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A Coinduction Principle for Recursively Defined Domains
 THEORETICAL COMPUTER SCIENCE
, 1992
"... This paper establishes a new property of predomains recursively defined using the cartesian product, disjoint union, partial function space and convex powerdomain constructors. We prove that the partial order on such a recursive predomain D is the greatest fixed point of a certain monotone operator ..."
Abstract

Cited by 40 (3 self)
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This paper establishes a new property of predomains recursively defined using the cartesian product, disjoint union, partial function space and convex powerdomain constructors. We prove that the partial order on such a recursive predomain D is the greatest fixed point of a certain monotone operator associated to D. This provides a structurally defined family of proof principles for these recursive predomains: to show that one element of D approximates another, it suffices to find a binary relation containing the two elements that is a postfixed point for the associated monotone operator. The statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains. Following Milner and Tofte [10], the method of proof is called coinduction. It closely resembles the way bisimulations are used in concurrent process calculi [9]. Two specific instances of the coinduction principle already occur in work of Abramsky [2, 1] in the form of `internal full abstraction' theorems for denotational semantics of SCCS and the lazy lambda calculus. In the first case postfixed binary relations are precisely Abramsky's partial bisimulations, whereas in the second case they are his applicative bisimulations. The coinduction principle also provides an apparently useful tool for reasoning about equality of elements of recursively defined datatypes in (strict or lazy) higher order functional programming languages.
A Coinduction Principle for Recursively De ned Domains, Theoretical Computer Science
, 1992
"... This paper establishes a new property of predomains recursively de ned using the cartesian product, disjoint union, partial function space and convex powerdomain constructors. We prove that the partial order on such a recursive predomain D is the greatest xed point of a certain monotone operator ass ..."
Abstract

Cited by 5 (1 self)
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This paper establishes a new property of predomains recursively de ned using the cartesian product, disjoint union, partial function space and convex powerdomain constructors. We prove that the partial order on such a recursive predomain D is the greatest xed point of a certain monotone operator associated to D. This provides a structurally de ned family of proof principles for these recursive predomains: to show that one element of D approximates another, it su ces to nd a binary relation containing the two elements that is a post xed point for the associated monotone operator. The statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains. Following Milner and Tofte [10], the method of proof is called coinduction. It closely resembles the way bisimulations are used in concurrent process calculi [9]. Two speci c instances of the coinduction principle already occur in work of Abramsky [2, 1] in the form of `internal full abstraction ' theorems for denotational semantics of SCCS and the lazy lambda calculus. In the rst case postxed binary relations are precisely Abramsky's partial bisimulations, whereas in the second case they are his applicative bisimulations. The coinduction principle also provides an apparently useful tool for reasoning about equality of elements of recursively de ned datatypes in (strict or lazy) higher order functional programming languages. 1