Monitor Logics for Quantitative Monitor Automata

In the last years, there has been increasing interest in quantitative features of the specification and analysis of systems. Such quantitative aspects include the consumption of a certain resource or the output of a benefit. Both weighted automata and weighted logics [7] are means to achieve this quantitative description of systems. They can be employed for both finite and infinite input. Very recently, Chatterjee et al. introduced a new automaton model operating on infinite words [4]. Quantitative Monitor Automata are equipped with a finite number of monitor counters. At each transition, a counter can be started, terminated, or the value of the counter can be increased or decreased. The term “monitor” stems from the fact that the values of the counters do not influence the behavior of the automaton. The values of the counters when they are terminated provide an infinite sequence of weights, which is evaluated into a single weight using a valuation function. Quantitative Monitor Automata possess several interesting features. They are expressively equivalent to a subclass of Nested Weighted Automata [3], an automaton model which for many valuation functions has decidable emptiness and universality problems. Quantitative Monitor Automata are also very expressive. As an example, imagine a storehouse with a resource which is restocked at regular intervals. Between restocks, demands can remove one unit of this resource at a time. Such a succession of restocks and demands can be modeled as an infinite sequence over the alphabet {restock, demand}. Interesting quantitative properties of such a sequence include the long-term average demand, the minimum demand and the maximum demand between restocks. These properties can be described using Quantitative Monitor Automata. At every restock a counter is started, counting the number of demands until the next restock. An appropriate valuation function then computes the desired property. For the average demand, this can be achieved with the Cesàro mean which was introduced to automata theory by Chatterjee et. al in [2]. Note that behaviors like these cannot be modeled using weighted Büchi-automata [12, 13] or their extension with valuation functions [8]. In the latter model, the Cesàro mean of any sequence is bounded by the largest transition weight in the automaton. This is not the case for Quantitative Monitor Automata. In this paper, we develop a logic which is expressively equivalent to Quantitative Monitor Automata. Our main results are the following.