Paul-André Melliès

• Senior Researcher at French National Centre for Scientific Research

In his recent and exploratory work on template games and linear logic, Melli\`es defines sequential and concurrent games as categories with positions as objects and trajectories as morphisms, labelled by a specific synchronization template. In the present paper, we bring the idea one dimension higher and advocate that template games should not be j...

Lawvere observed in his celebrated work on hyperdoctrines that the set-theoretic schema of comprehension can be elegantly expressed in the functorial language of categorical logic, as a comprehension structure on the functor $p:\mathscr{E}\to\mathscr{B}$ defining the hyperdoctrine. In this paper, we formulate and study a strictly ordered hierarchy...

An old dream of concurrency theory and programming language semantics has been to uncover the fundamental synchronization mechanisms which regulate situations as different as game semantics for higher-order programs, and Hoare logic for concurrent programs with shared memory and locks. In this paper, we establish a deep and unexpected connection be...

Game semantics is the art of interpreting types as games and programs as strategies interacting in space and time with their environment. In order to reflect the interactive behavior of programs, strategies are required to follow specific scheduling policies. Typically, in the case of a purely sequential programming language, the program (Player) a...

Concurrent separation logic (CSL) is a specification logic for concurrent imperative programs with shared memory and locks. In this paper, we develop a concurrent and interactive account of the logic inspired by asynchronous game semantics. To every program $C$, we associate a pair of asynchronous transition systems $[C]_S$ and $[C]_L$ which descri...

Concurrent separation logic (CSL) is a specification logic for concurrent imperative programs with shared memory and locks. In this paper, we develop a concurrent and interactive account of the logic inspired by asynchronous game semantics. To every program C, we associate a pair of asynchronous transition systems [C]S and [C]L which describe the o...

We introduce a topologically-aware version of tensorial logic, called ribbon tensorial logic. To every proof of the logic, we associate a ribbon tangle which tracks the flow of tensorial negations inside the proof. The translation is functorial: it is performed by exhibiting a correspondence between the notion of dialogue category in proof theory a...

The purpose of this paper is to define in a clean and conceptual way a non-deterministic and sheaf-theoretic variant of the category of simple games and deterministic strategies. One thus starts by associating to every simple game a presheaf category of non-deterministic strategies. The bicategory of simple games and non-deterministic strategies is...

Dialogue games were introduced by Melliès as an attempt to unify two historical paradigms of game semantics: concrete data structures and arena games. The definition of dialogue games relies on the idea that a move m of an arena game can be decomposed as a pair m=(α,v) consisting of a cell α and of a value v. Consequently, a dialogue game is define...

In this paper, we develop a game-theoretic account of concurrent separation logic. To every execution trace of the Code confronted to the Environment, we associate a specification game where Eve plays for the Code, and Adam for the Environment. The purpose of Eve and Adam is to decompose every intermediate machine state of the execution trace into...

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family of model structures on the fibers $\mathcal E_A$ and on the basis category $\mathcal B$ combines into a model...

The exponential modality of linear logic associates to every formula A a commutative comonoid ! A which can be duplicated in the course of reasoning. Here, we explain how to compute the free commutative comonoid ! A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor...

Tensorial logic is a primitive logic of tensor and negation which refines linear logic by relaxing the hypothesis that linear negation is involutive. Thanks to this mild modification, tensorial logic provides a type-theoretic account of game semantics where innocent strategies are portrayed as temporal refinements of traditional proof-nets in linea...

In this invited talk, I will review five basic concepts of Axiomatic Rewriting Theory, an axiomatic and diagrammatic theory of rewriting started 25 years ago in a LICS paper with Georges Gonthier and Jean-Jacques L\'evy, and developed along the subsequent years into a full-fledged 2-dimensional theory of causality and residuation in rewriting. I wi...

Combining insights from the study of type refinement systems and of monoidal closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of presheaves using a full and faithful embedding into a monoidal closed bifibration living now over the compact closed category of small categories and distributors. Besides revealing dualities which a...

We initiate a formal theory of graded monads whose purpose is to adapt and to extend the formal theory of monads developed by Street in the early 1970’s. We establish in particular that every graded monad can be factored in two different ways as a strict action transported along a left adjoint functor. We also explain in what sense the first constr...

Combining insights from the study of type refinement systems and of monoidal closed chiralities, we show how to reconstruct Lawvere's hyperdoctrine of presheaves using a full and faithful embedding into a monoidal closed bifibration living now over the compact closed category of small categories and distributors. Besides revealing dualities which a...

