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 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.