An Extension of Game Logic with Parallel Operators
Iouri Netchitailov
Abstract:
In this thesis we extend the semantics of Game Logic and introduce
some axioms to allow the description of parallel games. Game Logic,
introduced by Parikh (1985), is an extension of Propositional Dynamic
Logic. Game Logic is used to study the general structure of arbitrary
determined two-player games. The syntax of Game Logic contains the
operators of test, choice, sequential composition, duality, and
iteration. In particular, one can find examples of the semantics,
syntax and axiomatics of Game Logic in works of Parikh (1985) and
Pauly (1999). Reasoning about games is similar to reasoning about
programs or processes behaviour, which is supported by such formalisms
as Propositional Dynamic Logic, or Process Algebra. However, Game
Logic does not accept all operators which are involved, for instance,
in Process Algebra; in particular, Game Logic does not contain the
parallel operators, such as parallel composition, or left merge.
Thus, our idea is to introduce parallel operators for Game Logic. To
realise this we explore two versions of parallelism represented in
Process Algebra and Linear Logic.
Process Algebra was the first system that attracted our attention. It
contains alternative and sequential compositions in the basic part
that closely resemble respectively the operator of choice and
sequential composition of Game Logic. Besides, general Process Algebra
contains several sorts of parallel operators which we are looking
for. However, it turns out that it is not easy to incorporate these
operators into Game Logic immediately. There are several difficulties:
one of them is that the semantics of Process Algebra does not care by
which agent the processes execute: this demands a sophisticated
technique to convert it into the semantics of a two-player game. The
other is that the semantics of Process Algebra has a poor support of
truth-values, that is to say, while one might connect falsum with
deadlock, the operator of negation or duality has no an
analogue. Still, a look at Process Algebra is useful, because the
semantics of the parallel operators contains some features that do not
appear directly in Linear Logic, the next formal system which we
traced on the matter of parallel operators. Namely, it gives an option
to distinguish between communicative and non-communicative parallel
operators, and among different ways of parallel execution.
Additive Linear Logic resembles a Game Logic without the operators of
test, sequential composition and iteration. In that sense it looks
more removed from Game Logic than Basic Process
Algebra. Multiplicative Linear Logic brings us a couple of parallel
operators: tensor and par, and some operators of repetition: 'of
course' and 'why not'. Taking into account that Blass (1992) and
Abramsky, Jagadeesan (1994) introduced a two-player game semantics for
Linear Logic, our problem of translating this semantics to Game Logic
becomes more evident. However, even in that case there is a
difficulty, which appears due to the fact that the semantics of Game
Logic uses a description in terms of \phi^1-strategies, whereas the
semantics of Linear Logic refers to winning strategies. They could be
connected with each other in such a manner that a \phi^1-strategy can be
considered as a winning strategy as well as a loosing strategy, of one
player, of the other player or even of both of them. So the use of
\phi^1-strategies gives rise to a more complex structure in game semantics:
nevertheless, we introduce an extension of the semantics of Game Logic
by using analogues of the parallel operators of Linear Logic.
We explore the extension of Game Logic with parallel operators in the
thesis in four chapters, besides the introduction (Chapter 1). In
Chapter 2 we consider the models and the definitions of key operators
of Process Algebra, emphasising the parallel one. Chapter 3 we devote
to the semantics of game operators for Linear Logic based on the works
of Blass (1992) and Abramsky, Jagadeesan (1994). In Chapter 4, the
main part of the thesis, we introduce the semantics for parallel
operators of Game Logic based on the analogues of tensor and par of
Linear Logic. We also describe standard Game Logic operators in terms
of the proposed semantics. Moreover, we propose some axioms for the
parallel operators and offer proofs of soundness. In Chapter 5 the
discussion and conclusion can be found.