We explain how to construct in two different ways a cartesian closed fibration of higher-order regular languages in the sense of Salvati. In the first construction, we use fibrational techniques to derive the cartesian closed fibration from the various categories of regular languages of λ-terms associated to finite sets of ground states. In the second construction, we take advantage of the recent notion of profinite λ-calculus to define the cartesian closed fibration by a change-of-base from the fibration of clopen subsets over the category of Stone spaces, using an elegant idea coming from Hermida. We illustrate the expressive power of the cartesian closed fibration by generalizing the notion of Brzozowski derivative to higher-order regular languages, using an Isbell-like adjunction in the sense of Melliès and Zeilberger.

We develop fibrational perspectives on context-free grammars and on nondeterministic finite-state automata over categories and operads. A generalized CFG is a functor from a free colored operad (aka multicategory) generated by a pointed finite species into an arbitrary base operad: this encompasses classical CFGs by taking the base to be a certain operad constructed from a free monoid, as an instance of a more general construction of an \emph{operad of spliced arrows} $\mathcal{W}\,\mathcal{C}$ for any category $\mathcal{C}$. A generalized NFA is a functor from an arbitrary bipointed category or pointed operad satisfying the unique lifting of factorizations and finite fiber properties: this encompasses classical word automata and tree automata without $\epsilon$-transitions, but also automata over non-free categories and operads. We show that generalized context-free and regular languages satisfy suitable generalizations of many of the usual closure properties, and in particular we give a simple conceptual proof that context-free languages are closed under intersection with regular languages. Finally, we observe that the splicing functor $\mathcal{W} : Cat \to Oper$ admits a left adjoint $\mathcal{C}: Oper \to Cat$, which we call the \emph{contour category} construction since the arrows of $\mathcal{C}\,\mathcal{O}$ have a geometric interpretation as oriented contours of operations of $\mathcal{O}$. A direct consequence of the contour / splicing adjunction is that every pointed finite species induces a universal CFG generating a language of \emph{tree contour words.} This leads us to a generalization of the Chomsky-Sch\"utzenberger Representation Theorem, establishing that a subset of a homset $L \subseteq \mathcal{C}(A,B)$ is a CFL of arrows if and only if it is a functorial image of the intersection of a $\mathcal{C}$-chromatic tree contour language with a regular language.

Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product.

The language of Algebraic Geometry combines two complementary and dependent levels of discourse: on the geometric side, schemes define spaces of the same cohesive nature as manifolds ; on the vectorial side, every scheme X comes equipped with a symmetric monoidal category of quasicoherent modules, which may be seen as generalised vector bundles on the scheme X. In this paper, we use the functor of points approach to Algebraic Geometry developed by Grothendieck in the 1970s to establish that every covariant presheaf X on the category of commutative rings — and in particular every scheme X — comes equipped “above it” with a symmetric monoidal closed category PshModX of presheaves of modules. This category PshModX defines moreover a model of intuitionistic linear logic, whose exponential modality is obtained by glueing together in an appropriate way the Sweedler dual construction on ring algebras.

Large-scale software verification relies critically on the use of compositional languages, semantic models, specifications, and verification techniques. Recent work on certified abstraction layers synthesizes game semantics, the refinement calculus, and algebraic effects to enable the composition of heterogeneous components into larger certified systems. However, in existing models of certified abstraction layers, compositionality is restricted by the lack of encapsulation of state. In this paper, we present a novel game model for certified abstraction layers where the semantics of layer interfaces and implementations are defined solely based on their observable behaviors. Our key idea is to leverage Reddy's pioneer work on modeling the semantics of imperative languages not as functions on global states but as objects with their observable behaviors. We show that a layer interface can be modeled as an object type (i.e., a layer signature) plus an object strategy. A layer implementation is then essentially a regular map, in the sense of Reddy, from an object with the underlay signature to that with the overlay signature. A layer implementation is certified when its composition with the underlay object strategy implements the overlay object strategy. We also describe an extension that allows for non-determinism in layer interfaces. After formulating layer implementations as regular maps between object spaces, we move to concurrency and design a notion of concurrent object space, where sequential traces may be identified modulo permutation of independent operations. We show how to express protected shared object concurrency, and a ticket lock implementation, in a simple model based on regular maps between concurrent object spaces.

