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A Note on Logical Relations Between Semantics and Syntax
, 1997
"... This note gives a new proof of the `operational extensionality' property of Abramsky's lazy lambda calculusnamely the coincidence of contextual equivalence with a coinductively defined notion of `applicative bisimilarity'. This purely syntactic result is here proved using a logical relation (due ..."
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This note gives a new proof of the `operational extensionality' property of Abramsky's lazy lambda calculusnamely the coincidence of contextual equivalence with a coinductively defined notion of `applicative bisimilarity'. This purely syntactic result is here proved using a logical relation (due to Plotkin) between the syntax and its denotational semantics. The proof exploits a mixed inductive/coinductive characterisation of the logical relation recently discovered by the author.
Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory
 In Proc. 17th IEEE Symposium on Logic in Computer Science
, 2003
"... This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown ..."
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This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic ZermeloFraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambdacalculus with callbyvalue operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domaintheoretic models.
Relating Operational and Denotational Semantics for Input/Output Effects
, 1999
"... We study the longstanding problem of semantics for input/output (I/O) expressed using sideeffects. Our vehicle is a small higherorder imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational se ..."
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We study the longstanding problem of semantics for input/output (I/O) expressed using sideeffects. Our vehicle is a small higherorder imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity. We prove bisimilarity is a congruence using Howe's method. Next, we define a metalanguage M in which we may give a denotational semantics to O. M generalises Crole and Pitts' FIXlogic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with an operational semantics and a domaintheoretic semantics in the category CPPO of cppos (bottompointed posets with joins of !chains) and Scott continuous functions. We use the CPPO semantics to prove that M is computationally...
Monads in Action
"... In functional programming, monadic characterizations of computational effects are normally understood denotationally: they describe how an effectful program can be systematically expanded or translated into a larger, pure program, which can then be evaluated according to an effectfree semantics. An ..."
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In functional programming, monadic characterizations of computational effects are normally understood denotationally: they describe how an effectful program can be systematically expanded or translated into a larger, pure program, which can then be evaluated according to an effectfree semantics. Any effectspecific operations expressible in the monad are also given purely functional definitions, but these definitions are only directly executable in the context of an already translated program. This approach thus takes an inherently Churchstyle view of effects: the nominal meaning of every effectful term in the program depends crucially on its type. We present here a complementary, operational view of monadic effects, in which an effect definition directly induces an imperative behavior of the new operations expressible in the monad. This behavior is formalized as additional operational rules for only the new constructs; it does not require any structural changes to the evaluation judgment. Specifically, we give a smallstep operational semantics of a prototypical functional language supporting programmerdefinable, layered effects, and show how this semantics naturally supports reasoning by familiar syntactic techniques, such as showing soundness of a Currystyle effecttype system by the progress+preservation method.
Classifying Categories for Partial Equational Logic
, 2002
"... Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions. ..."
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Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions.
Full Abstraction by Translation
 Proc., 3rd Workshop in Theory and Formal Methods
, 1996
"... This paper shows how a fully abstract model for a rich metalanguage like FPC can be used to prove theorems about other languages. In particular, we use results obtained from a game semantics of FPC to show that the natural translation of the lazy calculus into the metalanguage is fully abstract, th ..."
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This paper shows how a fully abstract model for a rich metalanguage like FPC can be used to prove theorems about other languages. In particular, we use results obtained from a game semantics of FPC to show that the natural translation of the lazy calculus into the metalanguage is fully abstract, thus obtaining a new full abstraction result from an old one. The proofs involved are very easyall the hard work was done in giving the original games model. So far we have been unable to prove the completeness of our translation without recourse to the denotational model; we therefore have an indication of the worth of such fully abstract models. 1 Introduction Plotkin, in his CSLI notes [18], showed how denotational semantics can be viewed as a twostage process. First one defines a metalanguage which describes elements of the intended semantic model, usually some category of domains. Then to give semantics to a language L it suffices to translate it into the metalanguage. While this is ...
