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Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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Cited by 12 (6 self)
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
Coherence and Consistency in Domains
 IN THIRD ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1990
"... Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different conditionthat of coherencewhich has its origins in topology and log ..."
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Cited by 8 (4 self)
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Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different conditionthat of coherencewhich has its origins in topology and logic. In particular, we concentrate on those posets whose principal ideals are algebraic lattices and whose topologies are coherent. These form a cartesian closed category which has fixed points for domain equations. It is shown that a "universal domain" exists. Since the construction of this domain seems to be of general significance, a categorical treatment is provided and applied to other classes of domains. Universal domains constructed in this fashion enjoy an additional property: they are saturated. We show that there is exactly one such domain in each of the classes under consideration.
Lower Bag Domains
 Fundamenta Informaticae
, 1995
"... . Two lower bag domain constructions are introduced: the initial construction which gives free lower monoids, and the final construction which is defined explicitly in terms of second order functions. The latter is analyzed closely. For sober dcpo's, the elements of the final lower bag domains can b ..."
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Cited by 7 (3 self)
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. Two lower bag domain constructions are introduced: the initial construction which gives free lower monoids, and the final construction which is defined explicitly in terms of second order functions. The latter is analyzed closely. For sober dcpo's, the elements of the final lower bag domains can be described concretely as bags. For continuous domains, initial and final lower bag domains coincide. They are continuous again and can be described via a basis which is constructed from a basis of the argument domain. The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions [13, 15, 16] were introduced to describe the denotational semantics of nondeterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with some families of continuous operations. If X is the semantic domain ...
Disjunctive Systems and LDomains
 Proceedings of the 19th International Colloquium on Automata, Languages, and Programming (ICALP’92
, 1992
"... . Disjunctive systems are a representation of Ldomains. They use sequents of the form X ` Y , with X finite and Y pairwise disjoint. We show that for any disjunctive system, its elements ordered by inclusion form an Ldomain. On the other hand, via the notion of stable neighborhoods, every Ldomain ..."
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Cited by 6 (2 self)
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. Disjunctive systems are a representation of Ldomains. They use sequents of the form X ` Y , with X finite and Y pairwise disjoint. We show that for any disjunctive system, its elements ordered by inclusion form an Ldomain. On the other hand, via the notion of stable neighborhoods, every Ldomain can be represented as a disjunctive system. More generally, we have a categorical equivalence between the category of disjunctive systems and the category of Ldomains. A natural classification of domains is obtained in terms of the style of the entailment: when jXj = 2 and jY j = 0 disjunctive systems determine coherent spaces; when jY j 1 they represent Scott domains; when either jXj = 1 or jY j = 0 the associated cpos are distributive Scott domains; and finally, without any restriction, disjunctive systems give rise to Ldomains. 1 Introduction Discovered by Coquand [Co90] and Jung [Ju90] independently, Ldomains form one of the maximal cartesian closed categories of algebraic cpos. Tog...
A Characterisation of the LeastFixedPoint Operator By Dinaturality
, 1993
"... The paper addresses the question of when the leastfixedpoint operator, in a cartesian closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a sufficient condition on a cartesian closed full subcategory ..."
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Cited by 4 (1 self)
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The paper addresses the question of when the leastfixedpoint operator, in a cartesian closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a sufficient condition on a cartesian closed full subcategory of the category of algebraic cpos for the characterisation to hold. The condition is quite mild, and the leastfixedpoint operator is so characterised in many of the most commonly used categories of domains. By using retractions, the characterisation extends to the associated cartesian closed categories of continuous cpos. However, dinaturality does not always characterise the leastfixedpoint operator. We show that in cartesian closed full subcategories of the category of continuous lattices the characterisation fails. 1 Introduction Mulry [7] showed that, under general conditions on a category of domains, the leastfixedpoint operator, lfp D : D D ! D, is a dinatural transformation ...
A Cartesian Closed Category of Event Structures with Quotients
, 2006
"... We introduce a new class of morphisms for event structures. The category obtained is cartesian closed, and a natural notion of quotient event structure is defined within it. We study in particular the topological space of maximal configurations of quotient event structures. We introduce the compress ..."
