On Liveness Verification of Reversal-bounded Multicounter Machines with a Free Counter

We investigate a number of 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 develop its model-checking theories for NCMFs. NCMFs can be used to model reactive systems with counts on external events.