Syntactic Expressibility

Overview

In the previous chapters we described a variety of proof rules for proving fair termination. All these rules were described independently of any formal assertion language in which the various predicates needed for the proof might be expressed. In particular, the completeness proofs establish the existence of the sets of states needed without their representation by a formula in a calculus.

In this chapter we deal with the issue of syntactic expressibility in a formalized calculus. We introduce an assertion language, which we call L _μ , based on the (positive) μ-calculus and show that this language is sufficiently powerful for our needs. This language has already been used in the past to express predicates needed in termination proofs of recursive procedures [HP 73, deB 80].

The main concern is to show that in this calculus it is possible to express formally the weakest precondition for fair termination (fwp), characterizing the set of all the (initial) states for which no infinite fair computations of a given program exist. The first discussion of such a predicate transformer for weak fairness appears in [PA 81]; for unconditional and strong fairness we follow the ideas of [deR 81], and, mainly, of [SdeRG 84].

The version of the μ-calculus used is augmented with (constants denoting) all the recursive ordinals, an odering relation (on ordinals) and the characteristic predicate for the closing ordinal (of a structure). A similar language is used also by [AP 82] in the context of random assignments. A source book for the mathematics of monotone operators and inductivity is [MO 74].