In this paper, we consider a two-sided notion of dialogue category – called dialogue chirality – formulated as an adjunction between a mo-noidal category A of proofs and a monoidal category B of refutations equivalent to its opposite category A op(0,1) . The two-sided formulation is compared to the original one-sided formulation of dialogue categor...

Every dialogue category comes equipped with a continuation monad defined by applying the negation functor twice. In this paper, we advocate that this double negation monad should be understood as part of a larger parametric monad (or a lax action) with parameter taken in the opposite of the dialogue category. This alternative point of view has one...

Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming and *context effects* in linear logic, we study logical aspects of thi...

In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate intersection type system associated to the automaton A. The purpose of this article is to establish a clean con...

This article is concerned with semantic methods for higher-order verification. Given an alternating parity automaton A, we introduce a variant of the Scott semantics of linear logic extended with a colouring modality and a fixed point operator. We prove that the interpretation of a higher-order recursion scheme in the resulting model of the lambdaY...

We revisit the type-theoretic account of higher-order model-checking by Kobayashi and Ong in the light of linear logic and of its relational semantics. We start from the well-known fact that every higher-order recursion scheme (HORS) on a given signature may be seen after Church encoding as a lambda Y-term of a given second-order type. We then obse...

The standard reading of type theory through the lens of category theory is based on the idea of viewing a type system as a category of well-typed terms. We propose a basic revision of this reading: rather than interpreting type systems as categories, we describe them as functors from a category of typing derivations to a category of underlying term...

The standard reading of type theory through the lens of category theory is based on the idea of viewing a type system as a category of well-typed terms. We propose a basic revision of this reading: rather than interpreting type systems as categories, we describe them as functors from a category of typing derivations to a category of underlying term...

In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the re...

We establish that the local state monad introduced by Plotkin and Power is a monad with graded arities in the category [Inj,Set]. From this, we deduce that the local state monad is associated to a graded Lawvere theory which is presented by generators and relations, depicted in the graphical language of string diagrams.

The concept of_refinement_ in type theory is a way of reconciling the "intrinsic" and the "extrinsic" meanings of types. We begin with a rigorous analysis of this concept, settling on the simple conclusion that the type-theoretic notion of "type refinement system" may be identified with the category-theoretic notion of "functor". We then use this c...

We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done...

About ten years ago, B. Day and R. Street [Fields Inst. Commun. 43, 187–225 (2004; Zbl 1067.18006)] discovered a beautiful and unexpected connection between the notion of *-autonomous category in proof theory and the notion of Frobenius algebra in mathematical physics. The purpose of the present paper is to clarify the logical content of this conne...

A dialogue category is a symmetric monoidal category equipped with a notion of tensorial negation. We establish that the free dialogue category is a category of dialogue games and total innocent strategies. The connection clarifies the algebraic and logical nature of dialogue games, and their intrinsic connection to linear continuations. The proof...

After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As an appl...

Every finitary monad T on the category of sets is described by an algebraic theory whose n-ary operations are the elements of the free algebra Tn generated by n letters. This canonical presentation of the monad (called its Lawvere theory) offers a precious guideline in the search for an intuitive presentation of the monad by generators and relation...

The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of...

We study the algebraic structure of a programming language with higher-order store, in the style of ML references. Instead of working directly on the operational semantics of the language, we consider its fully abstract game semantics defined by Abramsky, Honda and McCusker one decade ago. This alternative description of the language is nice and co...

The exponential modality of linear logic associates a commutative comonoid !A to every formula A in order to duplicate it. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We then apply this...

One fundamental aspect of Lawvere's categorical semantics is that every algebraic theory (eg. of monoid, of Lie algebra) induces a free construction (eg. of free monoid, of free Lie algebra) computed as a Kan extension. Unfortunately, the principle fails when one shifts to linear variants of algebraic theories, like Adams and Mac Lane's PROPs, and...

We show how to construct the category of games and innocent strategies from a more primitive category of games. On that category we define a comonad and monad with the former distributing over the latter. Innocent strategies are the maps in the induced two-sided Kleisli category. Thus the problematic composition of innocent strategies reflects the...

The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of a misleading conception: the belief that linear logic is more primitive than game semantics. We advocate the contrary here: that game semantics is conceptually more primitive than linear logic. Starting from this revised point...

