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We show how to automatically construct a system that satisfies a givenlogical specification and has an optimal average behavior with respect to aspecification with ratio costs.  When synthesizing a system from a logical specification, it is often the casethat several different systems satisfy the specification. In this case, it isusually not easy for the user to state formally which system she prefers. Priorwork proposed to rank the correct systems by adding a quantitative aspect tothe specification. A desired preference relation can be expressed with (i) aquantitative language, which is a function assigning a value to every possiblebehavior of a system, and (ii) an environment model defining the desiredoptimization 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) aquantitative language given by an automaton with a ratio cost function, and(ii) an environment model given by a labeled Markov decision process. Theobjective of the system is to minimize the expected (ratio) costs. The solutionis based on a reduction to Markov Decision Processes with ratio cost functionswhich do not require that the costs in the denominator are strictly positive.We find an optimal strategy for these using a fractional linear program.

Synthesizing Systems with Optimal Average-Case Behavior for Ratio Objectives