On Relative and Probabilistic Finite Counterability
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A {\em counterexample\/} to the satisfaction of a linear property $\psi$ in a system $\S$ is an infinite computation of $\S$ that violates $\psi$. When $\psi$ is a safety property, a counterexample to its satisfaction need not be infinite. Rather, it is a {\em bad-prefix\/} for $\psi$: a finite word all whose extensions violate $\psi$. The existence of finite counterexamples is very helpful in practice. Liveness properties do not have bad-prefixes and thus do not have finite counterexamples.
We extend the notion of finite counterexamples to non-safety properties. We study {\em counterable languages\/} -- ones that have at least one bad-prefix. Thus, a language is counterable iff it is not liveness. Three natural problems arise: (1) Given a language, decide whether it is counterable, (2) study the length of minimal bad-prefixes for counterable languages, and (3) develop algorithms for detecting bad-prefixes for counterable languages. We solve the problems for languages given by means of LTL formulas or nondeterministic B{\"u}chi automata. In particular, our EXPSPACE-completeness proof for the problem of deciding whether a given LTL formula is counterable, and hence also for deciding liveness, settles a long-standing open problem.
In addition, we make finite counterexamples more relevant and helpful by introducing two variants of the traditional definition of bad-prefixes. The first adds a {\em probabilistic\/} component to the definition. There, a prefix is bad if almost all its extensions violate the property. The second makes it {\em relative\/} to the system. There, a prefix is bad if all its extensions in the system violate the property. We also study the combination of the probabilistic and relative variants. Our framework suggests new variants also for safety and liveness languages. We solve the above three problems for the different variants. Interestingly, the probabilistic variant not only increases the chances to return finite counterexamples, but also makes the solution of the three problems exponentially easier.