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Axiomatic Rewriting Theory IV  A stability theorem in Rewriting Theory
, 1998
"... One key property of the calculus is that there exists a minimal computation (the headreduction) M e \Gamma! V from a term M to the set of its headnormal forms. Minimality here means categorical "reflectivity " i.e. that every reduction path M f \Gamma! W to a headnormal form W fa ..."
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One key property of the calculus is that there exists a minimal computation (the headreduction) M e \Gamma! V from a term M to the set of its headnormal forms. Minimality here means categorical "reflectivity " i.e. that every reduction path M f \Gamma! W to a headnormal form W factors (up to redex permutation) to a path M e \Gamma! V h \Gamma! W . This paper establishes a stability `a la Berry or polyreflectivity theorem [D, La, T] which extends the minimality property to Rewriting Systems with critical pairs. The theorem is proved in the setting of Axiomatic Rewriting Systems where sets of headnormal forms are characterised by their frontier property in the spirit of [GK]. 1 Axiomatic Rewriting Theory Rewriting is a versatile model of computation which stretches from Turing Machines and Petri nets to  calculus and ßcalculus. This versatility has generated in the past a variety of theories which are still poorly interconnected. Axiomatic Rewriting Theory [GLM, M, 1,...
Information Categories
 Applied Categorical Structures
, 1991
"... \Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes o ..."
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\Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of domains. In previous work, the present authors introduced the notion of an Icategory as an abstraction and enhancement of this idea, with emphasis on the solution of domain equations of the form D = F (D), with F a functor. An important feature of the work is that we are not conned to domains of computation as usually understood; other classes of spaces, more familiar to mathematicians in general, become also accessible. Here we present the idea in terms of what we call information categories, which are concrete Icategories in which the objects are structured sets of \tokens" and morphisms are relations between tokens. This is more in the spirit of information system work, and...
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ..."
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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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
Some Monoidal Closed Categories of Stable Domains and Event Structures
 Mathematical Structures in Computer Science
, 1993
"... This paper introduces the following new constructions on stable domains and event structures: the tensor product, the linear function space, and the exponential. It results in a monoidal closed category of dIdomains as well as one of stable event structures which can be used to interpret intuitioni ..."
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This paper introduces the following new constructions on stable domains and event structures: the tensor product, the linear function space, and the exponential. It results in a monoidal closed category of dIdomains as well as one of stable event structures which can be used to interpret intuitionistic linear logic. Finally, the usefulness of the category of stable event structures for modeling concurrency and its relation to other models are discussed. 1 Introduction In [GL88], Girard and Lafont introduced intuitionistic linear logic with two (equivalent) formalisms: the sequent calculus and the combinators. In terms of category theory, as suggested in their paper, the combinator formulation corresponds to a linear category, i.e., a symmetric monoidal closed category with finite products and coproducts [GL88], [MM89]. One of the wellknown monoidal closed categories is the one of coherent spaces (with linear maps) [Gi87a]. In fact, it is in this category that Girard discovered line...
Stable and Sequential Functions on Scott Domains
, 1992
"... The search for a general semantic characterization of sequential functions is motivated by the full abstraction problem for sequential programming languages such as PCF. We present here some new developments towards such a theory of sequentiality. We give a general definition of sequential functions ..."
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The search for a general semantic characterization of sequential functions is motivated by the full abstraction problem for sequential programming languages such as PCF. We present here some new developments towards such a theory of sequentiality. We give a general definition of sequential functions on Scott domains, characterized by means of a generalized form of topology, based on sequential open sets. Our notion of sequential function coincides with the KahnPlotkin notion of sequential function when restricted to distributive concrete domains, and considerably expands the class of domains for which sequential functions may be defined. We show that the sequential functions between two dIdomains, ordered stably, form a dIdomain. The analogous property fails for KahnPlotkin sequential functions. Our category of dIdomains and sequential functions is not cartesian closed, because application is not sequential. We attribute this to certain operational assumptions underlying our notio...
