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An Axiomatic Approach to Adequacy
 University of Aarhus
, 1996
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Cited by 26 (1 self)
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Dissertation Series. Copies may be obtained by contacting: BRICS
Bidomains and full abstraction for countable nondeterminism
 In Proceedings of FoSSaCS’06, number 3921 in LNCS
, 2006
"... Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to mayandmust testing. Our model is based on biordered sets similar to Berry’s bid ..."
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Abstract. We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to mayandmust testing. Our model is based on biordered sets similar to Berry’s bidomains, and stable, monotone functions. However, (as in prior models of unbounded nondeterminism) these functions may not be continuous. Working in a biordered setting allows us to exploit the different properties of both extensional and stable orders to construct a Cartesian closed category of sequential, discontinuous functions, with least and greatest fixpoints having strong enough properties to prove computational adequacy. We establish full abstraction of the semantics by showing that it contains a simple, firstorder “universal typeobject ” within which all types may be embedded using functions defined by (countable) ordinal induction. 1
From Strategies to Profunctors
"... Abstract. A lax functor from a bicategory of very general nondeterministic concurrent strategies on concurrent games to the bicategory of profunctors is presented. The lax functor provides a fundamental connection between two approaches to generalizations of domain theory to forms of intensional dom ..."
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Abstract. A lax functor from a bicategory of very general nondeterministic concurrent strategies on concurrent games to the bicategory of profunctors is presented. The lax functor provides a fundamental connection between two approaches to generalizations of domain theory to forms of intensional domain theories, one based on game and strategies, and the other on presheaf categories and profunctors. The lax functor becomes a pseudo functor on the subbicategory of rigid strategies which includes ‘simple games ’ (underlying AJM and HO games) and stable spans (specializing to Berry’s stable functions, in the deterministic case). In general, the lax functor illustrates how composition of strategies is obtained from that of profunctors by restricting to ‘reachable ’ elements. The results are based on a new characterization of concurrent strategies, which exhibits concurrent strategies as certain discrete fibrations, or equivalently presheaves, over configurations of the game. Finally, the characterization suggests how to extend the definition of strategy to that of strategy on and between categories with a factorization system, an idea that relates to earlier work on bistructures and bidomains. 1
Stable Bistructure Models of PCF
, 1994
"... Stable bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. They represent Berry's bidomains. The ..."
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Stable bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. They represent Berry's bidomains. The representation can proceed in two stages. Bistructures form a categorical model of Girard's linear logic consisting of a linear category together with a comonad. The comonad has a coKleisli category which is equivalent to a cartesianclosed full subcategory of Berry's bidomains. A main motivation for bidomains came from the full abstraction problem for Plotkin's functional language PCF. However, although the bidomain model incorporates both the Berry stable order and the Scott pointwise order, its PCF theory (those inequalities on terms which hold in the bidomain model) does not include that of the Scott model. With a simple modification we can obtain a new model of PCF, combining the Berry and Scott orders, which does not have this inadequacy.
Sequentiality in Bounded Biorders
 FUNDAMENTA INFORMATICAE
, 2005
"... We study a notion of bounded stable biorder, showing that the monotone and stable functions on such biorders are sequential. We construct bounded biorder models of a range of sequential, higherorder functional calculi, including unary PCF, (typed and untyped) callbyvalue and lazy λcalculi, and ..."
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We study a notion of bounded stable biorder, showing that the monotone and stable functions on such biorders are sequential. We construct bounded biorder models of a range of sequential, higherorder functional calculi, including unary PCF, (typed and untyped) callbyvalue and lazy λcalculi, and nondeterministic SPCF. We prove universality and full abstraction results for these models by reduction to the case of unary PCF, for which we give a simple new argument to show that any orderextensional and sequential model is universal.
Full abstraction for recursive types with control and countable nondeterminism
, 2007
"... Abstract. We describe fully abstract denotational models for a functional language with firstclass continuations, recursive types and countable nondeterminism — a λµcalculus with recursive types and countable choice. In this setting we may consider problems relating countablenondeterminism w ..."
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Abstract. We describe fully abstract denotational models for a functional language with firstclass continuations, recursive types and countable nondeterminism — a λµcalculus with recursive types and countable choice. In this setting we may consider problems relating countablenondeterminism with fairness, such as defining the fair merge of infinite streams, allowing a semantic analysis of these longstanding problems. We give separate musttesting and maytesting models using meet and join biorders. In the musttesting case, this requires new notions of countably complete biorder, and the development of existing techniques for solving domain equations without ω0continuity. For example, we show that the (musttesting) denotation of the type of streams over the oneelement datatype in our model contains uncountably many elements, ordered linearly. We prove full abstraction for both models by showing that, despite the expresive power of the language it has a very simple universal type: every type is a retract of nat → 1. 1
Preliminary draft
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS
Fully Abstract Bidomain Models of the λCalculus
, 2001
"... We present a proof that the canonical models of the untyped λcalculus  with callbyvalue and lazy callbyname evaluation  in the category of bidomains and continuous and stable functions are fully abstract. This is achieved by showing that bidomains yield a fully abstract model of a versio ..."
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We present a proof that the canonical models of the untyped λcalculus  with callbyvalue and lazy callbyname evaluation  in the category of bidomains and continuous and stable functions are fully abstract. This is achieved by showing that bidomains yield a fully abstract model of a version of Plotkin's FPC in which the constructor for sum types is restricted to its unary form  lifting. It is shown that full abstraction for this model can be reduced to denability for the fragment corresponding to "unary PCF". An algorithm devised by SchmidtSchau is used to show that the bidomain model of this fragment is fully abstract.
On dialogue games and coherent strategies ˚
"... We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended w ..."
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We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature.