Fragments of ML Decidable by Nested Data Class Memory Automata

The call-by-value language RML may be viewed as a canonical restriction of Standard ML to ground-type references, augmented by a "bad variable" construct in the sense of Reynolds. We consider the fragment of (finitary) RML terms of order at most 1 with free variables of order at most 2, and identify two subfragments of this for which we show observational equivalence to be decidable. The first subfragment consists of those terms in which the P-pointers in the game semantic representation are determined by the underlying sequence of moves. The second subfragment consists of terms in which the O-pointers of moves corresponding to free variables in the game semantic representation are determined by the underlying moves. These results are shown using a reduction to a form of automata over data words in which the data values have a tree-structure, reflecting the tree-structure of the threads in the game semantic plays. In addition we show that observational equivalence is undecidable at every third- or higher-order type, every second-order type which takes at least two first-order arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument

Conway games, algebraically and coalgebraically

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science.Comment: 30 page

Block Structure vs. Scope Extrusion: Between Innocence and Omniscience

We study the semantic meaning of block structure using game semantics and introduce the notion of block-innocent strategies, which turns out to characterise call-by-value computation with block-allocated storage through soundness, finitary definability and universality results. This puts us in a good position to conduct a comparative study of purely functional computation, computation with block storage and dynamic memory allocation respectively. For example, we show that dynamic variable allocation can be replaced with block-allocated variables exactly when the term involved (open or closed) is of base type and that block-allocated storage can be replaced with purely functional computation when types of order two are involved. To illustrate the restrictive nature of block structure further, we prove a decidability result for a finitary fragment of call-by-value Idealized Algol for which it is known that allowing for dynamic memory allocation leads to undecidability.Supported by the Engineering and Physical Sciences Research Council (EP/C539753/1, EP/F067607/1) and the Royal Academy of Engineering (RF:Tzevelekos)

Verifying Data Independent Programs Using Game Semantics

International audienceWe address the problem of verification of program terms parameterized by a data type X, such that the only operations involving X a program can perform are to input, output, and assign values of type X, as well as to test for equality such values. Such terms are said to be data independent with respect to X. Logical relations for game semantics of terms are defined, and it is shown that the Basic Lemma holds for them. This proves that terms are predicatively parametrically polymorphic, and it provides threshold collections, i.e. sufficiently large finite interpretations of X, for the problem of verification of observational-equivalence, approximation, and safety of parameterized terms for all interpretations of X. In this way we can verify terms with data independent infinite integer types. The practicality of the approach is evaluated on several examples

Applying game semantics to compositional software modeling and verification

Abstract. We describe a software model checking tool founded on game semantics, highlight the underpinning theoretical results and discuss several case studies. The tool is based on an interpretation algorithm defined compositionally on syntax and thus can also handle open programs. Moreover, the models it produces are equationally fully abstract. These features are essential in the modeling and verification of software components such as modules and turn out to lead to very compact models of programs. 1 Introduction and Background Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntax-independent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with non-functional features such as control operators and locally-scoped references [1-6]