In  quantitative  verification, system states/transitions have associated payoffs, and these are used to associate  mean-payoffs  with infinite behaviors. In this paper, we propose to define  ***  -languages via Boolean queries over mean-payoffs. Requirements concerning averages such as "the number of messages lost is negligible" are not  ***  -regular, but specifiable in our framework. We show that, for closure under intersection, one needs to consider multi-dimensional payoffs. We argue that the acceptance condition needs to examine the set of  accumulation points  of sequences of mean-payoffs of prefixes, and give a precise characterization of such sets. We propose the class of  multi-threshold mean-payoff languages  using acceptance conditions that are Boolean combinations of inequalities comparing the minimal or maximal accumulation point along some coordinate with a constant threshold. For this class of languages, we study expressiveness, closure properties, analyzability, and Borel complexity.
['Rajeev Alur', 'Aldric Degorre', 'Oded Maler', 'Gera Weiss']
On Omega-Languages Defined by Mean-Payoff Conditions