Undecidable Cases of Model Checking Probabilistic Temporal-Epistemic Logic

TL;DR

This paper demonstrates that slight generalizations in probabilistic temporal-epistemic logic model checking lead to undecidability, clarifying the limits of existing decidability results in the field.

Contribution

It shows that minimal extensions to known decidable logics cause undecidability, resolving open questions about the boundaries of model checking in probabilistic logics.

Findings

Generalizations from finite to infinite temporal operators cause undecidability.

Extensions involving polynomial probability combinations lead to undecidability.

Adding monadic second order quantification within probability operators results in undecidability.

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.