The notion of innocent strategy was introduced by Hyland and Ong in order to capture the interactive behaviour of lambda-terms and PCF programs. An innocent strategy is defined as an alternating strategy with partial memory, in which the strategy plays according to its view. Extending the definition to non-alternating strategies is problematic, bec...

We present a model of recursive and impredicatively quan- tied types with mutable references. We interpret in this model all of the type constructors needed for typed inter- mediate languages and typed assembly languages used for object-oriented and functional languages. We establish in this purely semantic fashion a soundness proof of the typing s...

One fundamental aspect of linear logic is that its conjunction behaves in the same way as a tensor product in linear algebra. Guided by this intuition, we investigate the algebraic status of disjunction -- the dual of conjunction -- in the presence of linear continuations. We start from the observation that every monoidal category equipped with a t...

We present a simply typed -term whose computation in the -calculus does not always terminate.

String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in proof-theory (like Jean-Yves Girard’s proof-nets) and in concurrency theory (like Robin Milner’s bigraphs). This is not the case howev...

In game semantics, the higher-order value passing mechanisms of the λ-calculus are decomposed as sequences of atomic actions exchanged by a Player and its Opponent. Seen from this angle, game semantics is reminiscent of trace semantics in concurrency theory, where a process is identified to the sequences of requests it generates in the course of ti...

Algebraic type systems provide a general framework for the study of the interaction between typed -calculi and typed rewriting systems. A major problem in the development of a general theory for algebraic type systems is to prove that typing is preserved under reduction (Subject Reduction lemma). In this paper, we propose a general technique to pro...

Intuitionistic proofs and PCF programs may be interpreted as functions between domains, or as strategies on games. The two kinds of interpretation are inherently different: static vs. dynamic, extensional vs. intentional. It is thus extremely instructive to compare and to connect them. In this article, we investigate the extensional content of the...

We construct a realizability model of recursive polymorphic types, starting from an untyped language of terms and contexts. An orthogonality relation e⊥π indicates when a term e and a context π may be safely combined in the language. Types are interpreted as sets of terms closed by biorthogonality. Our main result states that recursive types are ap...

We construct a denotational model of propositional linear logic based on asynchronous games and winning uniform innocent strategies. Every formula A is interpreted as an asynchronous game [A] and every proof π of A is interpreted as a winning uniform innocent strategy [π] of the game [A]. We show that the resulting model is fully complete: every wi...

Since its early days, deterministic sequential game semantics has been limited to linear or polarized fragments of linear logic. Every attempt to extend the semantics to full propositional linear logic has bumped against the so-called Blass problem, which indicates (misleadingly) that a category of sequential games cannot be self-dual and cartesian...

By extending nondeterministic transition systems with concurrency and copy mechanisms, Axiomatic Rewriting Theory provides a uniform framework for a variety of rewriting systems, ranging from higher-order systems to Petri nets and process calculi. Despite its generality, the theory is surprisingly simple, based on a mild extension of transition sys...

In game semantics, one expresses the higher-order value passing mechanisms of the λ-calculus as sequences of atomic actions exchanged by a Player and its Opponent in the course of time. This is reminiscent of trace semantics in concurrency theory, in which a process is identified to the sequences of requests it generates. We take as working hypothe...

We show that two models and of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories and are related by a pair of monoidal functors and transformations and , and (2) their exponentials and are related by distributive laws and commuting to the promotion rule. The key ingredient of the proof is a notion...

We present a generalization of the ideal model for recursive polymorphic types. Types are defined as sets of terms instead of sets of elements of a semantic domain. Our proof of the existence of types (computed by fixpoint of a typing operator) does not rely on metric properties, but on the fact that the identity is the limit of a sequence of proje...

We formulate Girard's long trip criterion for multiplicative linear logic(MLL) in a topological way, by associating a ribbon diagram to everyswitching, and requiring that it is homeomorphic to the disk. Then, weextend the well-known planarity criterion for multiplicative cyclic linearlogic (McyLL) to multiplicative non-commutative logic (MNL) and s...

We present a generalization of the ideal model for recursive polymorphic types. Types are defined as sets of terms instead of sets of elements of a semantic domain. Our proof of the existence of types (computed by fixpoint of a typing operator) does not rely on metric properties, but on the fact that the identity is the limit of a sequence of proje...

