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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...
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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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...
A Powerdomain of Possibility Measures
, 1997
"... We provide a domaintheoretic framework for possibility theory by studying possibility measures on the lattice of opens of a topological space. The powerspaces Poss(X) and Poss [0;1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valu ..."
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We provide a domaintheoretic framework for possibility theory by studying possibility measures on the lattice of opens of a topological space. The powerspaces Poss(X) and Poss [0;1] (X) of all such maps extend to functors in the natural way. We may think of possibility measures as continuous valuations by replacing `+' with `' in their modular law. The functors above send continuous maps to supmaps and continuous domains to completely distributive lattices; in the latter case they are locally continuous. Finite suprema of scalar multiples of point valuations form a basis of the powerdomains above if O(X) is the Scotttopology of a continuous domain. The notion of [0; 1] and [0; 1]modules corresponds to that of continuous cones if addition on the reals and on the module is replaced by suprema. The powerdomain Poss(D) is the free [0; 1]module and Poss [0;1] (D) the free [0; 1]module over a continuous domain D. 1 Possibility Measures This extended abstract attempts to recast som...
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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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...
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. 0 Introduction In this section, we provide a common framework for the summarized papers. In Subsection 0.1, Moggi's approach to specify denotational semantics by means of strong monads is introduced. In Subsection 0.2, we specialize this approach to languages with a binary choice construct. Strong monads can be obtained in at least two ways: as free constructions w.r.t. algebraic theories (Subsection 0.3), and by using second order functions (Subsection 0.4). Finally, formal definitions of those concepts which are used in all...
A Denotional Semantics for . . .
, 2007
"... We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by ..."
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We provide a denotational model for a functional programming language for exact real number computation. A well known difficulty in real number computation is that the tests x = y and x ≤ y are undecidable and hence cannot be used to control the execution flow of programs. One solution, proposed by Boehm and Cartwright, is to use a nondeterministic test. For any two rational numbers p < q and any real number x, at least one of the relations p < x or x < q can be determined to hold; thus, an operator rtest is used, whose evaluation never diverges when x is a real number: 1. rtestp,q(x) evaluates to true or to false, 2. rtestp,q(x) may evaluate to true iff x < q and 3. rtestp,q(x) may evaluate to false iff p < x. Since a program can in general produce different results in different runs, Escardó and MarcialRomero took the view in previous work that programs of realnumber type denote sets of real numbers, and the question arose as to which power domains would be suitable for modelling the behaviour of rtest. It was shown that, among the known power domains,
InverseLimit and Topological Aspects of Abstract Interpretation
"... We develop abstractinterpretation domain construction in terms of the inverselimit construction of denotational semantics and topological principles: We define an abstract domain as a “structural approximation ” of a concrete domain if the former exists as a finite approximant in the inverselimit ..."
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We develop abstractinterpretation domain construction in terms of the inverselimit construction of denotational semantics and topological principles: We define an abstract domain as a “structural approximation ” of a concrete domain if the former exists as a finite approximant in the inverselimit construction of the latter, and we extract the appropriate Galois connection domain denote (basic) open sets from the concrete domain’s Scott topology, and we hypothesize that every abstract domain, even nonstructural approximations, defines a weakened form of topology on its corresponding concrete domain. We implement this observation by relaxing the definitions of topological open set and continuity; key results still hold. We show that families of closed and open sets defined by abstract domains generate post and precondition analyses, respectively, and Giacobazzi’s forwards and backwardscomplete functions of abstractinterpretation theory are the topologically closed and continuous maps, respectively. Finally, we show that Smyth’s upper and lower topologies for powerdomains induce the overapproximating and underapproximating transition functions used for abstractmodel checking.