Operational game semantics (OGS) is a method for interpreting programs as strategies in suitable games, or more precisely as labelled transition systems over suitable games, in the sense of Levy and Staton. Such an interpretation is called sound when, for any two given programs, weak bisimilarity of associated strategies entails contextual equivalence. OGS has been applied to a variety of languages, with rather tedious soundness proofs. In this paper, we contribute to the unification and mechanisation of OGS. Indeed, we propose an abstract notion of language with evaluator, for which we construct a generic OGS interpretation, which we prove sound. Our framework covers a variety of simply-typed and untyped lambda-calculi with various evaluation strategies. These calculi notably feature recursive definitions, first-class continuations, and a wide variety of datatypes. All constructions and proofs are entirely mechanised in the Coq proof assistant.

This artifact report is a companion to the ESOP’25 paper An abstract, certified account of operational game semantics [3]. The paper describes the construction of a sound model for an abstract notion of language. The model is built using a semantic technique named Operational Game Semantics (OGS). All our results are mechanised in the Coq proof assistant: this mechanisation (The proof artifact is archived at https://doi.org/10.5281/zenodo.14697618 .) constitutes the artifact we discuss in the present document. More specifically, our mechanisation covers our main result, the soundness of the abstract OGS model w.r.t. substitution equivalence (Theorem 8), as well as four example calculi: two variants of call-by-value λ -calculus and two variants of μμ̃ -calculus [4, 5]. The only axiom used is the Axiom K [17] for equality proof irrelevance (To ease dependent pattern matching due to the intrinsically scoped representation.). The README explains the installation process and the structure of the code. An online rendering ( https://lapin0t.github.io/ogs/esop25/Readme.html .) is available thanks to Alectryon [12]. Furthermore, the main paper provides systematic hyperlinks from statements to their Coq counterparts. We encourage the interested reader to use these tools to navigate the code. In this document, we focus first on users: how to read and instantiate our main result. We then detail salient technical aspects of our mechanisation.

We prove a general congruence result for bisimilarity in higher-order languages, which generalises previous work to languages specified by a labelled transition system in which programs may occur as labels, and which may rely on operations on terms other than capture-avoiding substitution. This is typically the case for PCF, $\lambda$-calculus with delimited continuations, and early-style bisimilarity in higher-order process calculi.

We introduce a category-theoreticabstraction of a syntax with auxiliary functions, called an admissiblemonad morphism. Relying on an abstract form of structural recursion,we then design generic tools to construct admissible monad morphismsfrom basic data. These tools automate ubiquitous standard patternslike (1) defining auxiliary functions in successive, potentiallydependent layers, and (2) proving properties of auxiliary functions byinduction on syntax. We cover significant examples from theliterature, including the standard lambda-calculus withcapture-avoiding substitution, a lambda-calculus with bindingevaluation contexts, the lambda-mu-calculus with named substitution, andthe differential lambda-calculus.

Applicative bisimilarity is a coinductive characterisation of observational equivalence in call-by-name lambda-calculus, introduced by Abramsky (1990). Howe (1996) gave a direct proof that it is a congruence, and generalised the result to all languages complying with a suitable format. We propose a categorical framework for specifying operational semantics, in which we prove that (an abstract analogue of) applicative bisimilarity is automatically a congruence. Example instances include standard applicative bisimilarity in call-by-name, call-by-value, and call-by-name non-deterministic $\lambda$-calculus, and more generally all languages complying with a variant of Howe's format.

By abstracting over well-known properties of De Bruijn's representation with nameless dummies, we design a new theory of syntax with variable binding and capture-avoiding substitution. We propose it as a simpler alternative to Fiore, Plotkin, and Turi's approach, with which we establish a strong formal link. We also show that our theory easily incorporates simple types and equations between terms.

This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as lambda-bar-mu-calculus, pi-calculus, Positive GSOS specifications, differential lambda-calculus, and the big-step, simply-typed, call-by-value lambda-calculus. Moreover, we design a suitable notion of signature for transition monads.

We introduce a categorical framework for operational semantics, in which we define substitution-closed bisimilarity, an abstract analogue of the open extension of Abramsky’s applicative bisimilarity. We furthermore prove a congruence theorem for substitution-closed bisimilarity, following Howe’s method. We finally demonstrate that the framework covers the call-by-name and call-by-value variants of λ-calculus in big-step style. As an intermediate result, we generalise the standard framework of Fiore et al. for syntax with variable binding to the skew-monoidal case. CCS Concepts: • Theory of computation → Semantics and reasoning; Categorical semantics; Operational semantics.

We introduce a categorical framework for operational semantics, in which we define substitution-closed bisimilarity, an abstract analogue of the open extension of Abramsky's applicative bisimilarity. We furthermore prove a congruence theorem for substitution-closed bisimilarity, following Howe's method. We finally demonstrate that the framework covers the call-by-name and call-by-value variants of λ-calculus in big-step style. As an intermediate result, we generalise the standard framework of Fiore et al. for syntax with variable binding to the skew-monoidal case.

