Spoiler-Duplicator games are used in finite model theory to examine the expressive power of logics. Their strategies have recently been reformulated as coKleisli maps of game comonads over relational structures, providing new results in finite model theory via categorical techniques. We present a novel framework for studying Spoiler-Duplicator games by viewing them as event structures. We introduce a first systematic method for constructing comonads for all one-sided Spoiler-Duplicator games: game comonads are now realised by adjunctions to a category of games, generically constructed from a comonad in a bicategory of game schema (called signature games). Maps of the constructed categories of games are strategies and generalise coKleisli maps of game comonads; in the case of one-sided games they are shown to coincide with suitably generalised homomorphisms. Finally, we provide characterisations of strategies on two-sided Spoiler-Duplicator games; in a common special case they coincide with spans of event structures.

The article bridges between two major paradigms in computation, the functional, at basis computation from input to output, and the interactive, where computation reacts to its environment while underway. Central to any compositional theory of interaction is the dichotomy between a system and its environment. Concurrent games and strategies address the dichotomy in fine detail, very locally, in a distributed fashion, through distinctions between Player moves (events of the system) and Opponent moves (those of the environment). A functional approach has to handle the dichotomy more ingeniously, via its blunter distinction between input and output. This has led to a variety of functional approaches, specialised to particular interactive demands. Through concurrent games we can see what separates and connects the differing paradigms, and show how:•to lift functions to strategies; how to turn functional dependency to causal dependency and so exploit functional techniques.•several approaches of functional programming and logic arise naturally as full subcategories of concurrent games, including stable domain theory; nondeterministic dataflow; geometry of interaction; the dialectica interpretation; lenses and optics, and their extensions to containers in dependent lenses and optics.•the enrichments of strategies (e.g. to probabilistic, quantum or real-number computation) specialise to the functional cases.

In the simplest form of event structure, a prime event structure, an event is associated with a unique causal history, its prime cause. However, it is quite common for an event to have disjunctive causes in that it can be enabled by any one of multiple sets of causes. Sometimes the sets of causes may be mutually exclusive, inconsistent one with another, and sometimes not, in which case they coexist consistently and constitute parallel causes of the event. The established model of general event structures can model parallel causes. On occasion however such a model abstracts too far away from the precise causal histories of events to be directly useful. For example, sometimes one needs to associate probabilities with different, possibly coexisting, causal histories of a common event. Ideally, the causal histories of a general event structure would correspond to the configurations of its causal unfolding to a prime event structure; and the causal unfolding would arise as a right adjoint to the embedding of prime in general event structures. But there is no such adjunction. However, a slight extension of prime event structures remedies this defect and provides a causal unfolding as a universal construction. Prime event structures are extended with an equivalence relation in order to dissociate the two roles, that of an event and its enabling; in effect, prime causes are labelled by a disjunctive event, an equivalence class of its prime causes. With this enrichment a suitable causal unfolding appears as a pseudo right adjoint. The adjunction relies critically on the central and subtle notion of extremal causal realisation as an embodiment of causal history. Finally, we explore subcategories which support parallel causes as well the key operations needed in developing probabilistic distributed strategies with parallel causes.

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.

Quantum programming languages permit a hardware independent, high-level description of quantum algo rithms. In particular, the quantum lambda-calculus is a higher-order programming language with quantum primitives, mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum lambda-calculus is a challenging problem that has attracted significant interest in the past few years. Several models have been proposed but for those that address the whole quantum λ-calculus, they either do not represent the dynamics of computation, or they lack the compositionality one often expects from denotational models. In this paper, we give the first compositional and interactive model of the full quantum lambda-calculus, based on game semantics. To achieve this we introduce a model of quantum games and strategies, combining quantum data with a representation of the dynamics of computation inspired from causal models of concurrent systems. In this model we first give a computationally adequate interpretation of the affine fragment. Then, we extend the model with a notion of symmetry, allowing us to deal with replication. In this refined setting, we interpret and prove adequacy for the full quantum lambda-calculus. We do this both from a sequential and a parallel interpretation, the latter representing faithfully the causal independence between sub-computations.

The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten. We present here new categories of games, not suffering from this bias. In our first category, we achieve this by avoiding hiding altogether; instead morphisms are uncovered strategies (with neutral events) up to weak bisimulation . Then, we show that by hiding only certain events dubbed inessential we can consider strategies up to isomorphism , and still get a category – this partial hiding remains sound up to weak bisimulation, so we get a concrete representations of programs (as in standard concurrent games) while avoiding the angelic bias. These techniques are illustrated with an interpretation of affine nondeterministic PCF which is adequate for weak bisimulation; and may, must and fair convergences.