In his recent and exploratory work on template games and linear logic, Melliès 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 just defined as 1-dimensional categories but as 2-dimensional categories of positions, trajectories and reshufflings (or reschedulings) as 2-cells. In order to achieve the purpose, we take seriously the parallel between asynchrony in concurrency and the Gray tensor product of 2-categories. One technical difficulty on the way is that the category $\mathbb{S} = 2$-Cat of small 2-categories equipped with the Gray tensor product is monoidal, and not cartesian. This prompts us to extend the framework of template games originally formulated by Melliès in a category $\mathbb{S}$ with finite limits, and to upgrade it in the style of Aguiar’s work on quantum groups to the more general situation of a monoidal category $\mathbb{S}$ with coreflexive equalizers, preserved by the tensor product componentwise. We construct in this way an asynchronous template game semantics of multiplicative additive linear logic (MALL) where every formula and every proof is interpreted as a labelled 2-category equipped, respectively, with the structure of Gray comonoid for asynchronous template games, and of Gray bicomodule for asynchronous strategies.

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 of three notions of comprehension structure on a given functor $p:\mathscr{E}\to\mathscr{B}$, which we call (i) comprehension structure, (ii) comprehension structure with section, and (iii) comprehension structure with image. Our approach is 2-categorical and we thus formulate the three levels of comprehension structure on a general morphism $p:\mathrm{\mathbf{E}}\to\mathrm{\mathbf{B}}$ in a 2-category $\mathscr{K}$. This conceptual point of view on comprehension structures enables us to revisit the work by Fumex, Ghani and Johann on the duality between comprehension structures and quotient structures on a given functor $p:\mathscr{E}\to\mathscr{B}$. In particular, we show how to lift the comprehension and quotient structures on a functor $p:\mathscr{E}\to\mathscr{B}$ to the categories of algebras or coalgebras associated to functors $F_{\mathscr{E}}:\mathscr{E}\to\mathscr{E}$ and $F_{\mathscr{B}}:\mathscr{B}\to\mathscr{B}$ of interest, in order to interpret reasoning by induction and coinduction in the traditional language of categorical logic, formulated in an appropriate 2-categorical way.

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. We establish a deep and unexpected connection between two recent lines of work on concurrent separation logic (CSL) and on template game semantics for differential linear logic (DiLL). Thanks to this connection, we reformulate in the purely conceptual style of template games for DiLL the asynchronous and interactive interpretation of CSL designed by Melliès and Stefanesco in a recent work. We believe that the analysis reveals something important about the secret anatomy of CSL, and more specifically about the subtle interplay, of a categorical nature, between sequential composition, parallel product, errors and locks.

We extend our recent template game model of multiplicative additive linear logic (MALL) with an exponential modality of linear logic (LL) derived from the standard categorical construction Sym of the free symmetric monoidal category. We obtain in this way the first game semantics of differential linear logic (DiLL) in its classical form. The construction of the model relies on a careful and healthy comparison with the model of generalised species designed ten years ago by Fiore, Gambino, Hyland and Winskel. Besides the resolution of an old open problem of game semantics, the study reveals an unexpected and promising convergence between linear logic and homotopy theory.

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) and its environment (Opponent) will play one after the other, in a strictly alternating way. On the other hand, in the case of a concurrent language, Player and Opponent will be allowed to play several moves in a row, in a non-alternating way. In both cases, the scheduling policy is designed very carefully in order to ensure that the strategies synchronize properly and compose well when plugged together. A longstanding conceptual problem has been to understand when and why a given scheduling policy works and is compositional in that sense. In this paper, we exhibit a number of simple and fundamental combinatorial structures which ensure that a given scheduling policy encoded as synchronization template defines a symmetric monoidal closed (and in fact star-autonomous) bicategory of games, strategies and simulations. To that purpose, we choose to work at a very general level, and illustrate our method by constructing two template game models of linear logic with different flavors (alternating and non-alternating) using the same categorical combinatorics, performed in the category of small categories. As a whole, the paper may be seen as a hymn in praise of synchronization, building on the notion of synchronization algebra in process calculi and adapting it smoothly to programming language semantics, using a combination of ideas at the converging point of game semantics and of categorical algebra.

