Partial, total and general correctness

  • Authors:

  • Walter Guttmann

  • Affiliations:

  • Institut für Programmiermethodik und Compilerbau, Universität Ulm, Ulm, Germany

  • Venue:
  • MPC'10 Proceedings of the 10th international conference on Mathematics of program construction
  • Year:
  • 2010

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Abstract

We identify weak semirings, which drop the right annihilation axiom a0 = 0, as a common foundation for partial, total and general correctness. It is known how to extend weak semirings by operations for finite and infinite iteration and domain. We use the resulting weak omega algebras with domain to define a semantics of while-programs which is valid in all three correctness approaches. The unified, algebraic semantics yields program transformations at once for partial, total and general correctness. We thus give a proof of the normal form theorem for while-programs, which is a new result for general correctness and extends to programs with non-deterministic choice. By adding specific axioms to the common ones, we obtain partial, total or general correctness as a specialisation. We continue our previous investigation of axioms for general correctness. In particular, we show that a subset of these axioms is sufficient to derive a useful theory, which includes the Egli-Milner order, full recursion, correctness statements and a correctness calculus. We also show that this subset is necessary.