We define a new games model of Probabilistic PCF (PPCF) by enriching thin concurrent games with symmetry, recently introduced by Castellan et al, with probability. This model supports two interpretations of PPCF, one sequential and one parallel. We make the case for this model by exploiting the causal structure of probabilistic concurrent strategies. First, we show that the strategies obtained from PPCF programs have a deadlock-free interaction, and therefore deduce that there is an interpretation-preserving functor from our games to the probabilistic relational model recently proved fully abstract by Ehrhard et al. It follows that our model is intensionally fully abstract. Finally, we propose a definition of probabilistic innocence and prove a finite definability result, leading to a second (independent) proof of full abstraction.

We imagine a team Player engaging a team Opponent in a distributed game. Such games and their strategies have been formalised within event structures. However there are limitations in founding strategies on traditional event structures. Sometimes a probabilistic distributed strategy relies on benign races where, intuitively, several members of team Player may race each other to make a common move. Although there exist event structures which support such parallel causes, in which an event is enabled in several compatible ways, they do not support an operation of hiding central to the composition of strategies; nor do they support probability adequately. An extension of traditional event structures is devised which supports parallel causes and hiding, as well as the mix of probability and nondeterminism needed to account for probabilistic distributed strategies. The extension is located within existing models for concurrency and tested in the construction of a bicategory of probabilistic distributed strategies with parallel causes.

Distributed/concurrent strategies have been introduced as special maps of event structures. As such they factor through their "rigid images," themselves strategies. By concentrating on such "rigid image" strategies we are able to give an elementary account of distributed strategies and their composition, resulting in a category of games and strategies. This is in contrast to the usual development where composition involves the pullback of event structures explicitly and results in a bicategory. It is shown how, in this simpler setting, to extend strategies to probabilistic strategies; and indicated how through probability we can track nondeterministic branching behaviour, that one might otherwise think lost irrevocably in restricting attention to "rigid image" strategies.

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.

We describe a framework for truly concurrent game semantics of programming languages, based on Rideau and Winskel's concurrent games on event structures. The model supports a notion of innocent strategy that permits concurrent and non-deterministic behaviour, but which coincides with traditional Hyland-Ong innocent strategies if one restricts to the deterministic sequential case. In this framework we give an alternative interpretation of Plot kin's PCF, that takes advantage of the concurrent nature of strategies and formalizes the idea that although PCF is a sequential language, certain sub-computations are independent and can be computed in a parallel fashion. We show that just as Hyland and Ong's sequential interpretation of PCF, our parallel interpretation yields a model that is intensionally fully abstract for PCF.

A generalised framework of site graphs is introduced in order to provide the first fully semantic definition of the side-effect-free core of the rule-based language Kappa. This formalisation allows the use of types either to confirm that a rule respects a certain invariant or to guide a restricted refinement process that allows us to constrain its run-time applicability.

In this paper, we introduce a novel way of constructing concise causal histories (pathways) to represent how specified structures are formed during simulation of systems represented by rule-based models. This is founded on a new, clean, graph-based semantics introduced in the first part of this paper for Kappa, a rule-based modelling language that has emerged as a natural description of protein-protein interactions in molecular biology [Bachman 2011]. The semantics is capable of capturing the whole of Kappa, including subtle side-effects on deletion of structure, and its structured presentation provides the basis for the translation of techniques to other models. In particular, we give a notion of trajectory compression, which restricts a trace culminating in the production of a given structure to the actions necessary for the structure to occur. This is central to the reconstruction of biochemical pathways due to the failure of traditional techniques to provide adequately concise causal histories, and we expect it to be applicable in a range of other modelling situations.

A bicategory of concurrent games, where nondeterministic strategies are formalized as certain maps of event structures, was introduced recently. This paper studies an extension of concurrent games by winning conditions, specifying players' objectives. The introduction of winning conditions raises the question of whether such games are determined, that is, if one of the players has a winning strategy. This paper gives a positive answer to this question when the games are well-founded and satisfy a structural property, race-freedom, which prevents one player from interfering with the moves available to the other. Uncovering the conditions under which concurrent games with winning conditions are determined opens up the possibility of further applications of concurrent games in areas such as logic and verification, where both winning conditions and determinacy are most needed. A concurrent-game semantics for predicate calculus is provided as an illustration.