Axioms for Definability and Full Completeness
 in Proof, Language and Interaction: Essays in Honour of Robin
, 2000
"... ion problem for PCF (see [BCL86, Cur93, Ong95] for surveys). The importance of full abstraction for the semantics of programming languages is that it is one of the few quality filters we have. Specifically, it provides a clear criterion for assessing how definitive a semantic analysis of some langu ..."
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ion problem for PCF (see [BCL86, Cur93, Ong95] for surveys). The importance of full abstraction for the semantics of programming languages is that it is one of the few quality filters we have. Specifically, it provides a clear criterion for assessing how definitive a semantic analysis of some language is. It must be admitted that to date the quest for fully abstract models has not yielded many obvious applications; but it has generated much of the deepest work in semantics. Perhaps it is early days yet. Recently, game semantics has been used to give the first syntaxindependent constructions of fully abstract models for a number of programming languages, including PCF [AJM96, HO96, Nic94], richer functional languages [AM95, McC96b, McC96a, HY97], and languages with nonfunctional features such as reference types and nonlocal control constructs [AM97c, AM97b, AM97a, Lai97]. A noteworthy feature is that the key definability results for the richer languages are proved by a reduction to...
A Simple Adequate Categorical Model for PCF
 In Proceedings of Third International Conference on Typed Lambda Calculi and Applications
, 1997
"... Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that nontermination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for ..."
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Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that nontermination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for PCF in the sense that we will introduce categorical axioms enabling an adequate semantics to be given. We assume the presence of certain "bottom" maps with the role of being the interpretation of nonterminating terms, but the orderstructure is left out. This is different from previous approaches where some kind of ordertheoretic structure has been considered as part of an adequate categorical model for PCF. We take the point of view that partiality is the fundamental notion from which orderstructure should be derived, which is corroborated by the observation that our categorical model induces an ordertheoretic model for PCF in a canonical way.
Recursive Types in Games: Axiomatics and Process Representation (Extended Abstract)
 IN PROCEEDINGS O.F LICS'98. IEEE COMPUTER
, 1998
"... This paper presents two basic results on gamebased semantics of FPC, a metalanguage with sums, products, exponentials and recursive types. First we give an axiomatic account of the category of games G introduced in [15], offering a fundamental structural analysis of the category as well as a transp ..."
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This paper presents two basic results on gamebased semantics of FPC, a metalanguage with sums, products, exponentials and recursive types. First we give an axiomatic account of the category of games G introduced in [15], offering a fundamental structural analysis of the category as well as a transparent way to prove computational adequacy. As a consequence we obtain an intensional fullabstraction result through a standard definability argument. Next we extend the category G by introducing a category of games G i with optimised strategies; we show that the denotational semantics in G i gives a compilation of FPC terms into core Pict codes (the asynchronous polyadic calculus without summation). The process representation follows a pioneering idea of Hyland and Ong [18]. However, we advance their representation by introducing semantically wellfounded optimisation techniques; we also exte...
Logical Relations and Data Abstraction
 Proc. Computer Science Logic, CSL 2000, Fischbachau. Springer LNCS 1862
, 1996
"... We prove, in the context of simple type theory, that logical relations are sound and complete for data abstraction as given by equational specifications. Specifically, we show that two implementations of an equationally specified abstract type are equivalent if and only if they are linked by a suita ..."
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We prove, in the context of simple type theory, that logical relations are sound and complete for data abstraction as given by equational specifications. Specifically, we show that two implementations of an equationally specified abstract type are equivalent if and only if they are linked by a suitable logical relation. This allows us to introduce new types and operations of any order on those types, and to impose equations between terms of any order. Implementations are required to respect these equations up to a general form of contextual equivalence, and two implementations are equivalent if they produce the same contextual equivalence on terms of the enlarged language. Logical relations are introduced abstractly, soundness is almost automatic, but completeness is more difficult, achieved using a variant of Jung and Tiuryn's logical relations of varying arity. The results are expressed and proved categorically.