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.