Categorical Combinatorics for Innocent Strategies

Harmer, Russell; Hyland, Martin; Melliès, Paul-André

doi:10.1109/lics.2007.14

Cited by 43 publications

(70 citation statements)

References 29 publications

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“…For example, in recent years languages with nominal features (in the sense of [78]) have been studied [4,82,59]. Interest in foundational aspects of Game Semantics was always high, such as understanding the connections between strategies and abstract machines [29], or finding better formulations of game-semantic constructs and constraints [28,66,46,67].…”

Section: Chronology Methodology Ideologymentioning

confidence: 99%

Applications of Game Semantics: From Program Analysis to Hardware Synthesis

Ghica¹

2009

2009 24th Annual IEEE Symposium on Logic in Computer Science

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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. Chronology, methodology, ideologyGame 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 are not sets and functions, as the reader familiar with denotational semantics in the tradition of Scott and Strachey might expect [81].To understand how Game Semantics works it is perhaps easiest to start with the fundamental notion of observable action: the simplest kind of event that a program, or term in general, can be involved in (as producer or consumer) during the course of its execution. For example, the program that interacts with its environment in the simplest possible form can be implicated in two events: starting execution and finishing it. If we construe the notions of input and output broadly enough, the former is an input and the latter is an output. Then, from this point of view there can only be two semantically distinct behaviours of this kind, one that starts then finishes, and one that starts and does not finish. A marginally more complex interaction would be the case of a program that computes a boolean value for which it must be able to produce two distinguishable results. In this class of programs we should be able to find three distinct behaviours, associated with the following three sequences of observable actions: start · true, start · false, start.The idea of observable action can be extended to programs of higher-order type, inductively on the structure of the type. The observable actions of a function are those of the function body and those of its arguments. For example, a (call-by-name) function of type (bool × bool) → bool has nine observable actions. They are the three distinct actions for boolean programs taken thrice, once for each argument and once for the result. Let us tag argument actions with a1 and a2 and result actions with r to prevent confusion. The set of possible behaviours is richer now, including for example interactions such asBoth interactions above represent a computation in which a function starts executing, evaluates one of the arguments, then the other, then produces a result. In both cases the arguments produce true, as does the function. However, these two interactions arise in two semantically distinct functions, which evaluate their arguments in different orders.In a programming language with side-effects it is of course relevant in what precise order a function evaluates its arguments. And modelling a (program) function as a sched...

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Section: Chronology Methodology Ideologymentioning

confidence: 99%

Applications of Game Semantics: From Program Analysis to Hardware Synthesis

Ghica¹

2009

2009 24th Annual IEEE Symposium on Logic in Computer Science

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“…A heap is a finite tree with a total order on its nodes refining the tree order. This structure is common throughout logic and computer science, and the manner in which it appears here is highly reminiscent of its role in the study of (logical) game semantics and innocent strategies [9]. We hope to explore this link further in future work.…”

Section: Introductionmentioning

confidence: 90%

“…(in fact a comonad) on a category Gam of games, whose meaning involves the addition of backtracking: thus, "weakening = backtracking". There are three main differences between our setting and that of [9], all relating to the category on which the endofunctor subsists. Modulo these three differences, the endofunctors are completely identical in form.…”

Section: 1mentioning

confidence: 99%

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Combinatorial structure of type dependency

Garner

2015

Journal of Pure and Applied Algebra

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Abstract. We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of all dependent sequent calculi, and exhibiting it as the category of algebras for a monad on a presheaf category. The objects of the presheaf category encode the basic judgements of a dependent sequent calculus, while the action of the monad encodes the deduction rules; so by giving an explicit description of the monad, we obtain an explicit account of the combinatorics of type dependency. We find that this combinatorics is controlled by a particular kind of decorated ordered tree, familiar from computer science and from innocent game semantics. Furthermore, we find that the monad at issue is of a particularly well-behaved kind: it is local right adjoint in the sense of Street-Weber. In future work, we will use this fact to describe nerves for dependent type theories, and to study the coherence problem for dependent type theory using the tools of two-dimensional monad theory.

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“…In that context, type inhabitants are seen as the interpretation of winning strategies in the arena given by a particular typing. From a categorical point of view, this relation has been established between dialogue games and innocent strategies [17]. Although the two method were developed within di↵erent backgrounds, there are several similarities between the concepts employed.…”

Section: Introductionmentioning

confidence: 99%

A short note on type-inhabitation: Formula-trees vs. game semantics

Alves

Broda

2015

Information Processing Letters

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Type-inhabitation is a topic of major importance, due to its close relationship to provability in logical systems and has been studied from di↵erent perspectives over the years. In 2000 a new proof method has been presented, evidencing the close relationship between the structure of types and their inhabitants. More recently, in 2011, another method has been given in the context of game semantics. In this paper we clarify the similarities between the two approaches.

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Categorical Combinatorics for Innocent Strategies

Abstract: Abstract

Cited by 43 publications

References 29 publications

Applications of Game Semantics: From Program Analysis to Hardware Synthesis

Applications of Game Semantics: From Program Analysis to Hardware Synthesis

Combinatorial structure of type dependency

A short note on type-inhabitation: Formula-trees vs. game semantics

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