Undecidable Cases of Model Checking Probabilistic Temporal-Epistemic Logic (Extended Abstract)

TL;DR

This paper explores the boundaries of decidability in model checking probabilistic temporal-epistemic logics, revealing that slight generalizations lead to undecidability, thus clarifying limitations in existing approaches.

Contribution

It demonstrates that minor extensions to known decidable cases of model checking for probabilistic logics cause undecidability, clarifying the scope of previous results.

Findings

Finite reach operators are decidable, infinite reach operators are undecidable.

Extensions involving polynomial probability combinations are undecidable.

Adding monadic second order quantification makes model checking undecidable.

Abstract

We investigate the decidability of model-checking logics of time, knowledge and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observed discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular, for a synchronous perfect recall, a generalization from temporal operators with finite reach to operators with infinite reach renders model checking undecidable. The case of the clock semantics is closely related to a monadic second order logic of time and probability that is known to be decidable, except on a set of measure zero. We show that two distinct extensions of this logic make model checking undecidable. One of these involves polynomial combinations of probability terms, the other involves monadic second order quantification into the scope of probability operators. These results explain some of the restrictions in previous work.