-calculus. By analyzing David & Pys result, we exhibit an extension of Parigots /spl lambda//spl mu/-calculus, the /spl Lambda//spl mu/-calculus, for which the Separation Property holds and which is built as an intermediate language between Parigots and de Grootes /spl lambda//spl mu/-calculi. We prove the theorem and describe how /spl Lambda//spl mu/-calculus can be considered as a calculus of terms and streams. We then illustrate Separation in showing how in /spl Lambda//spl mu/-calculus it is possible to separate the counter-example used by David & Py.

A Game Semantics for Proof Search: Preliminary Results

We describe an ongoing project in which we attempt to describe a neutral approach to proof and refutation. In particular, we present a language of neutral expressions which contains one element for each de Morgan pair of connectives in (linear) logic. Our goal is then to describe, in a neutral fashion, what it means to prove or refute. For this, we use games where moves are described as transitions between positions built with neutral expressions. In some settings, we can then relate winning a game with provability or with validity.

Typing streams in the Λμ-calculus

Λμ-calculus is a Böhm-complete extension of Parigots Λμ-calculus closely related with delimited control in functional programming. In this article, we investigate the meta-theory of untyped Λμ-calculus by proving confluence of the calculus and characterizing the basic observables for the Separation theorem, <i>canonical normal forms</i>. Then, we define Λ_<i>s</i>, a new type system for Λμ-calculus that contains a special type construction for streams, and prove that strong normalization and type preservation hold. Thanks to the new typing discipline of Λ_<i>s</i>, new computational behaviors can be observed, which were forbidden in previous type systems for λμ-calculi. Those new typed computational behaviors witness the stream interpretation of Λμ-calculus.

Proof and refutation in MALL as a game

We present a setting in which the search for a proof of B or a refutation of B (i.e., a proof of ¬B) can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates to a refutation of the formula. The game is described for multiplicative and additive linear logic (MALL). A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win (that is, neither B nor ¬B might be provable).

Infinitary proof theory : the multiplicative additive case

Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly underdeveloped. In particular, very little is known about the computational behavior of such proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a proof system: cut elimination and focalization. The first one is only known to hold for restricted (purely additive) infinitary calculi, thanks to the work of Santocanale and Fortier; the second one has never been studied in infinitary systems. In this paper, we consider the infinitary proof system muMALLi for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results. We thus establish muMALLi as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.

Separation with streams in the

The /spl lambda//spl mu/-calculus is an extension of the /spl lambda/-calculus introduced in 1992 by Parigot (M. Parigot, 1992) in order to generalize the Curry-Howard isomorphism to classical logic. Two versions of the calculus are usually considered in the literature: Parigots original syntax and an alternative syntax introduced by de Groote. In 2001, David and Py (R. David, 2001) proved that the Separation Property (also referred to as Bohm theorem) fails for Parigots /spl lambda//spl mu/-calculus. By analyzing David & Pys result, we exhibit an extension of Parigots /spl lambda//spl mu/-calculus, the /spl Lambda//spl mu/-calculus, for which the Separation Property holds and which is built as an intermediate language between Parigots and de Grootes /spl lambda//spl mu/-calculi. We prove the theorem and describe how /spl Lambda//spl mu/-calculus can be considered as a calculus of terms and streams. We then illustrate Separation in showing how in /spl Lambda//spl mu/-calculus it is possible to separate the counter-example used by David & Py.