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 then obtained by a construction inspired by the recent work by Melliès and Zeilberger on type refinement systems. We show that the resulting bicategory is symmetric monoidal closed and cartesian. We also define a 2-comonad which adapts the Curien-Lamarche exponential modality of linear logic to the 2-dimensional and non deterministic framework. We conclude by discussing in what sense the bicategory of simple games defines a model of non deterministic intuitionistic linear logic.

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 and the notion of ribbon category in knot theory. Our main result is that the translation is also faithful: two proofs are equal modulo the equational theory of ribbon tensorial logic if and only if the associated ribbon tangles are equal up to topological deformation. This "proof-as-tangle" theorem may be understood as a coherence theorem for balanced dialogue categories, and as a mathematical foundation for topological game semantics.

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 operational behavior of the Code when confronted to its Environment or Frame --- both at the level of machine states (S) and of machine instructions and locks (L). We then establish that every derivation tree π of a judgment Γ ⊢ {P}C{Q} defines a winning and asynchronous strategy [π]Sep with respect to both asynchronous semantics [C]S and [C]L. From this, we deduce an asynchronous soundness theorem for CSL, which states that the canonical map ℒ: [C]S~[C]L, from the stateful semantics [C]S to the stateless semantics [C]L satisfies a basic fibrational property. We advocate that this provides a clean and conceptual explanation for the usual soundness theorem of CSL, including the absence of data races.

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 product. We apply this general recipe to a series of models of linear logic, typically based on coherence spaces, Conway games and finiteness spaces. This algebraic description unifies for the first time a number of apparently different constructions of the exponential modality in spaces and games. It also sheds light on the duplication policy of linear logic, and its interaction with classical duality and double negation completion.

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 andcontext effectsin linear logic, we study logical aspects of this ‘positive’ (covariant) representation, as well as of an associated ‘negative’ (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day's embedding theorem for monoidal closed categories. Then, we establish that the positive and negative representations satisfy an Isbell-style duality. As corollaries, we derive two different formulas for the positive representation of a pushforward (inspired by the classical negative translations of proof theory), which express it either as the dual of a pullback of a dual or as the double dual of a pushforward. Besides explaining how these constructions on refinement systems generalize familiar category-theoretic ones (by viewing categories as special refinement systems), our main running examples involve representations of Hoare logic and linear sequent calculus.

We introduce a notion of higher-order parity automaton which extends to infinitary simply-typed λ-terms the traditional notion of parity tree automaton on infinitary ranked trees. Our main result is that the acceptance of an infinitary λ-term by a higher-order parity automaton A is decidable, whenever the infinitary λ-term is generated by a finite and simply-typed λY-term. The decidability theorem is established by combining ideas coming from linear logic, from denotational semantics and from infinitary rewriting theory.

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 main conceptual benefit: it reveals that the strength of the continuation monad is the fragment of a more fundamental and symmetric structure – provided by a distributivity law between the parametric continuation monad and the canonical action of the dialogue category over itself. The purpose of this work is to describe the formal properties of this parametric continuation monad and of its distributivity law.

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 are not immediately apparent in the traditional presentation of the presheaf hyperdoctrine, this reconstruction leads us to an axiomatic treatment of directed equality predicates (modelled by hom presheaves), realizing a vision initially set out by Lawvere (1970). It also leads to a simple calculus of string diagrams (representing presheaves) that is highly reminiscent of C. S. Peirce's existential graphs for predicate logic, refining an earlier interpretation of existential graphs in terms of Boolean hyperdoctrines by Brady and Trimble. Finally, we illustrate how this work extends to a bifibrational setting a number of fundamental ideas of linear logic.

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.

One main challenge of the theory of computational effects is to understand how to combine various notions of effects in a meaningful way. Here, we study the particular case of the local state monad, which we would like to express as the result of combining together a family of global state monads parametrized by the number of available registers. To that purpose, we develop a notion of indexed monad which refines and generalizes Power's recent notion of indexed Lawvere theory. One main achievement of the paper is to integrate the block structure necessary to encode allocation as part of the resulting notion of indexed state monad. We then explain how to recover the local state monad from the functorial data provided by our notion of indexed state monad. This reconstruction is based on the guiding idea that an algebra of the indexed state monad should be defined as a section of a 2-categorical notion of fibration associated to the indexed state monad by a Grothendieck construction.

