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 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 MultiplicativeAdditive Linear Logic is proved for this semantics. 1 Introduction This paper contains two main contributions: ffl the introduction of a new form of game semantics, which we call concurrent games. ffl a proof of full completeness of this semantics for Multiplicative-Additive Linear Logic. We explain the significance of each of these in turn. Concurrent games Traditional forms of game semantics which have appeared in logic and computer science have been sequential in format: a play of the game is formalized as a sequence of moves. The key feature of this sequential format is the existence of a global schedule (or polarization) : in each (finite) position, it is (exactly) one player's turn to move 1 . This seq...

A cornerstone of the theory of proof nets for unit-free multiplicative linear logic (MLL) is the abstract representation of cut-free proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cut-free monomial proof nets can correspond to the same cut-free proof. Thus the problem of finding a satisfactory notion of proof net for unit-free multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory. 1

Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown since the early nineties. In this note, we review the relation between definability and full abstraction, and we put a few old and recent results of this kind in perspective.

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.-Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...

We present a game description of free symmetric monoidal closed categories, which can also be viewed as a fully complete model for the Intuitionistic Multiplicative Linear Logic with the tensor unit. We model the unit by a distinguished one-move game called Joker. Special rules apply to the joker move. Proofs are modelled by what we call conditionally exhausting strategies, which are deterministic and total only at positions where no joker move exists in the immediate neighbourhood, and satisfy a kind of reachability condition called P-exhaustion. We use the model to give an analysis of a counting problem in free autonomous categories which generalises the Triple Unit Problem. 1 Introduction We aim to construct a fully complete game model for IMLL with unit, the intuitionistic multiplicative (\Omega ; (;?)-fragment of Linear Logic (we write ? for the tensor unit). The notion of full completeness [2] is best formulated in terms of a categorical model of the logic, in which formulas (or...

This paper introduces a model of IMLAL, the intuitionistic multiplicative ( ; (; x ; ! )-fragment of Light Ane Logic, based on games and discreet strategies. We dene a generalized notion of threads, so that a play of a game (of depth k) may be regarded as a number of interwoven threads (of depths ranging from 1 to k). To constrain the way threads communicate with each other, we organize them into networks at each depth (up to k), in accord with a protocol: A network comprises an O-thread (which can only be created by O) and nitely many P-threads (which can only be created by P).

ion for PCF Motivated by the full completeness results, it became of compelling interest to re-examine perhaps the best-known "open problem" in the semantics of programming languages, namely the "Full Abstraction problem for PCF", using the new tools provided by game semantics. 2 PCF is a higher-order functional programming language; modulo issues of the parameterpassing strategies, it forms a fragment of any programming language with higher-order procedures (which includes any reasonably expressive object-oriented language). The aspect of the Full Abstraction problem I personally found most interesting was: to construct a syntax-independent model in which every element is the denotation of some program (note the analogy with full completeness, whose definition had in turn been motivated in part by this aspect of full abstraction). This is not how the problem was originally formulated, but by "general abstract nonsense", given such a model one can always quotient it to get a fully ab...

Game semantics is an unusual denotational semantics in that it captures the intensional (or algorithmic) and dynamical aspects of the computation. This makes it an ideal semantical framework in which to seek to unify analyses of both the qualitative (correctness) as well as the quantitative (efficiency) properties of programming languages. This paper reports work arising from a recent construction of an order (or inequationally) fully abstract model for Scott's functional programming language pcf based on a kind of two-person (Player and Opponent) dialogue game of questions and answers [HO94]. In this model types are interpreted as games and terms as innocent strategies. The fully abstract game model may be said to be canonical for the semantical analysis of sequential functional languages. Unfortunately even for relatively simple pcf-terms, precise description of their denotations as strategies in [HO94] very rapidly becomes unwieldy and opaque. What is needed to remedy the situation ...