Controller synthesis for MDPs and Frequency LTL$\setminus$GU

TL;DR

This paper introduces a new decidability result for controller synthesis in Markov decision processes with frequency LTL, specifically for the fragment excluding the until operator within G, using a novel automata translation.

Contribution

It extends the decidability of controller synthesis for frequency LTL to a broader fragment by developing a new automata-based translation method.

Findings

Decidability established for the fragment fLTL\setminus GU.

Automata translation method for frequency LTL formulas.

Framework supports probabilistic system control with frequency guarantees.

Abstract

Quantitative extensions of temporal logics have recently attracted significant attention. In this work, we study frequency LTL (fLTL), an extension of LTL which allows to speak about frequencies of events along an execution. Such an extension is particularly useful for probabilistic systems that often cannot fulfil strict qualitative guarantees on the behaviour. It has been recently shown that controller synthesis for Markov decision processes and fLTL is decidable when all the bounds on frequencies are 1. As a step towards a complete quantitative solution, we show that the problem is decidable for the fragment fLTL$\setminus$GU, where U does not occur in the scope of G (but still F can). Our solution is based on a novel translation of such quantitative formulae into equivalent deterministic automata.