Sequentiality vs. Concurrency in Games and Logic

We use Games and Logic as a mirror to understand an aspect of the sequentiality/concurrency distinction. We begin with the simple, intuitive notion of polarized games due to Blass (Blass 1972;Blass 1992), which prefigured many of the ideas in Linear Logic (Girard 1987), and which can be seen as a polarized version of ideas familiar from process calculi such as CCS (Milner 1999) (synchronization trees, prefixing, summation, the Expansion theorem). We analyze the 'shocking' fact that this very clear and intuitive idea leads to a non-associative composition; a kind of incompatibility between a purely sequential model and logic in a classical format. Two ways of addressing this issue have been found. One is to modify the syntax, by studying a 'hyper-sequentialized' version of sequent calculus, in which the current focus of attention in the proof is explicitly represented. This is the approach taken in Girard's Ludics (Girard 2001). The other is to broaden the notion of game to encompass 'truly concurrent games'. In such games there is no longer a global polarization (we can have positions in which both players can move concurrently), although there are still local polarizations (each local decision is made by just one of the players). This idea of concurrent games was used by Abramsky and Melliès (Abramsky and Melliès 1999) to give a fully complete model for Linear Logic in its original form (i.e. not 'hyper-sequentialized'), and indeed the correspondence is with proof-nets, the 'parallel syntax for proof theory' in Girard's phrase (Girard 1995). In this way, the distinction between sequentiality and concurrency is reflected at a fundamental level, in the analysis of the 'space of proofs'.

The main aim of the present paper is to expose some of the conceptual issues underlying recent technical work on games and logic. The presentation is deliberately elementary in style, in the hope of making the discussion accessible both to concurrency theorists, and to those interested in the semantics of proofs-and of exhibiting a significant point of contact betwen these two fields.

The analysis of Blass games and the problem of non-associativity of composition, and the connection of this issue to the interleaving/true concurrency distinction, were first presented by the author in a lecture given at the Isaac Newton Institute for the Mathematical Sciences at Cambridge in 1995, during the programme on Semantics of Computation.

For the reader’s convenience, some material on Linear Logic (specifically, the sequent calculus presentation of propositional Multiplicative-Additive Linear Logic (MALL)) is recalled in an appendix.

Acknowledgement The comments made by the journal referees suggested a number of improvements to the presentation.

In this section we will view games as polarized processes. More precisely, we will develop a correspondence between certain 2-person games of perfect information and polarized deterministic synchronization trees.

We can describe (well-founded) deterministic synchronization trees inductively, as given by expressions of the form i∈I a i .P i (i = j ⇒ a i = a j )

i.e. as the least set closed under the operation of disjointly guarded summation (Milner 1989).

It is understood that the summation, as in CCS, is associative and commutative (idempotence does not arise because of the disjointness condition). The basic case of the inductive definition is given by the empty summation, written 0 (the ‘NIL’ process of CCS). There are labelled transitions i∈I a i .P i ai -→ P i for each i ∈ I, giving the arcs from the root of the synchronization tree to its immediate sub-trees. See (Winskel and Nielsen 1995) for useful background on synchronization trees and related models. We could accomodate infinite branches in such trees by using a coinductive rather than an inductive definition. This issue is not important for our purposes here, for which it will be quite sufficient to consider only finite trees.

We can define interleaving (non-communicating) parallel composition on the synchronization trees thus: if P = i∈I a i .P i , and Q = j∈J b j .Q j , then

(In order to preserve the disjointness property in the summation, we require that the sorts of P and Q (i.e. the sets of labels appearing anywhere in the synchronization trees for P and Q respectively) are disjoint (Milner 1989)-this will be tacitly assumed in the sequel.) This is the Expansion Theorem (Milner 1989) in an appropriate version. Note that it can be taken as an (inductive or coinductive) definition of parallel composition as an operation on synchronization trees. It shows how to eliminate parallel composition in favour of purely sequential constructs. As such, it expresses the essence of the interleaving view of concurrency. Note that, in Milner’s classification of the operations of process algebra (Milner 1989), guarded summation is built from the dynamic operations, while parallel composition is the key static operation. So the interleavi

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