Real-Time Synthesis is Hard!

TL;DR

This paper explores the limits of reactive synthesis for Metric Interval Temporal Logic, establishing undecidability in many cases and providing an efficient algorithm for a specific decidable subproblem.

Contribution

It precisely delineates the decidability boundary for MITL synthesis and introduces an efficient on-the-fly algorithm for the BRRS subcase.

Findings

RS is undecidable on finite words.

Many restrictions remain undecidable.

An efficient algorithm for BRRS is proposed.

Abstract

We study the reactive synthesis problem (RS) for specifications given in Metric Interval Temporal Logic (MITL). RS is known to be undecidable in a very general setting, but on infinite words only; and only the very restrictive BRRS subcase is known to be decidable (see D'Souza et al. and Bouyer et al.). In this paper, we precise the decidability border of MITL synthesis. We show RS is undecidable on finite words too, and present a landscape of restrictions (both on the logic and on the possible controllers) that are still undecidable. On the positive side, we revisit BRRS and introduce an efficient on-the-fly algorithm to solve it.