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 terms. Then, turning this around, we explain how in fact any functor gives rise to a generalized type system, with an abstract notion of typing judgment, typing derivations and typing rules. This leads to a purely categorical reformulation of various natural classes of type systems as natural classes of functors.

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 in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature.

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 of the statement is based on an algebraic presentation of dialogue categories inspired by knot theory, and a factorization theorem established by rewriting techniques.

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 relations. Hence, much work has been devoted to extend this correspondence between monads and theories to situations of semantic interest, like enriched categories and countable monads. In this paper, we clarify the conceptual nature of these extended Lawvere theories by investigating the change-of-base mechanisms which underlie them. Our starting point is the Segal condition recently established by Weber for a general notion of monad with arities. Our first step is to establish the Segal condition a second time, by reducing it to the Linton condition which characterizes the algebras of a monad as particular presheavesover the category of free algebras. This reduction is achieved by a relevant change-of-base from the category of interest to its subcategory of arities. This conceptual approach leads us to an abstract notion of Lawvere theory with arities, which extends to every class of arity the traditional correspondence in Set between Lawvere theories and finitary monads. Finally, we illustrate the benefits of Lawvere's ideas by describing how the concrete presentation of the state monad recently formulated by Plotkin and Power is ultimately validated by a rewriting property on sequences of updates and lookups.

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 use of the distributive law. The composition of simple strategies, and the combinatorics of pointers used to give the comonad and monad are themselves described in categorical terms. The notions of view and of legal play arise naturally in the explanation of the distributivity. The category-theoretic perspective provides a clear discipline for the necessary combinatorics.

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 of view, we design a categorical model of resources in game semantics, and construct an arena game model where the usual notion of bracketing is extended to multi-bracketing in order to capture various resource policies: linear, affine and exponential.

International audience

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 /spl pi/ of A is interpreted as a winning uniform innocent strategy [/spl pi/] of the game [A]. We show that the resulting model is fully complete: every winning uniform innocent strategy /spl sigma/ of the asynchronous game [A] is the denotation [/spl pi/] of a proof /spl pi/ of the formula A.

We construct a realizability model of recursive polymorphic types, starting from an untyped language of terms and contexts. An orthogonality relation e/spl perp//spl pi/ indicates when a term e and a context /spl pi/ 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 approximated by converging sequences of interval types. Our proof is based on a "type-directed" approximation technique, which departs from the "language-directed" approximation technique developed by MacQueen, Plotkin and Sethi in the ideal model. We thus keep the language elementary (a call-by-name /spl lambda/-calculus) and unstratified (no typecase, no reduction labels). We also include a short account of parametricity, based on an orthogonality relation between quadruples of terms and contexts.

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 projection terms. This establishes a connection with the work of Pitts on relational properties of domains. This also suggests that ideals are better understood as closed sets of terms defined by orthogonality with respect to a set of contexts.

We construct a double category [Dscr ] 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 module (π1; π2) factors out as the separate and interacting computations of the two subcomponents π1 and π2. This enables us to trace MLL modules in the course of cut-elimination, and analyze their behaviour in time.

Every needed strategy is normalizing in the λ-calculus. Here, we extend the result to the λσ-calculus, a λ-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 count for every term M the number of its normalizing paths, up to Lévy permutation equivalence. We deduce from standardization that every needed strategy normalizes when this number is finite. The number is zero or one in the λ-calculus, and we show that it is finite in the λσ-calculus.

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 /spl lambda/-calculus is that there exists a minimal computation (the head-reduction) M/spl rarr//sup e/V from a /spl lambda/-term M to the set of its head-normal forms. Minimality here means categorical "reflectivity" i.e. that every reduction path M/spl rarr//sup f/W to a head-normal form W factors (up to redex permutation) to a path M/spl rarr//sup e/V/spl rarr//sup h/W. This paper establishes a stability a la Berry or poly-reflectivity theorem [D, La, T] which extends the minimality property to rewriting systems with critical pairs. The theorem is proved in the setting of axiomatic rewriting systems where sets of head-normal forms are characterised by their frontier property in the spirit of J. Glauert and Z. Khasidashvili (1996).

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 redexes are also treated. The axioms include stability in the sense given by G. Berry (Ph.D. thesis, Univ. of Paris, 1979), proving it to be an intrinsic notion of deterministic calculi.>