One key property of the -calculus is that there exists a minimal computation (the head-reduction) 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 head-normal 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 poly-reflectivity 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 head-normal 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 ß-calculus. This versatility has generated in the past a variety of theories which are still poorly interconnected. Axiomatic Rewriting Theory [GLM, M, 1,...

\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 I-category 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 I-categories 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...

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This paper shows that Axiom d and Axiom I are important when one works within the realm of Scott-domains. In particular, it has been shown that (i) if [D ! s D] has a countable basis, then D must be finitary, for any Scott-domain D; (ii) if [D ! s D] is bounded complete, then D must be distributive, for any finitary Scott-domain D. Therefore, the category of dI-domains is the largest cartesian closed category within omega-algebraic, 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

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 Kahn-Plotkin 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 dI-domains, ordered stably, form a dI-domain. The analogous property fails for Kahn-Plotkin sequential functions. Our category of dI-domains and sequential functions is not cartesian closed, because application is not sequential. We attribute this to certain operational assumptions underlying our notio...

ion for a Sub-language of PCF Stephen Brookes Shai Geva April 1993 CMU-CS-93-163 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 non-trivial sub-language 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 sub-language, with respect to the usual notion of program behavior. This research was supported in part by National Science Foundation grant CCR-9006064. The views and...

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 dI-domains 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 well-known monoidal closed categories is the one of coherent spaces (with linear maps) [Gi87a]. In fact, it is in this category that Girard discovered line...

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the &-semantics, especially call-by-value and call-byname semantics, are highlighted. Furthermore, a property can be validated for all instances of the &-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the &-semantics. We present and apply means for very simple proofs of equivalence with the denotational &-semantics for a large class of reduction-based &-semantics. Our basis are simple first-order constructorbased functional programs with patterns. Keyw...

The problem of decomposing domains into sensible factors is addressed and solved for the case of dI-domains. A decomposition theorem is proved which allows the represention of a large subclass of dI-domains in a product of flat domains. Direct product decompositions of Scott-domains are studied separately. 1 Introduction This work was initiated by Peter Buneman's interest in generalizing relational databases, see [6]. He --- quite radically --- dismissed the idea that a database should be forced into the format of an n-ary relation. Instead he allowed it to be an arbitrary anti-chain in a Scott-domain. The reason for this was that advanced concepts in database theory, such as `null values', `nested relations', and `complex objects' force one to augment relations and values with a notion of information order. Following Buneman's general approach, the question arises how to define basic database theoretic concepts such as `functional dependency' for anti-chains in Scott-domains. For...

This paper describes a multirelation semantics for a fragment of the Extended Entity-Relationship schemata formalism based on the bicategorical definition of multirelations. The approach we follow is elementary---we try to introduce as few notions as possible. We claim that bicategorical algebra handles gracefully multirelations and their operations, and that multirelations are essential in a number of applications; moreover, the bicategorical composition of multirelations turns out to correspond to natural joins. From the formal semantics we derive an algorithm that can establish statically the possibility of building parallel ownership paths of weak entities. The ideas described in this paper have been implemented in a free tool, ERW, which lets users edit sets and multirelations instantiating an EER schema via a sophisticated web interface.