The Satisfiability and Validity Problems for Probabilistic Computational Tree Logic Are Highly Undecidable

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Miroslav Chodil, Antonín Kucera

Published: 01 Jan 2025, Last Modified: 23 Jul 2025ICALP 2025EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: The Probabilistic Computational Tree Logic (PCTL) is the main specification formalism for discrete probabilistic systems modeled by Markov chains. Despite serious research attempts, the decidability of PCTL satisfiability and validity problems remained unresolved for 30 years. We show that both problems are highly undecidable, i.e., beyond the arithmetical hierarchy. Consequently, there is no sound and complete deductive system for PCTL.