Synthesizing Systems with Optimal Average-Case Behavior for Ratio Objectives

TL;DR

This paper presents a method to automatically synthesize systems that satisfy logical specifications while optimizing average-case ratio costs, using reductions to Markov decision processes and fractional linear programming.

Contribution

It introduces a novel synthesis approach for systems with ratio-based cost functions and environment models, enabling optimal average-case behavior.

Findings

Successfully reduces synthesis problem to Markov decision processes with ratio costs.

Employs fractional linear programming to find optimal strategies.

Achieves systems with minimized expected ratio costs under specified models.

Abstract

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 ratio 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 ratio cost function, and (ii) an environment model given by a labeled Markov decision process. The objective of the system is to minimize the expected (ratio) costs. The solution is based on a reduction to Markov Decision Processes with ratio 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.