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Axioms and (Counter)examples in Synthetic Domain Theory
 Annals of Pure and Applied Logic
, 1998
"... this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the p ..."
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Cited by 9 (7 self)
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this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the purposes of the axiomatic part of this paper, we believe that it would also be
Enrichment and Representation Theorems for Categories of Domains and Continuous Functions
, 1996
"... This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported b ..."
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Cited by 8 (5 self)
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This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported by SERC grant RR30735 and EC project Programming Language Semantics and Program Logics grant SC1000 795 they explain the ordertheoretic structure). Our aim is 1. to provide a justification of Scott's original consideration of ordered structures, and 2. to deepen our understanding of the notion of passage to the limit
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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Cited by 7 (3 self)
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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...
Inductive Reasoning About Effectful Data Types
 In Proceedings of the ACM SIGPLAN International Conference on Functional Programming
, 2007
"... We present a pair of reasoning principles, definition and proof by rigid induction, which can be seen as proper generalizations of lazydatatype induction to monadic effects other than partiality. We further show how these principles can be integrated into logicalrelations arguments, and obtain as ..."
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Cited by 7 (1 self)
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We present a pair of reasoning principles, definition and proof by rigid induction, which can be seen as proper generalizations of lazydatatype induction to monadic effects other than partiality. We further show how these principles can be integrated into logicalrelations arguments, and obtain as a particular instance a general and principled proof that the successstream and failurecontinuation models of backtracking are equivalent. As another application, we present a monadic model of general search trees, not necessarily traversed depthfirst. The results are applicable to both lazy and eager languages, and we emphasize this by presenting most examples in both Haskell and SML.
Hybrid PartialTotal Type Theory
, 1995
"... In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be memb ..."
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Cited by 5 (0 self)
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In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ¸ on computations, and a fixed point induction principle. The resulting theory is thus intended as a generalpurpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...
An Enrichment Theorem for an Axiomatisation of Categories of Domains and Continuous Functions
, 1996
"... This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that they explain the order ..."
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Cited by 4 (4 self)
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This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that they explain the ordertheoretic structure). Our aim is y
Fibrational Induction Meets Effects
"... Abstract. This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs ’ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by G ..."
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Abstract. This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs ’ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type µ(T F) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that µ(T F) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and Støvring. 1
A Computational Formalization for Partial Evaluation (Extended Version)
, 1996
"... We formalize a partial evaluator for Eugenio Moggi's computational metalanguage. This formalization gives an evaluationorder independent view of bindingtime analysis and program specialization, including a proper treatment of call unfolding, and enables us to express the essence of " ..."
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We formalize a partial evaluator for Eugenio Moggi's computational metalanguage. This formalization gives an evaluationorder independent view of bindingtime analysis and program specialization, including a proper treatment of call unfolding, and enables us to express the essence of "controlbased bindingtime improvements" for let expressions. Specifically,
Factoring an adequacy proof (preliminary report
 Functional Programming, Glasgow 1993, Workshops in Computing
, 1993
"... This paper contributes to the methodology of using metalogics for reasoning about programming languages. As a concrete example we consider a fragment of ML corresponding to callbyvalue PCF and translate it into a metalogic which contains (amongst other types) computation types and a fixpoint type. ..."
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This paper contributes to the methodology of using metalogics for reasoning about programming languages. As a concrete example we consider a fragment of ML corresponding to callbyvalue PCF and translate it into a metalogic which contains (amongst other types) computation types and a fixpoint type. The main result is a soundness property (⋆): if the denotations of two programs are provably equal in the metalogic, they have the same operationally observable behaviour. As usual, this follows from a computational adequacy result. In early notes, Plotkin showed how such proofs could be factored into two stages, the first nontrivial and the second (essentially) routine; our contribution is to rework his suggestion within a new framework. We define a metalogic, which incorporates computation and fixpoint types, and specify a modular translation of the ML fragment. Our proof of (⋆) factors into two parts. First, the term language of the metalogic is equipped with an operational semantics and a (generic) computational adequacy result obtained. Second, a simple syntactic argument establishes a correspondence between the operational behaviour of an object program and of its denotation. The first part is not routine but is proved once and for all. The second is a detailed but essentially trivial calculation that is easily adaptable to other object languages. Such a factored proof is important because it promises to scale up more easily than a monolithic one. We show that it may be adapted to an object language with callbyname functions and one with a simple exception mechanism. 1
Retraction Approach to CPS Transform
 HIGHERORDER AND SYMBOLIC COMPUTATION
, 1998
"... We study the continuation passing style (CPS) transform and its generalization, the computational transform, in which the notion of computation is generalized from continuation passing to an arbitrary one. To establish a relation between direct style and continuation passing style interpretation ..."
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Cited by 2 (0 self)
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We study the continuation passing style (CPS) transform and its generalization, the computational transform, in which the notion of computation is generalized from continuation passing to an arbitrary one. To establish a relation between direct style and continuation passing style interpretation of sequential callbyvalue programs, we prove the Retraction Theorem which says that a lambda term can be recovered from its continuationized form via a definable retraction. The Retraction Theorem is proved in the logic of computational lambda calculus for the simply typable terms.