The Complexity of Mean-Payoff Automaton Expression

Quantitative languages are extension of Boolean languages that assign to each word a real number. With quantitative languages, systems and specifications can be formalized more accurately. For example, a system may use a varying amount of some resource (e.g., memory consumption, or power consumption) depending on its behavior, and a specification may assign a maximal amount of available resource to each behavior, or fix the long-run average available use of the resource. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. Mean-payoff automata forms a class of quantitative languages that is not robust, since it is not closed under the basic algebraic operations: min, max, sum and numerical complement. The class of mean-payoff automaton expressions, recently introduced by Chatterjee et al., is currently the only known class of quantitative languages that is robust, expressive and decidable. This class is defined as the closure of mean-payoff automata under the basic algebraic operations. In this work, we prove that all the classical decision problems for mean-payoff expressions are PSPACE-complete. Our proof improves the previously known 4EXPTIME upper bound. In addition, our proof is significantly simpler, and fully accessible to the automata-theoretic community.