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Closed and logical relations for over and underapproximation of powersets
 In SAS
, 2004
"... Abstract. We redevelop and extend Dams’s results on over and underapproximation with higherorder Galois connections: (1) We show how Galois connections are generated from UGLBLLUBclosed binary relations, and we apply them to lower and upper powerset constructions, which are weaker forms of powe ..."
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Abstract. We redevelop and extend Dams’s results on over and underapproximation with higherorder Galois connections: (1) We show how Galois connections are generated from UGLBLLUBclosed binary relations, and we apply them to lower and upper powerset constructions, which are weaker forms of powerdomains appropriate for abstraction studies. (2) We use the powerset types within a family of logical relations, show when the logical relations preserve UGLBLLUBclosure, and show that simulation is a logical relation. We use the logical relations to rebuild Dams’s mostprecise simulations, revealing the inner structure of overand underapproximation. (3) We extract validation and refutation logics from the logical relations, state their resemblance to HennesseyMilner logic and description logic, and obtain easy proofs of soundness and best precision. Almost all Galoisconnectionbased static analyses are overapproximating: For
Making Choices Lazily
 Proc. FPCA'95, ACM
, 1995
"... We present a natural semantics that models the untyped, normal order calculus plus McCarthy's amb in the context of callbyneed parameter passing. This results in a singular semantics for amb. Previous work on singular choice has concentrated on erratic choice, a less interesting nondetermin ..."
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We present a natural semantics that models the untyped, normal order calculus plus McCarthy's amb in the context of callbyneed parameter passing. This results in a singular semantics for amb. Previous work on singular choice has concentrated on erratic choice, a less interesting nondeterministic choice operator, and only in relation to callby value parameter passing, or callbyname restricted to deterministic terms. The natural semantics contains rules for both convergent and divergent behaviour, allowing it to distinguish programs that dier only in their divergent behaviour. As a result, it is more discriminating than current domaintheoretic models. This, and the fact that it models singular amb, makes the natural semantics suitable for reasoning about lazy, functional languages containing McCarthy's amb. 1 Introduction The need for nondeterminism in functional programming is apparent. There are parallel algorithms that are inherently nondeterministic, deterministic para...
Quantum Domain Theory  Definitions and Applications
 Proceedings of CCA’03
, 2003
"... Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum sett ..."
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Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 1
Observable Modules and Power Domain Constructions
 Semantics of Programming Languages and Model Theory, volume 5 of Algebra, Logic, and Applications
, 1993
"... An Rmodule M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable Rmodules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is ap ..."
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An Rmodule M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable Rmodules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is applied to the cases of the lower and the upper semiring. All lower modules are observable, whereas there are nonobservable upper modules. Accordingly, all known lower power constructions coincide, whereas there are at least three different upper power constructions. We show that they coincide for continuous ground domains, but differ on more general domains. 1 Introduction A power domain construction maps every domain X into a socalled power domain over X whose points represent sets of points of the ground domain. Power domain constructions were originally proposed to model the semantics of nondeterministic programming languages [Plo76, Smy78, HP79, Mai85]. Other motivations are the sema...
Natural Semantics for NonDeterminism
, 1993
"... We present a natural semantics for the untyped lazy calculus plus McCarthy's amb, a nondeterministic choice operator. The natural semantics includes rules for both convergent behaviour (dened inductively) and divergent behaviour (dened coinductively). This semantics is equivalent to a smal ..."
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We present a natural semantics for the untyped lazy calculus plus McCarthy's amb, a nondeterministic choice operator. The natural semantics includes rules for both convergent behaviour (dened inductively) and divergent behaviour (dened coinductively). This semantics is equivalent to a small step reduction semantics that corresponds closely to our operational intuitions about McCarthy's amb. We present equivalences for convergent and divergent behaviour based on the natural semantics and prove a Context Lemma for the convergence equivalence. We then give a theory l 8 , based on the equivalences for convergent and divergent behaviour. Since it is able to distinguish between programs that dier only in their divergent behaviour, the theory is more discriminating than equational theories based on current domaintheoretic models. It is therefore more suitable for reasoning about functional programs containing McCarthy's amb. Contents 1 Introduction 2 2 Related Work 3 3 ...
Comparing free algebras in Topological and Classical Domain Theory. Submitted, 2006. (Available from http://homepages.inf.ed.ac.uk/als/Research/topologicaldomaintheory.html) [6
 Math. Struct. of Comp. Science
, 2006
"... We compare how computational effects are modelled in Classical Domain Theory and Topological Domain Theory. Both of these theories provide powerful toolkits for denotational semantics: Classical Domain Theory being introduced by Scott, and wellestablished and developed since; Topological Domain Th ..."
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We compare how computational effects are modelled in Classical Domain Theory and Topological Domain Theory. Both of these theories provide powerful toolkits for denotational semantics: Classical Domain Theory being introduced by Scott, and wellestablished and developed since; Topological Domain Theory being a generalization in which topologies more general than the Scotttopology are admitted. Computational effects can be modelled using free algebra constructions, according to Plotkin and Power, and we show that for a wide range of computational effects, including all the classical powerdomains, this free algebra construction coincides in Classical and Topological Domain Theory, when restricted to countablybased continuous domains. 1
GabrielUlmer Duality and Lawvere Theories Enriched over a General Base
 UNDER CONSIDERATION FOR PUBLICATION IN J. FUNCTIONAL PROGRAMMING
"... Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as P oset, ωCpo, or fun ..."
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Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as P oset, ωCpo, or functor categories such as [Inj, Set] or [Inj, ωCpo]. That involves allowing the arities of Lawvere theories to be extended to being sizerestricted objects of the locally presentable category. We develop a body of theory at this level of generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends GabrielUlmer duality.
Two Probabilistic Powerdomains in Topological Domain Theory
"... We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given b ..."
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We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ωcontinuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domaintheoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countablybased spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces.
Instances of computational effects: an algebraic perspective
"... Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different ins ..."
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Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different instances of a particular computational effect. We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and Power; the analysis of restriction by Pitts; and the analysis of the pi calculus by Stark. I.
Semantics of Binary Choice Constructs
"... This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, ..."
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This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6.