Template games unify various approaches to game semantics, by exhibiting them as instances of a double-categorical variant of the slice construction. However, in the particular case of simple games, template games do not quite yield the standard (bi)category. We refine the construction using factorisation systems, obtaining as an instance a slight generalisation of simple games and strategies. Another instance is Day's convolution monoidal structure on the category of presheaves over a strict monoidal category, which answers a question raised in the first author's PhD thesis.

We propose a categorical framework for structural operational semantics, in which we prove that under suitable hypotheses bisimilarity is a congruence. We then refine the framework to prove soundness of bisimulation up to context, an efficient method for reducing the size of bisimulation relations. Finally, we demonstrate the flexibility of our approach by reproving known results in three variants of the π-calculus.

We unify previous constructions from our work on concurrent game semantics into a single categorical framework. From an operational description of positions and moves in some game, called a signature, we produce a pseudo double category, in which objects are positions and vertical morphisms are plays. The considered games are multi-player, so it makes sense to consider embeddings of positions: these are the horizontal morphisms. Finally, cells may be thought of as embeddings of plays preserving initial and final positions. In order to be suitable for game semantics, the obtained pseudo double category should enjoy a certain fibredness property. Under suitable hypotheses, we show that our construction actually produces such a fibred pseudo double category, from which we can define relevant categories of plays, and thus of strategies. We give a first necessary and sufficient criterion for this to hold and then a sufficient criterion that can be checked more easily.

We give a leisurely introduction to our abstract framework for operational semantics based on cellular monads on transition categories. Furthermore, we relate it for the first time to an existing format, by showing that all Positive GSOS specifications generate cellular monads whose free algebras are all compositional. As a consequence, we recover the known result that bisimilarity is a congruence in the generated labelled transition system.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. A practical type system for generalized recursion Tom Hirschowitz, Serguei Lenglet

Game semantics is a rich and successful class of denotational models for programming languages. Most game models feature a rather intuitive setup, yet surprisingly difficult proofs of such basic results as associativity of composition of strategies. We seek to unify these models into a basic abstract framework for game semantics, game settings. Our main contribution is the generic construction, for any game setting, of a category of games and strategies. Furthermore, we extend the framework to deal with innocence, and prove that innocent strategies form a subcategory. We finally show that our constructions cover many concrete cases, mainly among the early models [5, 23] and the recent, sheaf-based ones [40].

Game semantics is a rich and successful class of denotational models for programming languages. Most game models feature a rather intuitive setup, yet surprisingly difficult proofs of such basic results as associativity of composition of strategies. We seek to unify these models into a basic abstract framework for game semantics, game settings. Our main contribution is the generic construction, for any game setting, of a category of games and strategies. Furthermore, we extend the framework to deal with innocence, and prove that innocent strategies form a subcategory. We finally show that our constructions cover many concrete cases, mainly among the early models [5, 23] and the recent, sheaf-based ones [40].

Game semantics has provided adequate models for a variety of programming languages, in which types are interpreted as two-player games and programs as strategies. Melliès (2018) suggested that such categories of games and strategies may be obtained as instances of a simple abstract construction on weak double categories. However, in the particular case of simple games, his construction slightly differs from the standard category. We refine the abstract construction using factorisation systems, and show that the new construction yields the standard category of simple games and strategies. Another perhaps surprising instance is Day's convolution monoidal structure on the category of presheaves over a strict monoidal category.

In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that any two operations of the same shape agree. An important part of this work is the study of analytic functors between presheaf categories, which are a common generalisation of Joyal's analytic endofunctors on sets and of the parametric right adjoint functors on presheaf categories introduced by Diers and studied by Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among the analytic endofunctors, and may be characterised as the submonads of a universal analytic monad with exactly one operation of each shape. In fact, shapeliness also gives a way to define the data and axioms of a structure directly from its graphical calculus, by generating a free shapely monad on the basic operations of the calculus. In this paper we do this for some of the examples listed above; in future work, we intend to do so for graphical calculi such as Milner's bigraphs, Lafont's interaction nets, or Girard's multiplicative proof nets, thereby obtaining canonical notions of denotational model.

Game semantics is a class of models of programming languages in which types are interpreted as games and programs as strategies. Though originally designed for sequential languages, its scope has recently been extended to concurrent ones. A salient feature of game semantics is the notion of innocence, which requires strategies to be determined by their values on a certain class of plays, called views. In previous work, we have obtained a representation theorem for Tsukada and Ong's categories of views and plays, in particular by constructing an embedding V of views into a coslice of a certain presheaf category. We here exploit this result to exhibit an efficient categorical account of two crucial constructions of game semantics. First, we recover the interpretation of normal forms into innocent strategies as the singular functor associated to V. Second, the corresponding geometric realisation functor yields the standard definability result saying that any innocent strategy is (isomorphic to) the interpretation of a normal form.