In this draft manuscript, we reduce the coherence theorem for braided monoidal categories to the resolution of a word problem, and the construction of a category of fractions. The technique explicates the combinatorial nature of that particular coherence theorem.

We construct a double category [script D] of proof-nets in multiplicative linear logic (MLL). Its horizontal arrows are MLL modules (subnets of well-formed nets), its vertical arrows model side-effects, and its double cells interpret the cut-elimination procedure. The categorical model is modular in the sense that every computation of a composite m...

Residual theory is the algebraic theory of confluence for the λ-calculus, and more generally conflict-free rewriting systems (=without critical pairs). The theory took its modern shape in Lévy’s PhD thesis, after Church, Rosser and Curry’s seminal steps. There, Lévy introduces a permutation equivalence between rewriting paths, and establishes that...

In this survey, we review the existing categorical axiomatizations of linear logic, with a special emphasis on Seely and Lafont presentations. In a first part, we explain why Benton, Bierman, de Paiva and Hyland had to replace Seely categories by a more complicated axiomatization, and how a while later, Benton managed to simplify this axiomatizatio...

Every needed strategy is normalizing in the &lgr;-calculus. Here, we extend the result to the &lgr;&sgr;-calculus, a &lgr;-calculus with explicit substitutions. The extension requires considering rewriting systems with critical pairs, confluent or non-confluent, and developing for them a satisfactory theory of needed normalization. Our idea is to c...

A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is proved for this semantics. 1 Introduction This pap...

A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for Multiplicative-Additive Linear Logic is proved for this semantics

One key property of the λ-calculus is that there exists a minimal computation (the head-reduction) M→^eV from a λ-term M to the set of its head-normal forms. Minimality here means categorical “reflectivity” i.e. that every reduction path M→^fW to a head-normal form W factors (up to redex permutation) to a path M→^eV→<s...

. The article is mainly concerned with the Kruskal tree theorem and the following observation: there is a duality at the level of binary relations between well and noetherian orders. The first step here is to extend Kruskal theorem from orders to binary relations so that the duality applies. Then, we describe the theorem obtained by duality and sho...

Some computations on a symbolic term M are more judicious than others, for instance the leftmost outermost derivations in the λ-calculus. In order to characterise generically that kind of judicious computations, [M] introduces the notion of external derivations in its axiomatic description of Rewriting Systems: a derivation e : M → P is said to be...

This paper is concerned with three methodologies to construct a congruence in a language (L, O) among which Frege's specific solution stands. The first construction is derived from Leibniz's principle that indiscernible objects should be identified. The second construction is apparented to Frege's solution by its use of a referential model. The thi...

. Some computations on a symbolic term M are more judiciousthan others, for instance the leftmost outermost derivations in the-calculus. In order to characterise generically that kind of judiciouscomputations, [M] introduces the notion of external derivations in itsaxiomatic description of Rewriting Systems: a derivation e : M \Gamma! Pis said to b...

We prove the strong normalisation for any PTS, provided the existence of a certain-setA*(s) for every sort s of the system. The properties verified by the A*(s)"s depend of theaxiom and rules of the type system.1 Introduction1.1 Brief HistoryThis work is an attempt to deal with the structure of complex Type Theories. Historically,once Girard had tr...

An axiomatic version of the standardization theorem that shows the necessary basic properties between nesting of redexes and residuals is presented. This axiomatic approach provides a better understanding of standardization, and makes it applicable in other settings, such as directed acyclic graphs (dags) or interaction networks. conflicts between...

Proof theory is the result of a short and tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyn- cratic: sequent calculus, cut-elimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematician along a smooth and consistent path, investig...

We present a formalism for describing categories equipped with extra structure that involves covariant and contravariant functors re- lated by dinatural transformations. A typical example is the concept of 'cartesian closed category', or more generally 'monoidal closed cat- egory'.

This article opens a series of papers on asynchronousgames semantics, which aims at a concurrent and geometric account of interference andstates in programming languages. In order to develop our theory, we need to reformulatearena games in a simpler algebraic vocabulary, inspired by Girard's Geometry of Interactionand Abramsky, Jagadeesan and Malac...

One key property of the -calculus is that thereexists a minimal computation (the head-reduction)Me\Gamma! V from a -term M to the set of its headnormalforms. Minimality here means categorical "reflectivity" i.e. that every reduction path Mf\Gamma! Wto a head-normal form W factors (up to redex permutation)to a path Me\Gamma! Vh\Gamma! W . This paper...