 
Summary: Sequentiality vs. Concurrency in Games and Logic
Samson Abramsky
Oxford University Computing Laboratory
March 22, 2001
Abstract
Connections between the sequentiality/concurrency distinction and the semantics of proofs
are investigated, with particular reference to games and Linear Logic.
1 Introduction
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 [Bla72,
Bla92], which pregured many of the ideas in Linear Logic [Gir87], and which can be seen as
a polarized version of ideas familiar from process calculi such as CCS [Mil99] (synchronization
trees, prexing, summation, the Expansion theorem). We analyze the `shocking' fact that this
very clear and intuitive idea leads to a nonassociative 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 `hypersequentialized'
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 [Gir01]. 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
