Completeness and predicate-based abstract interpretation

  • Authors:

  • Alan Mycroft

  • Affiliations:

  • -

  • Venue:
  • PEPM '93 Proceedings of the 1993 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation
  • Year:
  • 1993

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Abstract

Traditionally, the theory of abstract interpretation has concentrated on the study of when one interpretation is sound (also safe or correct) with respect to another. We consider the dual notion of when one interpretation is complete with respect to another. Under the usual formulation of abstract interpretation, undecidability in general implies that a finitely computable sound abstraction of the standard interpretation is not complete. (For example, if we simplify 643 * (-192) to (+) * (-) using the “rule of signs” we cannot expect to retrieve -123456 from the resulting (1), even though we are certain that the result is negative.) Based on the idea that compilers can only depend on a finite number of program properties, we augment interpretations with predicate symbols specifying properties of interest (thereby replacing algebraic interpretations with logic interpretations). Interpretation J being sound (resp. complete) with respect to I is now phrased as “all questions (formulae) yielding true for J (resp. I) also yield true for I (resp. J)”.The traditional “rule of signs” turns out to be sound and complete for multiplication but only sound for addition.Sometimes abstract interpretations have spurious domain elements. The state minimisation algorithm for finite deterministic automata can be used to produce a canonical (simplest) abstract interpretation which is sound and complete with respect to any given finite abstract interpretation but possibly simpler to compute.A homomorphism always yields a sound and complete abstraction. Moreover, we show that a sound and complete abstraction map is not necessarily a homomorphism, but its composition with the natural map to the canonical interpretation is a homomorphism,One side-effect of our formulation of abstract interpretation is that it de-emphasises the ordering on the abstract domain which is relegated to an (optional) proof basis.