Infinitary Classical Logic: Recursive Equations and Interactive Semantics

In this paper, we present an interactive semantics for derivations in an infinitary extension of classical logic. The formulas of our language are possibly infinitary trees labeled by propositional variables and logical connectives. We show that in our setting every recursive formula equation has a unique solution. As for derivations, we use an infinitary variant of Tait-calculus to derive sequents. The interactive semantics for derivations that we introduce in this article is presented as a debate (interaction tree) between a test > (derivation candidate, Proponent) and an environment << not S >> (negation of a sequent, Opponent). We show a completeness theorem for derivations that we call interactive completeness theorem: the interaction between > (test) and > (environment) does not produce errors (i.e., Proponent wins) just in case > comes from a syntactical derivation of >.Comment: In Proceedings CL&C 2014, arXiv:1409.259

Infinitary λ\lambda-Calculi from a Linear Perspective (Long Version)

We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative $\lambda$-calculus and is universal for computations over infinite strings. What is particularly interesting about $\ell\Lambda_{\infty}$, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of $\ell\Lambda$ built around the principles of $\mathsf{SLL}$ and $\mathsf{4LL}$. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary $\lambda$-calculi

The Ackermann Award 2018

The Ackermann Award is the EACSL Outstanding Dissertation Award for Logic in Computer Science. It is presented during the annual conference of the EACSL (CSL\u27xx). This contribution reports on the 2018 edition of the award

On the meaning of focalization

Abstract In this paper, we use Girard&apos;s Ludics to analyze focalization, a fundamental property of linear logic. In particular, we show how this can be realized interactively thanks to section-retraction pairs (u αβ , f αβ ) between behaviours α ˆ(β Y ), X and αβ Y, X

Infinitary proof theory : the multiplicative additive case

Infinitary and regular proofs are commonly used in fixed point logics. Being natural intermediatedevices between semantics and traditional finitary proof systems, they are commonly found incompleteness arguments, automated deduction, verification, etc. However, their proof theoryis surprisingly underdeveloped. In particular, very little is known about the computationalbehavior of such proofs through cut elimination. Taking such aspects into account has unlockedrich developments at the intersection of proof theory and programming language theory. Onewould hope that extending this to infinitary calculi would lead, e.g., to a better understanding ofrecursion and corecursion in programming languages. Structural proof theory is notably basedon two fundamental properties of a proof system: cut elimination and focalization. The firstone is only known to hold for restricted (purely additive) infinitary calculi, thanks to the workof Santocanale and Fortier; the second one has never been studied in infinitary systems. Inthis paper, we consider the infinitary proof system μMALL ∞ for multiplicative and additivelinear logic extended with least and greatest fixed points, and prove these two key results. Wethus establish μMALL ∞ as a satisfying computational proof system in itself, rather than just anintermediate device in the study of finitary proof systems

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

Logics based on the µ-calculus are used to model induc-tive 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. Inductive and coinductive reasoning is pervasive in computer science to specify and reason about infinite data as well as reactive properties. Developing appropriate proof systems amenable to automated reasoning over (co)inductive statements is therefore important for designing programs as well as for analyzing computational systems. Various logical settings have been introduced to reason about such inductive and coinductive statements, both at the level of the logical languages modelling (co)induction (such as Martin Löf's inductive predicates or fixed-point logics, also known as µ-calculi) and at the level of the proof-theoretical framework considered (finite proofs with explicit (co)induction rulesà la Park [23] or infinite, non-wellfounded proofs with fixed-point unfold-ings) [6-8, 4, 1, 2]. Moreover, such proof systems have been considered over classical logic [6, 8], intuitionistic logic [9], linear-time or branching-time temporal logic [19, 18, 25, 26, 13-15] or linear logic [24, 16, 4, 3, 14]