After informally reviewing the main concepts from game semantics and placing the development of the field in a historical context we examine its main applications. We focus in particular on finite state model checking, higher order model checking and more recent developments in hardware design. 1. Chronology, methodology, ideology Game Semantics is a denotational semantics in the conventional sense: for any term, it assigns a certain mathematical object as its meaning, which is constructed compositionally from the meanings of its sub-terms in a way that is independent of the operational semantics of the object language. What makes Game Semantics particular, peculiar maybe, is that the mathematical objects it operates with

Introduction The articles in this volume concern operational semantics of higher order programming languages, mathematical techniques for developing the properties of such operational semantics, and applications of those techniques. In this Introduction we set the articles in the wider context of research into programming languages and bring out some of the themes and techniques that recur throughout the book. Operational Semantics The various approaches to giving meanings to programming languages fall broadly into three categories: denotational, axiomatic, and operational. In a denotational semantics, the meaning of programs is defined abstractly using elements of some suitable mathematical structure. In an axiomatic semantics, meaning is defined indirectly via the axioms and rules of some logic of program properties. In an operational semantics, the meaning of programs is defined in terms of their behaviour, for example the steps of computation they can take during

Semantics of Imperative Languages using Regular Expressions -- program not occur free P. No = else Representation Semantics of Imperative Languages using Regular Expressions -- program not occur free P. No = else Representation Independence bool true;P (!x, !x) int 1;P(!x > 0,x -!x) Game Semantics of Imperative Languages using Regular Expressions -- . Milne-Strachey "marked store" models. These can fail to validate garbage collection [2]. Functor-category models handle locality well. representation independence. Snapback remains problematic [6, 5]. "object spaces" handle snapback too. Full abstraction up to order 2 via the Yoneda embedding [8, 4]. reasoning [7] . semantics: full Semantics of Imperative Languages using Regular Expressions -- . Milne-Strachey "marked store" models. These can fail to validate garbage collection [2]. .

Reasoning about program equivalence is one of the oldest problems in semantics. In recent years, useful techniques have been developed, based on bisimulations and logical relations, for reasoning about equivalence in the setting of increasingly realistic languages—languages nearly as complex as ML or Haskell. Much of the recent work in this direction has considered the interesting representation independence principles enabled by the use of local state, but it is also important to understand the principles that powerful features like higher-order state and control effects disable. This latter topic has been broached extensively within the framework of game semantics, resulting in what Abramsky dubbed the “semantic cube”: fully abstract game-semantic characterizations of various axes in the design space of ML-like languages. But when it comes to reasoning about many actual examples, game semantics does not yet supply a useful technique for proving equivalences. In this paper, we marry the aspirations of the semantic cube to the powerful proof method of stepindexed Kripke logical relations. Building on recent work of Ahmed, Dreyer, and Rossberg, we define the first fully abstract logical relation for an ML-like language with recursive types, abstract types, general references and call/cc. We then show how, under orthogonal restrictions to the expressive power of our language—namely, the restriction to first-order state and/or the removal of call/cc—we can enhance the proving power of our possible-worlds model in correspondingly orthogonal ways, and we demonstrate this proving power on a range of interesting examples. Central to our story is the use of state transition systems to model the way in which properties of local state evolve over time. 1