Pierre Clairambault

The biequivalence of locally cartesian closed categories and Martin-Löf type theories

Seelys paper Locally cartesian closed categories and type theory (Seely 1984) contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Lof type theories with Pi, Sigma and extensional identity types. However, Seelys proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seelys theorem: that the Benabou-Hofmann interpretation of Martin-Lof type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development, we employ categories with families as a substitute for syntactic Martin-Lof type theories. As a second result, we prove that if we remove Pi-types, the resulting categories with families with only Sigma and extensional identity types are biequivalent to left exact categories.

Symmetry in concurrent games

Behavioural symmetry is introduced into concurrent games. It expresses when plays are essentially the same. A characterization of strategies on games with symmetry is provided. This leads to a bi-category of strategies on games with symmetry. Symmetry helps allay the perhaps overly-concrete nature of games and strategies, and shares many mathematical features with homotopy. In the presence of symmetry we can consider monads for which the monad laws do not hold on the nose but do hold up to symmetry. This broadening of the concept of monad has a dramatic effect on the types concurrent games can support and allows us, for example, to recover the replication needed to express and extend traditional game semantics.

The Parallel Intensionally Fully Abstract Games Model of PCF

We describe a framework for truly concurrent game semantics of programming languages, based on Rideau and Winskels concurrent games on event structures. The model supports a notion of innocent strategy that permits concurrent and non-deterministic behaviour, but which coincides with traditional Hyland-Ong innocent strategies if one restricts to the deterministic sequential case. In this framework we give an alternative interpretation of Plot kins PCF, that takes advantage of the concurrent nature of strategies and formalizes the idea that although PCF is a sequential language, certain sub-computations are independent and can be computed in a parallel fashion. We show that just as Hyland and Ongs sequential interpretation of PCF, our parallel interpretation yields a model that is intensionally fully abstract for PCF.

Imperfect Information in Logic and Concurrent Games

This paper builds on a recent definition of concurrent games as event structures and an application giving a concurrent-game model for predicate calculus. An extension to concurrent games with imperfect information, through the introduction of ‘access levels’ to restrict the allowable strategies, leads to a concurrent-game semantics for a variant of Hintikka and Sandu’s Independence-Friendly (IF) logic.

Böhm Trees as Higher-Order Recursive Schemes

Higher-order recursive schemes (HORS) are schematic representations of functional programs. They generate possibly infinite ranked labelled trees and, in that respect, are known to be equivalent to a restricted fragment of the lambda-Y-calculus consisting of ground-type terms whose free variables have types of the form o -> ... -> o (with o being a special case). In this paper, we show that any lambda-Y-term (with no restrictions on term type or the types of free variables) can actually be represented by a HORS. More precisely, for any lambda-Y-term M, there exists a HORS generating a tree that faithfully represents Ms (eta-long) Bohm tree. In particular, the HORS captures higher-order binding information contained in the Bohm tree. An analogous result holds for finitary PCF. As a consequence, we can reduce a variety of problems related to the lambda-Y-calculus or finitary PCF to problems concerning higher-order recursive schemes. For instance, Bohm tree equivalence can be reduced to the equivalence problem for HORS. Our results also enable MSO model-checking of Bohm trees, despite the general undecidability of the problem.

Bounding Skeletons, Locally Scoped Terms and Exact Bounds for Linear Head Reduction

Bounding skeletons were recently introduced as a tool to study the length of interactions in Hyland/Ong game semantics. In this paper, we investigate the precise connection between them and execution of typed λ-terms. Our analysis sheds light on a new condition on λ-terms, called local scope. We show that the reduction of locally scoped terms matches closely that of bounding skeletons. Exploiting this connection, we give upper bound to the length of linear head reduction for simply-typed locally scoped terms. General terms lose this connection to bounding skeletons. To compensate for that, we show that λ-lifting allows us to transform any λ-term into a locally scoped one. We deduce from that an upper bound to the length of linear head reduction for arbitrary simply-typed λ-terms. In both cases, we prove the asymptotical optimality of the upper bounds by providing matching lower bounds.

On Concurrent Games with Payoff

The paper considers an extension of concurrent games with a payoff, i.e. a numerical value resulting from the interaction of two players. We extend a recent determinacy result on concurrent games [Pierre Clairambault, Julian Gutierrez, and Glynn Winskel. The winning ways of concurrent games. In LICS. IEEE Computer Society, 2012] to a value theorem, i.e. a value that both players can get arbitrarily close to, whatever the behaviour of their opponent. This value is not reached in general, i.e. there is not always an optimal strategy for one of the players (there is for finite games). However when they exist, we show that optimal strategies are closed under composition, which opens up the possibility of computing optimal strategies for complex games compositionally from optimal strategies for their component games.

Causality vs. interleavings in concurrent game semantics

We investigate relationships between interleaving and causal notions of game semantics for concurrent programming languages, focusing on the existence of canonical compact causal representations of the interleaving game semantics of programs. We perform our study on an affine variant of Idealized Parallel Algol (IPA), for which we present two games model: and interleaving model (an adaptation of Ghica and Murawski’s fully abstract games model for IPA up to may-testing), and a causal model (a variant of Rideau and Winskel’s games on event structures). Both models are sound and adequate for affine IPA. Then, we relate the two models. First we give a causality-forgetting operation mapping functorially the causal model to the interleaving one. We show that from an interleaving strategy we can reconstruct a causal strategy, from which it follows that the interleaving model is the observational quotient of the causal one. Then, we investigate several reconstructions of causal strategies from interleaving ones, showing finally that there are programs which are inherently causally ambiguous, with several distinct minimal causal representations.

Bounding linear head reduction and visible interaction through skeletons

In this paper, we study the complexity of execution in higher-order programming languages. Our study has two facets: on the one hand we give an upper bound to the length of interactions between bounded P-visible strategies in Hyland-Ong game semantics. This result covers models of programming languages with access to computational effects like non-determinism, state or control operators, but its semantic formulation causes a loose connection to syntax. On the other hand we give a syntactic counterpart of our semantic study: a non-elementary upper bound to the length of the linear head reduction sequence (a low-level notion of reduction, close to the actual implementation of the reduction of higher-order programs by abstract machines) of simply-typed lambda-terms. In both cases our upper bounds are proved optimal by giving matching lower bounds. These two results, although different in scope, are proved using the same method: we introduce a simple reduction on finite trees of natural numbers, hereby called interaction skeletons. We study this reduction and give upper bounds to its complexity. We then apply this study by giving two simulation results: a semantic one measuring progress in game-theoretic interaction via interaction skeletons, and a syntactic one establishing a correspondence between linear head reduction of terms satisfying a locality condition called local scope and the reduction of interaction skeletons. This result is then generalized to arbitrary terms by a local scopization transformation.

Strong functors and interleaving fixpoints in game semantics

We describe a sequent calculus μLJ with primitives for inductive and coinductive datatypes and equip it with reduction rules allowing a sound translation of Godel’s system T. We introduce the notion of a μ -closed category , relying on a uniform interpretation of open μLJ formulas as strong functors. We show that any μ -closed category is a sound model for μLJ . We then turn to the construction of a concrete μ -closed category based on Hyland-Ong game semantics. The model relies on three main ingredients: the construction of a general class of strong functors called open functors acting on the category of games and strategies, the solution of recursive arena equations by exploiting cycles in arenas, and the adaptation of the winning conditions of parity games to build initial algebras and terminal coalgebras for many open functors. We also prove a weak completeness result for this model, yielding a normalisation proof for μLJ .