Inductive and Functional Types in Ludics

Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation

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

Bipolar Proof Nets for MALL

In this work we present a computation paradigm based on a concurrent and incremental construction of proof nets (de-sequentialized or graphical proofs) of the pure multiplicative and additive fragment of Linear Logic, a resources conscious refinement of Classical Logic. Moreover, we set a correspon- dence between this paradigm and those more pragmatic ones inspired to transactional or distributed systems. In particular we show that the construction of additive proof nets can be interpreted as a model for super-ACID (or co-operative) transactions over distributed transactional systems (typi- cally, multi-databases).Comment: Proceedings of the "Proof, Computation, Complexity" International Workshop, 17-18 August 2012, University of Copenhagen, Denmar

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]

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

Dancing before the Lord: Renaissance Ludics and Incarnational Discourse

Play is a manifestation of overflowing excess. When applied to the study of discourse, this bounty can be understood in terms of figurativeness and depth. If “degree-zero” discourse is the almost entirely unfigured language of an instruction manual, then verse lies near the other extreme: highly figured and elaborate language open to rich interpretive possibilities. I posit a further pole yet on this continuum: the hyperabundant texts of the Renaissance, when ludics were at a height partially quashed by the Enlightenment preference for the plain style. These ludic texts are not merely decorative but rather reflect the incarnational impulse of Renaissance Christian thought; they attempt to praise and to imitate the power of Divine language, in which Word is made Flesh in the West’s master model of superabundance, grace through Christ’s Incarnation and Sacrifice. This project conducts three case studies of playfully incarnational discourses during the Renaissance: in speech, in imagery, and in verse. First, it analyzes sermons by John Donne that reflect candidly on the power of Donne’s own ludic speech, concluding that his transgressive, gamelike rhetoric was oriented toward stimulating responsive action. Next, it examines period images through the lens of contemporary popular works that conceive of images as puzzles to be decoded, solved, and read, concluding that period anamorphoses and similar works were efforts to infuse images with lively presence in a way that helps to account for iconophobic and iconophilic strains in English Reformation thought. Finally, it reads George Herbert’s deceptively simple poem, “The Altar,” examining how the piece may be understood as an intervention into the shaped-verse tradition and how it reflects on period debates about Church fabric, concluding that the toylike or tricklike construction evokes the Eucharistic presence of the Divine in Herbert’s worshipful meditation. At stake are a greater appreciation for Renaissance artistry, a fuller understanding of the complexity of the English Reformation, and a richer vocabulary for play theorists working with ludic discourses. A conclusion considers these implications and explains why Renaissance thinkers might have chosen a ludic mode of imitative worship—God’s grace and creation are themselves forms of play

Writing Verdicts: French And Francophone Narratives Of Race And Racism

Inspired by Didier Eribon\u27s La société comme verdict, this dissertation examines how the novelistic representation of racism and racialization in French-language texts can push back against a collective social verdict in France that stigmatizes non-white peoples as lesser and as other. Though discussing the existence of race is still taboo in France, I show that this stigmatization is in fact a racial verdict, or one that operates through racialization, as opposed to a verdict predicated on sexuality, gender, or class. To do so, I analyze the representation of racial hierarchies and the experience of racism in six novels written in French: Daniel Biyaoula\u27s L\u27Impasse, Gisèle Pineau\u27s L\u27Exil selon Julia, Leïla Sebbar\u27s Le Chinois vert d\u27Afrique, Zahia Rahmani\u27s Musulman roman, Cyril Bedel\u27s Sale nègre, and Bessora\u27s 53 cm. As I argue, these authors resist racial verdicts by writing about and examining the place of the ethnic minority individual in French society. In so doing they also issue counter verdicts that condemn society, individuals, and the state for their complicity in maintaining an unjust status quo. I first demonstrate that the racial hierarchies introduced during the colonial era to justify the exploitation and domination of non-white peoples continue to mark French society, operating as a collective societal verdict that marks those perceived as colored according to the sign of stigma. This judgment is first and foremost tied to phenotype–the inescapable physical body–but also includes fluid markers like religion, speech, dress and culture. Second, I examine this racial verdict through three lenses– blackness, arabness, and whiteness –interrogating both how the authors present its impact on society and individuals–, thus proving its very existence–and how they refute it. Once exposed, verdict is countered through the rehabilitation of stigmatized identities such as blackness, the indictment of the impulse to categorize individuals based on race, the condemnation of the French state for its exclusion of its own citizens, and the revelation that whiteness–the source of the racial hierarchy–is both colored and founded on emptiness