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Relational Properties of Domains
 Information and Computation
, 1996
"... New tools are presented for reasoning about properties of recursively defined domains. We work within a general, categorytheoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a ..."
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Cited by 113 (6 self)
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New tools are presented for reasoning about properties of recursively defined domains. We work within a general, categorytheoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with userdeclared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality /finality property gives rise to the coinduct...
Explicit Cyclic Substitutions
, 1993
"... In this paper we consider rewrite systems that describe the lambdacalculus enriched with recursive and nonrecursive local definitions by generalizing the `explicit substitutions' used by Abadi, Cardelli, Curien, and Lévy [1] to describe sharing in lambdaterms. This leads to `explicit cyclic ..."
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Cited by 25 (2 self)
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In this paper we consider rewrite systems that describe the lambdacalculus enriched with recursive and nonrecursive local definitions by generalizing the `explicit substitutions' used by Abadi, Cardelli, Curien, and Lévy [1] to describe sharing in lambdaterms. This leads to `explicit cyclic substitutions' that can describe the mutual sharing of local recursive definitions. We demonstrate how this may be used to describe standard binding constructions (let and letrec)  directly using substitution and fixed point induction as well as using `smallstep' rewriting semantics where substitution is interleaved with the mechanics of the following betareductions. With this we hope to contribute to the synthesis of denotational and operational specifications of sharing and recursion.
Towards Lambda Calculus OrderIncompleteness
 Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science
"... After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially orde ..."
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Cited by 3 (3 self)
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After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially ordered model (orderincompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is orderincomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the orderincompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).
An Ideal Model for an Extended LambdaCalculus with Refinement
, 1991
"... In Computer Science, Lambda Calculus has been mainly used as the skeleton of functional programming languages. It has also been used as a higher order parameterization mechanism in some specification languages. In this paper we view calculus as both the applicative structure of a programming formal ..."
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Cited by 2 (1 self)
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In Computer Science, Lambda Calculus has been mainly used as the skeleton of functional programming languages. It has also been used as a higher order parameterization mechanism in some specification languages. In this paper we view calculus as both the applicative structure of a programming formalism and a lowlevel specification formalism. Considered as a programming formalism, its operational semantics is the usual one, mainly based on fireduction. Considered as a specification formalism calculus admits a precise notion of refinement between expressions. This refinement relation will stand for the correctness of a step in the incremental development of a program from a specification. The main goal of this paper is to show that calculus, extended with some set operators, can be interpreted as a specification formalism in a domain whose elements are a particular class of posets (partial ordered sets), the closed ideals. The main reason of such interpretation is that it allows to...
A Note on Absolutely Unorderable Combinatory Algebras
, 2001
"... Plotkin [18] has conjectured that there exists an absolutely unorderable combinatory algebra, i.e. an algebra which cannot be embedded in another algebra that admits a nontrivial compatible partial order. In this paper we prove that a wide class of combinatory algebras admits extensions with a non ..."
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Plotkin [18] has conjectured that there exists an absolutely unorderable combinatory algebra, i.e. an algebra which cannot be embedded in another algebra that admits a nontrivial compatible partial order. In this paper we prove that a wide class of combinatory algebras admits extensions with a nontrivial compatible partial order.