Probabilistic Synthesis and Verification for LTL on Finite Traces

We study synthesis and verification of probabilistic models and specifications over finite traces. Probabilistic models are formalized in this work as Markov Chains and Markov Decisions Processes. Motivated by the recent attention given to, and importance of, finite-trace specifications in AI, we use linear-temporal logic on finite traces as a specification formalism for properties of traces with finite but unbounded time horizons. Since there is no bound on the time horizon, our Markov chains generate infinite traces, and we consider two possible semantics: “existential (resp. universal) prefix- semantics” which says that the finite-trace property holds on some (resp. every) finite prefix of the trace. For both types of semantics, we study two computational problems: the verification problem — “does a given Markov chain satisfy the specification with probability one?”; and the synthesis problem — “find a strategy (if there is one) that ensures the Markov decision process satisfies the specification with probability one”. We provide optimal algorithms that follow an automata-theoretic approach, and prove that the complexity of the synthesis problem is 2EXPTIME-complete for both semantics, and that for the verification problem it is PSPACE-complete for the universal-prefix semantics, but EXPSPACE-complete for the existential-prefix semantics.