Abstract Nondeterministic concurrent strategies—those strategies compatible with copy-cat behaving as identity w.r.t. composition—have been characterised as certain maps of event structures. This leads to a bicategory of general concurrent games in which the maps are nondeterministic concurrent strategies. This paper explores the important sub-bicategory of deterministic concurrent strategies. It is shown that deterministic strategies in a game can be identified with certain subgames, with the benefit that the bicategory of deterministic games becomes equivalent to a technically-simpler order-enriched category. Via a characterisation, deterministic strategies are shown to coincide with the receptive ingenuous strategies of Melliès and Mimram. Deterministic strategies determine closure operators , in accord with an early definition of Abramsky and Melliès. Known subcategories appear as special cases: Berry’s order-enriched category of dI-domains and stable functions arises as a full subcategory in which the games comprise solely of Player moves; the `simple games’ of Hyland et al., a basis for much of game semantics, form a subcategory in which the games permit no concurrency, Player-Opponent moves alternate and Opponent always moves first.

A bi category of very general nondeterministic concurrent games and strategies is presented. The intention is to formalize distributed games in which both Player (or a team of players) and Opponent (or a team of opponents) can interact in highly distributed fashion, without, for instance, enforcing that their moves alternate.

The article discusses causal models, such as Petri nets and event structures, how they have been rediscovered in a wide variety of recent applications and why they are fundamental to computer science. A discussion of their present limitations leads to their extension with symmetry. The consequences, actual and potential, are discussed.

This paper establishes a bridge between presheaf models for concurrency and the more operationally-informative world of event structures. It concentrates on a particular presheaf category, consisting of presheaves over finite partial orders of events; such presheaves form a model of nondeterministic processes in which the computation paths have the shape of partial orders. It is shown how with the introduction of symmetry event structures represent all presheaves over finite partial orders. This is in contrast with plain event structures which only represent certain separated presheaves. Specifically a coreflection from the category of presheaves to the category of event structures with symmetry is exhibited. It is shown how the coreflection can be cut down to an equivalence between the presheaf category and the subcategory of graded event structures with symmetry. Event structures with strong symmetries are shown to represent precisely all the separated presheaves. The broader context and specific applications to the unfolding of higher-dimensional automata and Petri nets, and weak bisimulation on event structures are sketched.

The unfolding of (1-)safe Petri nets to occurrence nets is well understood. There is a universal characterization of the unfolding of a safe net which is part and parcel of a coreflection from the category of occurrence nets to the category of safe nets. The unfolding of general Petri nets, nets with multiplicities on arcs whose markings are multisets of places, does not possess a directly analogous universal characterization, essentially because there is an implicit symmetry in the multiplicities of general nets, and that symmetry is not expressed in their traditional occurrence net unfoldings. In the present paper, we show how to recover a universal characterization by representing the symmetry in the behaviour of the occurrence net unfoldings of general Petri nets. We show that this is part of a coreflection between enriched categories of general Petri nets with symmetry and occurrence nets with symmetry.

A compositional Petri net-based semantics is given to a simple language allowing pointer manipulation and parallelism. The model is then applied to give a notion of validity to the judgements made by concurrent separation logic that emphasizes the process-environment duality inherent in such rely-guarantee reasoning. Soundness of the rules of concurrent separation logic with respect to this definition of validity is shown. The independence information retained by the Petri net model is then exploited to characterize the independence of parallel processes enforced by the logic. This is shown to permit a refinement operation capable of changing the granularity of atomic actions.

A compositional Petri net based semantics is given to a simple pointer-manipulating language. The model is then applied to give a notion of validity to the judgements made by concurrent separation logic that emphasizes the processenvironment duality inherent in such rely-guarantee reasoning. Soundness of the rules of concurrent separation logic with respect to this definition of validity is shown. The independence information retained by the Petri net model is then exploited to characterize the independence of parallel processes enforced by the logic. This is shown to permit a refinement operation capable of changing the granularity of atomic actions.

We study the combination of probability and non-determinism from a categorical point of view. In category theory, non-determinism and probability are represented by suitable monads. However, these two monads do not combine well as they are. To overcome this problem, we introduce the notion of indexed valuations. This notion is used to define a new monad that can be combined with the usual non-deterministic monad via a categorical distributive law. We give an equational characterisation of our construction. We discuss the computational meaning of indexed valuations, and we show how they can be used by giving a denotational semantics of a simple imperative language.

The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higher-order processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higher-order processes. For this it is useful to generalise event structures to allow events, which "persist".

A path-based domain theory for higher-order processes is extended to allow name generation. The original domain theory is built around the monoidal-closed category Lin consisting of path orders with join-preserving functions between their domains of path sets. Name generation is adjoined by forming the functor category [I, Lin], where I consists of finite sets of names and injections. The functor category [I, Lin] is no longer monoidal-closed w.r.t. the tensor inherited pointwise from Lin. However, conditions are given under which function spaces exist. The conditions are preserved by a rich discipline of linear types, including those of new-HOPLA, a recent powerful language for higher-order processes with name generation.

