A game semantics of finite nondeterminism is proposed. In this model, a strategy may make a choice between different moves in a given situation; moreover, strategies carry extra information about their possible divergent behaviour. A Cartesian closed category is built and a model of a simple, higher-order nondeterministic imperative language is given. This model is shown to be fully abstract, with respect to an equivalence based on both safety and liveness properties, by means of a factorization theorem which states that every nondeterministic strategy is the composite of a deterministic strategy with a nondeterministic oracle. 1 Introductory remarks Nondeterminism, the notion that the behaviour of a computer system need not be completely determined by the behaviour of its environment, is a valuable abstraction in the analysis of programs. An unreliable hardware component, governed by laws of physics too complex to take into account, can be understood as a nondeterministic entity; mul...

Abstract Nondeterminism is a pervasive phenomenon in computation. Often it arises as an emergent property of a complex system, typically as the result of contention for access to shared resources. In such circumstances, we cannot always know, in advance, exactly what will happen. In other circumstances, nondeterminism is explicitly introduced as a means of abstracting away from implementation details such as precise command scheduling and control flow. However, the kind of behaviours exhibited by nondeterministic computations can be extremely subtle in comparison to those of their deterministic counterparts and reasoning about such programs is notoriously tricky as a result. It is therefore important to develop semantic tools to improve our understanding of, and aid our reasoning about, such nondeterministic programs. In this thesis, we extend the framework of game semantics to encompass nondeterministic computation. Game semantics is a relatively recent development in denotational semantics; its main novelty is that it views a computation not as a static entity, but rather as a dynamic process of interaction. This perspective makes the theory well-suited to modelling many aspects of computational processes: the original use of game semantics in modelling the simple functional language PCF has subsequently been extended to handle more complex control structures such as references and continuations.

We reveal a symmetric structure in the ho/n games model of innocent strategies, introducing rigid strategies, a concept dual to bracketed strategies. We prove a direct definability theorem of general innocent strategies with respect to a simply typed language of extended Bohm trees, which gives an operational meaning to rigidity in call-byname. A corresponding factorization of innocent strategies into rigid ones with some form of conditional as an oracle is constructed. 1