The UNITY rules for leads-to, based on totality of the commands and weak fairness, are generalized to specifications with nontotal commands and impartiality. The rules and the corresponding predicate transformers are proved to be sound and complete by elementary means. These results are subsequently extended to specifications where the liveness property also contains a finite number of strong fairness assumptions. This is illustrated by means of a proof of starvation freedom for the standard implementation of mutual exclusion by plain semaphores, with strong fairness for the P operations.
A paper Complete assertional proof rules for progress under weak and strong fairness is in print. A PVS dumpfile is available. PVS users can unzip it and undump it, and inspect the specification and proof files. It is made primarily for author confidence, not for communication. It is therefore not very instructive.
While finalizing the paper mentioned above, I was inspired by the reviews of that paper to go considerably further on this road.
UNITY logic serves to prove for a concurrent program progress properties of the form ``P leads to Q''. This logic is sound and complete with respect to the operational semantics. When a UNITY program is extended with axioms that postulate progress properties, the logic can use these axioms and remain sound. It is unknown whether the logic remains complete. It is shown here that the logic remains complete when the axioms added are moderate, where moderacy of an axiom "F leads to G" means that "F unless G" holds.
It is further shown that specifications with leads-to axioms that are not moderate, as well as specifications with strong fairness requirements, can be simulated faithfully by specifications with moderate leads-to axioms. A paper is in preparation. The results have been verified with PVS, see the PVS dumpfile. This dump file may be somewhat better accessible than the one above. They are independent. The completeness proof in the one above is redone here.
Comments and questions are welcome.
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Wim H. Hesselink