arXiv subjects

Paul-André Melliès

Publications and source records attributed to Paul-André Melliès.

At least 19 recordsLinked to original sources

arXiv2026

An explicit shuffle construction of the homotopy span model of differential linear logic

We give a direct and explicit construction of the homotopy span model of linear logic originally formulated by Melliès using abstract homotopy theory in the Quillen category Cat of small categories equipped with its natural model structure. Following the principles of homotopy theory, the formulas and proofs of linear logic are interpreted in the model as small categories up to categorical equivalence, the notion of weak homotopy equivalence underlying the natural Quillen model structure on Cat. Here, we explain how the original interpretation of proofs as fibrant spans can be replaced by a more liberal interpretation of proofs as separately fibrant spans. This relaxation from fibrant spans to separately fibrant spans enables us to give a simple and intuitive description of the homotopy span model, where the axiom and cut links are interpreted as identity spans, and where the symmetries of the exponential modality are captured by a shuffle structure on the interpretation of proofs.

cs.LO↗

arXiv2026

Syntax and semantics of focalisation with relative monads and comonads

The logical principles of focalisation and polarisation can be used to design well-behaved term syntaxes for sequent calculus, which play a role as meta-languages for describing effectful computation. On the semantics side, this corresponds to an axiomatic and polarised notion of model of computation stated in terms of adjunctions over non-associative categories. In this paper, we study the special and delicate cases of resource and effect modalities in a general intuitionistic and linear setting: an exponential comonad $!$ (refining $\square$) and a strong monad $\lozenge$. The starting point of our contribution is noticing that the completeness for a polarised syntax for $!$ and $\lozenge$ with respect to (co)monads in linear call-by-push-value models can be achieved if we move to relative (co)monads: more precisely, comonads relative to $\downarrow$ (the positive shift functor) for $!$ and monads relative to $\uparrow$ (the negative shift functor) for $\lozenge$. These specialisations of the concept of relative (co)monad to call-by-push-value adjunctions recently appeared. Yet the syntax we present arose from proof-theoretic consideration, without the link with relative (co)monads being noticed at the time. Our first remark is thus that (co)monads relative to a call-by-push-value adjunction have been motivated previously from a proof-theoretic perspective in the context of focalisation, which also provides a meta-language for these concepts in an effectful setting. We carry out the study of these modalities from the axiomatic, non-associative point of view. We recall the notion of adjunction over non-associative categories, and establish correspondence results between this notion of adjunction and that of relative adjunction. This correspondence is then extended to linear-non-linear and strong versions of adjunctions as needed to model $!$ and $\lozenge$.

cs.LO↗

arXiv2026

Definitional Inversion, Without Normalisation

We contribute a new proof technique, based on domain theory, to prove key meta-theoretic properties of dependent type systems: definitional inversion properties, i.e. injectivity and no-confusion of type constructors. This proof technique is independent of normalisation, and indeed applies even for the "type-in-type" rule of Martin-Löf's original type theory. Our proof is the first to establish injectivity of type constructors for such a system in the presence of $η$ laws. More generally, the technique is motivated by, and intended for, the metatheory of systems such as Idris, Lean, or dependent Haskell, whose underlying type theory is known to be non-normalising, as well as projects such as MetaRocq or Lean4Lean, where Gödel's second incompleteness theorem means we cannot show normalisation of the object logic in itself. We showcase the method on a small type theory, then explain how it extends to more ambitious extensions.

cs.LO↗

arXiv2026

The Latent Space of Equational Theories

Building on the collaborative Equational Theories project initiated by Terence Tao fifteen months ago, and combining it with ideas coming from machine learning and finite model theory, we construct a latent space of equational theories where each equational theory is located at a specific location, determined by its statistical behavior with respect to a large sample of finite magmas. This experiment enables us to observe for the first time how reasoning flows and produces surprisingly oriented and well-structured chains of logical implications in the latent space of equational theories.

cs.LO↗

arXiv2026

A cartesian closed fibration of higher-order regular languages

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.

cs.LO↗

arXiv2025

Classical notions of computation and the Hasegawa-Thielecke theorem (extended version)

In the spirit of the Curry-Howard correspondence between proofs and programs, we define and study a syntax and semantics for classical logic equipped with a computationally involutive negation, using a polarised effect calculus, the linear classical L-calculus. A main challenge in designing a denotational semantics for the calculus is to accommodate both call-by-value and call-by-name evaluation strategies, which leads to a failure of associativity of composition. In order to tackle this issue, we define a notion of adjunction between graph morphisms on non-associative categories, which we use to formulate polarized and non-associative notions of symmetric monoidal closed duploid and of dialogue duploid. We show that they provide a direct style counterpart to adjunction models: linear effect adjunctions for the (linear) call-by-push-value calculus and dialogue chiralities for linear continuations, respectively. In particular, we show that the syntax of the linear classical L-calculus can be interpreted in any dialogue duploid, and that it defines in fact a syntactic dialogue duploid. As an application, we establish, by semantic as well as syntactic means, the Hasegawa-Thielecke theorem, which states that the notions of central map and of thunkable map coincide in any dialogue duploid (in particular, for any double negation monad on a symmetric monoidal category).

