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Soundness, Idempotence and Commutativity of Set-Sharing

[Page last updated on 2002/04/16.]

Patricia M. Hill
School of Computing
The University of Leeds
Leeds LS2 9JT
England

Roberto Bagnara
Dipartimento di Matematica
Università di Parma
Via D'Azeglio, 85/A
I-43100 Parma
Italy

Enea Zaffanella
Dipartimento di Matematica
Università di Parma
Via D'Azeglio, 85/A
I-43100 Parma
Italy

Abstract:

It is important that practical data-flow analyzers are backed by reliably proven theoretical results. Abstract interpretation provides a sound mathematical framework and necessary generic properties for an abstract domain to be well-defined and sound with respect to the concrete semantics. In logic programming, the abstract domain Sharing is a standard choice for sharing analysis for both practical work and further theoretical study. In spite of this, we found that there were no satisfactory proofs for the key properties of commutativity and idempotence that are essential for Sharing to be well-defined and that published statements of the soundness of Sharing assume the occurs-check. This paper provides a generalization of the abstraction function for Sharing that can be applied to any language, with or without the occurs-check. Results for soundness, idempotence and commutativity for abstract unification using this abstraction function are proven.

Keywords: Abstract Interpretation; Logic Programming; Occurs-Check; Rational Trees; Set-Sharing.

Available: Gzipped postscript, BibTeX entry.