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Cited by 4 (0 self)
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We introduce a new class of morphisms for event structures. The category obtained is cartesian closed, and a natural notion of quotient event structure is defined within it. We study in particular the topological space of maximal configurations of quotient event structures. We introduce the compression of event structures as an example of quotient: the compression of an event structure E is a minimal event structure with the same space of maximal configurations as E.
The Largest Cartesian Closed Category of Stable Domains
 Theoretical Computer Science
"... This paper shows that Axiom d and Axiom I are important when one works within the realm of Scottdomains. In particular, it has been shown that (i) if [D ! s D] has a countable basis, then D must be finitary, for any Scottdomain D; (ii) if [D ! s D] is bounded complete, then D must be distributive, ..."
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Cited by 3 (0 self)
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This paper shows that Axiom d and Axiom I are important when one works within the realm of Scottdomains. In particular, it has been shown that (i) if [D ! s D] has a countable basis, then D must be finitary, for any Scottdomain D; (ii) if [D ! s D] is bounded complete, then D must be distributive, for any finitary Scottdomain D. Therefore, the category of dIdomains is the largest cartesian closed category within omegaalgebraic, bounded complete domains, with the exponential being the stable function space. 1 Introduction Among Scott's many insights which shaped the whole area of domain theory, one is that the partial ordering of a domain should be interpreted as the ordering about information. "Thus," wrote Scott [16], "x v y means that
Ldomains and Lossless Powerdomains
"... The category of Ldomains was discovered by A. Jung while solving the problem of finding maximal cartesian closed categories of algebraic CPO's and continuous functions. In this note we analyse properties of the lossless powerdomain construction, that is closed on the algebraic Ldomains. The powerd ..."
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The category of Ldomains was discovered by A. Jung while solving the problem of finding maximal cartesian closed categories of algebraic CPO's and continuous functions. In this note we analyse properties of the lossless powerdomain construction, that is closed on the algebraic Ldomains. The powerdomain is shown to be isomorphic to a collection of subsets of the domain on which the construction was done. The proof motivates a certain finiteness condition on the inconsistency relations of elements. It is shown that all algebraic CPO's D whose basis B(D) has property M satisfy the condition. In particular, the coherent L domains satisfy the condition. 1 Introduction Recent work by A. Jung and C. Gunter shows that the Ldomains discovered independently by A. Jung[6] and T. Coquand[1] form an interesting cartesian closed category. P. Buneman proposed the lossless powerdomain construction that is closed on Ldomains. Buneman's construction was based on intuitions from databases. We were...
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...
On Conditional Information in FeatureBased Theories
 University of Central Florida
, 1999
"... Two approaches to conditional information used in featurebased linguistic theories, especially headdriven phrase structure grammar, are compared and their interrelation is explicated on a formal and a conceptual level. For this purpose concepts of locale theory are introduced that allow to define f ..."
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Two approaches to conditional information used in featurebased linguistic theories, especially headdriven phrase structure grammar, are compared and their interrelation is explicated on a formal and a conceptual level. For this purpose concepts of locale theory are introduced that allow to define feature descriptions and structures in a unified manner and, in particular, to make the difference between both approaches transparent. In addition, a foundation for attributevalue logic as a predicatefunctor logic based on regimented and formalized descriptions is proposed. It turns out that the relative pseudocomplement version of conditional constraints as put forward by Pollard and Sag (1987) is the wrong choice. From a formal perspective, the mistake is due to the confusion of the category of Heyting algebras with that of frames. Only the former are equipped with an operation corresponding to the conditional, that is, with means to represent conditional information as an element within the algebra itself. With respect to cognitive processing the central point is that conditional information is (in general) not actual information acquired during cognitive action but background information applied in processing. An appropriate formalism to keep conditional and actual/observational information apart is given by geometric or observational logic (Vickers, 1989). Since this logic is on the other hand the logic of frames and locales it provides a promising framework for grammatical theory. 1