High-Quality Synthesis Against Stochastic Environments

TL;DR

This paper introduces a stochastic extension to LTL synthesis, aiming to generate systems that maximize expected quality in probabilistic environments, addressing practical concerns of system quality beyond mere correctness.

Contribution

It formalizes and solves the stochastic synthesis problem for LTL[F], extending classical synthesis to optimize expected quality under probabilistic input distributions.

Findings

The stochastic synthesis problem is 2EXPTIME-complete.

The approach handles extensions with quality guarantees and environment assumptions.

It provides a framework for high-quality system synthesis in probabilistic settings.

Abstract

In the classical synthesis problem, we are given an LTL formula psi over sets of input and output signals, and we synthesize a transducer that realizes psi. One weakness of automated synthesis in practice is that it pays no attention to the quality of the synthesized system. Indeed, the classical setting is Boolean: a computation satisfies a specification or does not satisfy it. Accordingly, while the synthesized system is correct, there is no guarantee about its quality. In recent years, researchers have considered extensions of the classical Boolean setting to a quantitative one. The logic LTL[F] is a multi-valued logic that augments LTL with quality operators. The satisfaction value of an LTL[F] formula is a real value in [0,1], where the higher the value is, the higher is the quality in which the computation satisfies the specification. Decision problems for LTL become search or optimization problems for LFL[F]. In particular, in the synthesis problem, the goal is to generate a transducer that satisfies the specification in the highest possible quality. Previous work considered the worst-case setting, where the goal is to maximize the quality of the computation with the minimal quality. We introduce and solve the stochastic setting, where the goal is to generate a transducer that maximizes the expected quality of a computation, subject to a given distribution of the input signals. Thus, rather than being hostile, the environment is assumed to be probabilistic, which corresponds to many realistic settings. We show that the problem is 2EXPTIME-complete, like classical LTL synthesis, and remains so in two extensions we consider: one that maximizes the expected quality while guaranteeing that the minimal quality is, with probability $1$, above a given threshold, and one that allows assumptions on the environment.