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Power Domain Constructions
 SCIENCE OF COMPUTER PROGRAMMING
, 1998
"... The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the alg ..."
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Cited by 23 (9 self)
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The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the algebraic properties of the basic operations empty set, union, singleton, and extension of functions. A host of derived operations is introduced and investigated algebraically. Every power construction is shown to be equipped with a characteristic semiring such that the resulting power domains become semiring modules. Power homomorphisms are introduced as a means to relate different power constructions. They also allow to define the notion of initial and final constructions for a fixed characteristic semiring. Such initial and final constructions are shown to exist for every semiring, and their basic properties are derived. Finally, the known power constructions are put into the general framewo...
Probabilistic Domains
 in Proc. CAAP ’94, LNCS
, 1997
"... We show the equivalence of several different axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a finite number of possibilities with arbitr ..."
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Cited by 22 (4 self)
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We show the equivalence of several different axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a finite number of possibilities with arbitrary probabilities, whereas the poorest one has binary choice with equal probabilities as the only operation. The remaining theories lie in between; one of them is the theory of binary choice by Graham [1]. 1 Introduction A probabilistic programming language could contain different kinds of language constructs to express probabilistic choice. In a rather poor language, there might be a construct x \Phi y, whose semantics is a choice between the two possibilities x and y with equal probabilities 1=2. The `possibilities' x and y can be statements in an imperative language or expressions in a functional language. A quite rich language could contain a construct [p 1 : x 1 ; : : : ; p n : x n ],...
Power Domains and Second Order Predicates
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the seco ..."
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Cited by 13 (7 self)
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Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the second order predicates.
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 nondeterministic ..."
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Cited by 8 (3 self)
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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...
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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Cited by 4 (1 self)
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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...
Power Domains Supporting Recursion and Failure
 In CAAP'92
, 1998
"... Following the program of Moggi, the semantics of a simple nondeterministic functional language with recursion and failure is described by a monad. We show that this monad cannot be any of the known power domain constructions, because they do not handle nontermination properly. Instead, a novel con ..."
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Cited by 3 (1 self)
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Following the program of Moggi, the semantics of a simple nondeterministic functional language with recursion and failure is described by a monad. We show that this monad cannot be any of the known power domain constructions, because they do not handle nontermination properly. Instead, a novel construction is proposed and investigated. It embodies both nondeterminism (choice and failure) and possible nontermination caused by recursion. 1 Introduction Following the proposals of Moggi [Mog89, Mog91b], functional languages with various notions of computations can be denotationally described by means of monads. Monads are constructions mapping domains of values into domains of computations for these values. Computations involving destructive assignments, for instance, are handled by the state transformer monad [Wad90], whereas computations by nondeterministic choice are handled by power domain constructions [Plo76, Smy78, Gun90]. All known power domain constructions and many others ar...
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 small ste ..."
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Cited by 2 (0 self)
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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 ...
Product Operations in Strong Monads
 Proceedings of the First Imperial College, Department of Computing, Workshop on Theory and Formal Methods, Workshops in Computing
, 1993
"... If a strong monad M is used to define the denotational semantics of a functional language with computations, a product operation \Gamma \Theta : MX \Theta MY !M (X \Theta Y ) is needed to define the semantics of pairing. Every strong monad is equipped with two standard products, which correspond t ..."
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Cited by 1 (1 self)
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If a strong monad M is used to define the denotational semantics of a functional language with computations, a product operation \Gamma \Theta : MX \Theta MY !M (X \Theta Y ) is needed to define the semantics of pairing. Every strong monad is equipped with two standard products, which correspond to lefttoright and righttoleft evaluation. We study the algebraic properties of these standard products in general. Then we define alternative products with similar properties for strict and parallel evaluation in the special case of strong monads in DCPO which are obtained as free constructions w.r.t. various theories of nondeterministic computation, including both classical and probabilistic theories. 1 Introduction Following the proposals of Moggi [12, 13], functional languages with various notions of computations can be denotationally described by means of (particular kinds of) monads. Informally, these monads are constructions mapping domains of values into domains of computations...