arXiv subjects

Simon Castellan

Publications and source records attributed to Simon Castellan.

8 recordsLinked to original sources

arXiv2026

Wiring the Pi-calculus to Denotational Semantics

We introduce a dialect of the Asynchronous pi-calculus, called AWpi, in which (1) an input name may be owned, at any time, by at most one process; (2) each name has either only the input or only the output capability. As a result, special processes called wires (aka forwarders, that is, processes that receive values at one name and re-transmit) behave as substitutions when composed with any AWpi process. Thus AWpi naturally yields a category, whose morphisms are AWpi processes (modulo the reference behavioural equivalence, barbed congruence) and whose objects are types; and where wires act as identity morphisms. We show that the category of processes can be further organised into (sub)categories with the structures needed for the interpretation of common higher-order language features in the literature by drawing on insights from game semantics; notably, we construct a relative Seely category, the categorical structure that concurrent game semantics has. At the same time, AWpi follows the tradition of ordinary pi-calculi in that expressiveness is preserved and the operational and algebraic theory are developed in a similar manner, notwithstanding substantial technical differences in their development and proofs. In short, the goal of AWpi is to remain faithful to the operational and algebraic tradition of the pi-calculi while connecting to the tradition of denotational models for programming languages.

cs.LO↗

arXiv2021

The Mays and Musts of Concurrent Strategies

Concurrent strategies based on event structures are examined from the viewpoint of 'may' and 'must' testing in traditional process calculi. In their pure form concurrent strategies fail to expose the deadlocks and divergences that can arise in their composition. This motivates an extension of the bicategory of concurrent strategies to treat the 'may' and 'must' behaviour of strategies under testing. One extension adjoins neutral moves to strategies but in so doing loses identities w.r.t. composition. This in turn motivates another extension in which concurrent strategies are accompanied by stopping configurations; the ensuing stopping strategies inherit the structure of a bicategory from that of strategies. The technical developments converge in providing characterisations of the 'may' and 'must' equivalences and preorders on strategies.

cs.LO↗

arXiv2021

Disentangling Parallelism and Interference in Game Semantics

Game semantics is a denotational semantics presenting compositionally the computational behaviour of various kinds of effectful programs. One of its celebrated achievement is to have obtained full abstraction results for programming languages with a variety of computational effects, in a single framework. This is known as the semantic cube or Abramsky's cube, which for sequential deterministic programs establishes a correspondence between certain conditions on strategies (''innocence'', ''well-bracketing'', ''visibility'') and the absence of matching computational effects. Outside of the sequential deterministic realm, there are still a wealth of game semantics-based full abstraction results; but they no longer fit in a unified canvas. In particular, Ghica and Murawski's fully abstract model for shared state concurrency (IA) does not have a matching notion of pure parallel program-we say that parallelism and interference (i.e. state plus semaphores) are entangled. In this paper we construct a causal version of Ghica and Murawski's model, also fully abstract for IA. We provide compositional conditions parallel innocence and sequentiality, respectively banning interference and parallelism, and leading to four full abstraction results. To our knowledge, this is the first extension of Abramsky's semantic cube programme beyond the sequential deterministic world.

cs.LO↗

arXiv2020

Game Semantics: Easy as Pi

Game semantics has proven to be a robust method to give compositional semantics for a variety of higher-order programming languages. However, due to the complexity of most game models, game semantics has remained unapproachable for non-experts. In this paper, we aim at making game semantics more accessible by viewing it as a syntactic translation into a session typed pi-calculus, referred to as metalanguage, followed by a semantics interpretation of the metalanguage into a particular game model. The syntactic translation can be defined for a wide range of programming languages without knowledge of the particular game model used. Simple reasoning on the model (soundness, and adequacy) can be done at the level of the metalanguage, escaping tedious technical proofs usually found in game semantics. We call this methodology programming game semantics. We design a metalanguage (PiDiLL) inspired from Differential Linear Logic (DiLL), which is concise but expressive enough to support features required by concurrent game semantics. We then demonstrate our methodology by yielding the first causal, non-angelic and interactive game model of CML, a higher-order call-by-value language with shared memory concurrency. We translate CML into PiDiLL and show that the translation is adequate. We give a causal and non-angelic game semantics model using event structures, which supports a simple semantics interpretation of PiDiLL. Combining both of these results, we obtain the first interactive model of a concurrent language of this expressivity which is adequate with respect to the standard weak bisimulation, and fully abstract for the contextual equivalence on second-order terms. We have implemented a prototype which can explore the generated causal object from a subset of OCaml.

cs.PL↗

arXiv2020

Categories with Families: Unityped, Simply Typed, and Dependently Typed

We show how the categorical logic of untyped, simply typed and dependently typed lambda calculus can be structured around the notion of category with family (cwf). To this end we introduce subcategories of simply typed cwfs (scwfs), where types do not depend on variables, and unityped cwfs (ucwfs), where there is only one type. We prove several equivalence and biequivalence theorems between cwf-based notions and basic notions of categorical logic, such as cartesian operads, Lawvere theories, categories with finite products and limits, cartesian closed categories, and locally cartesian closed categories. Some of these theorems depend on the restrictions of contextuality (in the sense of Cartmell) or democracy (used by Clairambault and Dybjer for their biequivalence theorems). Some theorems are equivalences between notions with strict preservation of chosen structure. Others are biequivalences between notions where properties are only preserved up to isomorphism. In addition to this we discuss various constructions of initial ucwfs, scwfs, and cwfs with extra structure.

cs.LO↗

arXiv2019

Thin Games with Symmetry and Concurrent Hyland-Ong Games

We build a cartesian closed category, called Cho, based on event structures. It allows an interpretation of higher-order stateful concurrent programs that is refined and precise: on the one hand it is conservative with respect to standard Hyland-Ong games when interpreting purely functional programs as innocent strategies, while on the other hand it is much more expressive. The interpretation of programs constructs compositionally a representation of their execution that exhibits causal dependencies and remembers the points of non-deterministic branching.The construction is in two stages. First, we build a compact closed category Tcg. It is a variant of Rideau and Winskel's category CG, with the difference that games and strategies in Tcg are equipped with symmetry to express that certain events are essentially the same. This is analogous to the underlying category of AJM games enriching simple games with an equivalence relations on plays. Building on this category, we construct the cartesian closed category Cho as having as objects the standard arenas of Hyland-Ong games, with strategies, represented by certain events structures, playing on games with symmetry obtained as expanded forms of these arenas.To illustrate and give an operational light on these constructions, we interpret (a close variant of) Idealized Parallel Algol in Cho.

cs.LO↗

arXiv2017

Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)

We show that a version of Martin-Löf type theory with an extensional identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type is a free category with families (supporting these type formers) both in a 1- and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic. Essentially the same construction also shows a slightly strengthened form of the result that equality in extensional Martin-Löf type theory with one universe is undecidable.

cs.LO↗

arXiv2017

Games and Strategies as Event Structures

In 2011, Rideau and Winskel introduced concurrent games and strategies as event structures, generalizing prior work on causal formulations of games. In this paper we give a detailed, self-contained and slightly-updated account of the results of Rideau and Winskel: a notion of pre-strategy based on event structures; a characterisation of those pre-strategies (deemed strategies) which are preserved by composition with a copycat strategy; and the construction of a bicategory of these strategies. Furthermore, we prove that the corresponding category has a compact closed structure, and hence forms the basis for the semantics of concurrent higher-order computation.

math.LO↗