Sequential Functions on Indexed Domains and Full Abstraction for a Sublanguage of PCF
 Proceedings of the 8th Annual Symposium on Mathematical Foundations of Program Semantics, volume 802 of Lecture Notes in Computer Science
, 1994
"... ion for a Sublanguage of PCF Stephen Brookes Shai Geva April 1993 CMUCS93163 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 To appear in Proceedings of Mathematical Foundations of Programming Semantics, New Orleans, 1993 (Springer Verlag Lecture Notes in Computer Sc ..."
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ion for a Sublanguage of PCF Stephen Brookes Shai Geva April 1993 CMUCS93163 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 To appear in Proceedings of Mathematical Foundations of Programming Semantics, New Orleans, 1993 (Springer Verlag Lecture Notes in Computer Science). Abstract We present a general semantic framework of sequential functions on domains equipped with a parameterized notion of incremental sequential computation. Under the simplifying assumption that computation over function spaces proceeds by successive application to constants, we construct a sequential semantic model for a nontrivial sublanguage of PCF with a corresponding syntactic restriction  that variables of function type may only be applied to closed terms. We show that the model is fully abstract for the sublanguage, with respect to the usual notion of program behavior. This research was supported in part by National Science Foundation grant CCR9006064. The views and...
Factorisation Systems on Domains
, 1996
"... We present a cartesian closed category of continuous domains containing the classical examples of Scottdomains with continuous functions and Berry's dIdomains with stable functions as full cartesian closed subcategories. Furthermore, the category is closed with respect to bilimits and there i ..."
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We present a cartesian closed category of continuous domains containing the classical examples of Scottdomains with continuous functions and Berry's dIdomains with stable functions as full cartesian closed subcategories. Furthermore, the category is closed with respect to bilimits and there is an algebraic and a generalised topological description of its morphisms. 1 Introduction There are two kinds of morphism that are studied in classical domain theory, Scottcontinuous ones and stable ones. Berry introduced stable maps to model sequentiality in the calculus [Ber78]. A stable map, in addition to being continuous (i.e. preserving directed suprema), also preserves bounded binary infima. So, for first order the stable functions are a subset of the continuous ones, but at higher order types the continuous and the stable function space become incomparable. Stability captures sequentiality to some extend, e.g. POR is not a stable function, yet the stable model of PCF fails to be fully...
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.
Bistability: an Extensional Characterization of Sequentiality
"... Abstract. We give a simple ordertheoretic construction of a cartesian closed category of sequential functions. It is based on biordered sets analogous to Berry's bidomains, except that the stable order is replaced with a new notion, the bistable order, and instead of preserving stably bounded ..."
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Abstract. We give a simple ordertheoretic construction of a cartesian closed category of sequential functions. It is based on biordered sets analogous to Berry's bidomains, except that the stable order is replaced with a new notion, the bistable order, and instead of preserving stably bounded greatest lower bounds, functions are required to preserve bistably bounded least upper bounds and greatest lower bounds. We show that bistable cpos and bistable and continuous functions form a CCC, yielding models of functional languages such as the simplytyped *calculus and SPCF. We show that these models are strongly sequential and use this fact to prove universality and full abstraction results. 1 Introduction A longstanding problem in domain theory has been to find a simple characterization of higherorder sequential functions which is wholly extensional in character. Typically, what is sought is some form of mathematical structure, such that settheoretic functions which preserve this structure are sequential and can be used to construct a cartesian closed category. Clearly, any solution to this problem is dependant on what one means by sequential. It has been closely associated with the full abstraction problem for PCF, although it is now known that PCF sequentiality cannot be characterized effectively in this sense [8, 11]. As an alternative, we have the strongly stable, or sequentially realizable functionals [2, 12]. Here the difficulty is perhaps to understand precisely the sense in which these are sequential at higher types, in the absence of direct connections to a sequential programming language or an explicitly sequential model.