This paper studies fundamental connections between profunctors (that is, distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and of how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ‘lifting’, and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation, but the results are likely to have more general applicability because of the ubiquitous nature of profunctors.

We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well-known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular, it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors; (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps; and (3) a treatment of higher-order dataflow as a biproduct, essentially by following the geometry of interaction programme.

The meaning and mathematical consequences of linearity (managing without a presumed ability to copy) are studied for a path-based model of processes which is also a model of affine-linear logic. This connection yields an affine-linear language for processes, automatically respecting open-map bisimulation, in which a range of process operations can be expressed. An operational semantics is provided for the tensor fragment of the language. Different ways to make assemblies of processes lead to different choices of exponential, some of which respect bisimulation.

The events of a security protocol and their causal dependency can play an important role in the analysis of security properties. This insight underlies both strand spaces and the inductive method. But neither of these approaches builds up the events of a protocol in a compositional way, so that there is an informal spring from the protocol to its model. By broadening the models to certain kinds of Petri nets, a restricted form of contextual nets, a compositional event-based semantics is given to an economical, but expressive, language for describing security protocols; so the events and dependency of a wide range of protocols are determined once and for all. The net semantics is formally related to a transition semantics, strand spaces and inductive rules, as well as trace languages and event structures, so unifying a range of approaches, as well as providing conditions under which particular, more limited, models are adequate for the analysis of protocols. The net semantics allows the derivation of general properties and proof principles which are demonstrated in establishing an authentication property, following a diagrammatic style of proof.

A process language for security protocols is presented together with a semantics in terms of sets of events. The denotation of process is a set of events, and as each event specifies a set of pre and postconditions, this denotation can be viewed as a Petri net. By means of an example we il-lustrate how the Petri-net semantics can be used to prove security properties. 1.

A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles, we are able to account for weak morphisms (roughly only required to preserve observable paths) and to derive a saturation monad (on the category of presheaves over the category of paths). Weak morphisms may be encoded as strong ones via the Kleisli construction associated to the saturation monad. A general notion of weak open-map bisimulation is introduced, and results relating various notions of strong and weak bisimulation are provided. The abstract theory is accompanied by fine concrete study of two key models for concurrency, the interleaving model of synchronisation trees and the independence model of labelled event structures.

We develop a 2-categorical theory for recursively defined domains. In particular we generalise the traditional approach based on order-theoretic structures to category-theoretic ones. A motivation for this development is the need of a domain theory for concurrency, with an account of bisimulation. Indeed, the leading examples throughout the paper are provided by recursively defined presheaf models for concurrent process calculi. Further we use the framework to study (open-map) bisimulation.

We present a proof system for determining satisfaction between processes in a fairly general process algebra and assertions of the modal /spl mu/-calculus. The proof system is compositional in the structure of processes. It extends earlier work on compositional reasoning within the modal /spl mu/-calculus and combines it with techniques from work on local model checking. The proof system is sound for all processes and complete for a class of finite-state processes.>

An abstract definition of bisimulation is presented. It allows a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronization trees, transition systems with independence (an abstraction from Petri nets), and labeled event structures are considered. On transition systems, the abstract definition readily specialises to Milner's (1989) strong bisimulation. On event structures, it explains and leads to a revision of the history-preserving bisimulation of Rabinovitch and Traktenbrot (1988), and Goltz and van Glabeek (1989). A tie-up with open maps in a (pre)topos brings to light a promising new model, presheaves on categories of pomsets, into which the usual category of labeled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has refinement operators, though further work is required to justify their appropriateness and understand their relation to previous attempts.>

An extension of Milner's CCS with a priority choice operator called prisum is investigated. This operator is very similar to the PRIALT construct of Occam. The binary prisum operator only allows execution of its second component in the case in which the environment is not ready to allow the first component to proceed. This dependency on the set of actions the environment is ready to perform goes beyond that encountered in traditional CCS. Its expression leads to a novel operational semantics in which transitions carry read-sets (of the environment) as well as the normal action symbols from CCS. A notion of strong bisimulation is defined on agents with priority by means of this semantics. It is a congruence and satisfies new equational laws (including a new expansion law) which are shown to be complete for finite agents with prisum. The laws are conservative over agents of traditional CCS.>

An attempt is made to cast labeled Petri nets and other models in an algebraic framework. One aim is to utilize the framework of categorical l to cast labeled Petri nets and other models in an algebraic framework. The other aim is to utilize the framework of categorical logic to systematize specification languages and the derivation of proof systems for parallel processes. A category of labeled nets is presented, and its categorical constructions are used to establish a compositional proof system. A category of properties of nets is used in forming the proof system.>