Computing Bounds for Counter Automata

Abstract

Qualitative formal verification, that seeks Boolean answers about the behavior of a system, is often insufficient for practical purposes. Observing quantitative information is of greater interest, e.g. for the calibration of a battery or a real-time scheduler. Historically, the focus has been on quantities in continuous domain, but recent years showed a renewed interest for discrete quantitative domains. Counter Automata (CA) is a quantitative extension of classical omega-automata. Recently a nice theory has been developed for them that extends the qualitative setting, with counterparts in terms of logics, automata and algebraic structure. We propose an adaptation, with plenty of practical applications,  of this formalism to express properties over discrete quantitative domains. The behavior of a Counter Automaton defines a function from infinite words to integers. Finding the bounds of such a function over a given set of words can be seen as an extension of qualitative universal and existential model-checking. Although the problem of determining whether such bounds are finite have already been addressed, efficient algorithms to compute their exact values still lack.  We propose an non-naive method for the computation of the exact values of these bounds. It relies on a generalization of the emptiness problem of omega-automata. To solve this generalized emptiness problem, we propose an algorithm that extends emptiness check algorithms based on SCC enumeration.