Liveness verification of reversal-bounded multicounter machines with a free counter (extended abstract)

We investigate the Presburger liveness problems for nondeterministic reversal-bounded multicounter machines with a free counter (NCMFs). We show the following: – The ∃-Presburger-i.o. problem and the ∃-Presburger-eventual problem are both decidable. So are their duals, the ∀-Presburger-almost-always problem and the ∀-Presburger-always problem. – The ∀-Presburger-i.o. problem and the ∀-Presburger-eventual problem are both undecidable. So are their duals, the ∃-Presburger-almost-always problem and the ∃-Presburger-always problem. These results can be used to formulate a weak form of Presburger linear temporal logic and developits model-checking theories for NCMFs. They can also be combined with [12] to study the same set of liveness problems on an extended form of discrete timed automata containing, besides clocks, a number of reversalbounded counters and a free counter.