Synthesizing Systems with Optimal Average-Case Behavior for Ratio Objectives

We show how to automatically construct a system that satisfies a given logical specification and has an optimal average behavior with respect to a specification with fractional costs. When synthesizing a system from a logical specification, it is often the case that several different systems satisfy the specification. In this case, it is usually not easy for the user to state formally which system she prefers. Prior work proposed to rank the correct systems by adding a quantitative aspect to the specification. A desired preference relation can be expressed with (i) a quantitative language, which is a function assigning a value to every possible behavior of a system, and (ii) an environment model defining the desired optimization criteria of the system, e.g., worst-case or average-case optimal. In this paper, we show how to synthesize a system that is optimal for (i) a quantitative language given by an automaton with a fractional cost function, and (ii) an environment model given by a labeled Markov decision process. The objective of the system is to minimize the expected (fractional) costs. The solution is based on a reduction to Markov Decision Processes with extended-fractional cost functions which do not require that the costs in the denominator are strictly positive. We find an optimal strategy for these using a fractional linear program.