Synthesis from Weighted Specifications with Partial Domains over Finite Words

TL;DR

This paper studies the synthesis of reactive systems from quantitative specifications over finite words, establishing decidability results and developing an infinite game framework for solving synthesis problems with weighted automata.

Contribution

It introduces a comprehensive framework for synthesis from weighted specifications with partial domains, using infinite games to address non-regular objectives.

Findings

Decidability results for threshold, best-value, and approximate synthesis objectives.

Development of infinite game methods for weighted automata with sum, discounted-sum, and average measures.

Framework applicable to systems with partial domain specifications.

Abstract

In this paper, we investigate the synthesis problem of terminating reactive systems from quantitative specifications. Such systems are modeled as finite transducers whose executions are represented as finite words in $(I\times O)^*$, where $I,O$ are finite sets of input and output symbols, respectively. A weighted specification $S$ assigns a rational value (or $-\infty$) to words in $(I\times O)^*$, and we consider three kinds of objectives for synthesis, namely threshold objectives where the system's executions are required to be above some given threshold, best-value and approximate objectives where the system is required to perform as best as it can by providing output symbols that yield the best value and $\varepsilon$-best value respectively w.r.t. $S$. We establish a landscape of decidability results for these three objectives and weighted specifications with partial domain over finite words given by deterministic weighted automata equipped with sum, discounted-sum and average measures. The resulting objectives are not regular in general and we develop an infinite game framework to solve the corresponding synthesis problems, namely the class of (weighted) critical prefix games.