cs.LO↗

arXiv2025

The categorical contours of the Chomsky-Schützenberger representation theorem

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 $ε$-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-Schützenberger 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.

math.CT↗

arXiv2023

Profinite lambda-terms and parametricity

Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite lambda-term which, we show, generalizes at every type the traditional notion of profinite word coming from automata theory. We start by defining the Stone space of profinite lambda-terms as a projective limit of finite sets of usual lambda-terms, considered modulo a notion of equivalence based on the finite standard model. One main contribution of the paper is to establish that, somewhat surprisingly, the resulting notion of profinite lambda-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite lambda-term is compositional by constructing a cartesian closed category of profinite lambda-terms, and we establish that the embedding from lambda-terms modulo beta-eta-conversion to profinite lambda-terms is faithful using Statman's finite completeness theorem. Finally, we prove that the traditional Church encoding of finite words into lambda-terms can be extended to profinite words, and leads to a homeomorphism between the space of profinite words and the space of profinite lambda-terms of the corresponding Church type.

cs.LO↗

arXiv2023

Parsing as a lifting problem and the Chomsky-Schützenberger representation theorem

We begin by explaining how any context-free grammar encodes a functor of operads from a freely generated operad into a certain "operad of spliced words". This motivates a more general notion of CFG over any category $C$, defined as a finite species $S$ equipped with a color denoting the start symbol and a functor of operads $p : Free[S] \to W[C]$ into the operad of spliced arrows in $C$. We show that many standard properties of CFGs can be formulated within this framework, and that usual closure properties of CF languages generalize to CF languages of arrows. We also discuss a dual fibrational perspective on the functor $p$ via the notion of "displayed" operad, corresponding to a lax functor of operads $W[C] \to Span(Set)$. We then turn to the Chomsky-Schützenberger Representation Theorem. We describe how a non-deterministic finite state automaton can be seen as a category $Q$ equipped with a pair of objects denoting initial and accepting states and a functor of categories $Q \to C$ satisfying the unique lifting of factorizations property and the finite fiber property. Then, we explain how to extend this notion of automaton to functors of operads, which generalize tree automata, allowing us to lift an automaton over a category to an automaton over its operad of spliced arrows. We show that every CFG over a category can be pulled back along a ND finite state automaton over the same category, and hence that CF languages are closed under intersection with regular languages. The last important ingredient is the identification of a left adjoint $C[-] : Operad \to Cat$ to the operad of spliced arrows functor, building the "contour category" of an operad. Using this, we generalize the C-S representation theorem, proving that any context-free language of arrows over a category $C$ is the functorial image of the intersection of a $C$-chromatic tree contour language and a regular language.

math.CT↗

arXiv2020

Comprehension and quotient structures in the language of 2-categories

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.

math.CT↗

arXiv2020

Concurrent Separation Logic Meets Template Games

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 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. 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.

cs.LO↗

arXiv2018

An Asynchronous soundness theorem for concurrent separation logic

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 $Γ\vdash\{P\}C\{Q\}$ 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 $\mathcal{L}:[C]_S\to[C]_L$ 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.

cs.PL↗

arXiv2017

A Game Semantics of Concurrent Separation Logic

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 three pieces: one piece for the Code, one piece for the Environment, and one piece for the available shared resources. We establish the soundness of concurrent separation logic by interpreting every derivation tree of the logic as a winning strategy of this specification game.

cs.LO↗

arXiv2017

On bifibrations of model categories

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 structure on the total category $\mathcal E$, such that the functor $p$ preserves cofibrations, fibrations and weak equivalences. Using this Grothendieck construction for model structures, we revisit the traditional definition of Reedy model structures, and possible generalizations, and exhibit their bifibrational nature.

math.CT↗

arXiv2016

Five Basic Concepts of Axiomatic Rewriting Theory

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 Lévy, and developed along the subsequent years into a full-fledged 2-dimensional theory of causality and residuation in rewriting. I will give a contemporary view on the theory, informed by my later work on categorical semantics and higher-dimensional algebra, and also indicate a number of current research directions in the field.

cs.LO↗

arXiv2016

A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine

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.

cs.LO↗

arXiv2015

An Isbell Duality Theorem for Type Refinement Systems

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 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.

cs.LO↗

arXiv2015

Relational semantics of linear logic and higher-order model-checking

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 his 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.

cs.LO↗