Limit Deterministic and Probabilistic Automata for LTL ∖ GU
Lecture Notes in Computer Science
LTL\GU is a fragment of linear temporal logic (LTL), where negations appear only on propositions, and formulas are built using the temporal operators X (next), F (eventually), G (always), and U (until, with the restriction that no until operator occurs in the scope of an always operator. Our main result is the construction of Limit Deterministic Büchi automata for this logic that are exponential in the size of the formula. One consequence of our construction is a new, improved EX-PTIME model
... cking algorithm (as opposed to the previously known doubly exponential time) for Markov Decision Processes and LTL\GU formulae. Another consequence is that it gives us a way to construct exponential sized Probabilistic Büchi Automata for LTL\GU . U (until), with the restriction that no U operator appears in the scope of a G operator. Our main result is a translation of LTL\GU formulae into nondeterministic Büchi automata of exponential size that is deterministic in the limitan automaton is deterministic in the limit (or limit deterministic) if the transitions from any state that is reachable from an accepting state are deterministic. This construction should be contrasted with the observation that any translation from LTL\GU to deterministic automata must in the worst case result in automata that are doubly exponential in size ; in fact, this lower bound applies to any fragment of LTL that has ∨, ∧, and F . Our construction of limit deterministic automata for LTL\GU proceeds in two steps. First we construct limit deterministic automata for LTL(F ,G) which is the LTL fragment without until, i.e., with just the temporal operators next, always, and eventually. Next, we observe that the automaton for ϕ ∈ LTL\GU can be seen as the composition of two limit deterministic automata: one automata for the formula ψ, where all the until-free subformulae of ϕ are replaced by propositions, and another automaton for the until-free subformulae of ϕ. This composition is reminiscent of the master-slave composition in  and the composition of temporal testers  but with some differences. Our construction of exponentially sized limit deterministic automata for LTL\GU has complexity theoretic consequences for model checking MDPs. Courcoubetis and Yannakakis  proved that the problem of model checking MDPs against LTL is 2EXPTIME-complete. Our automata construction, coupled with the algorithm outlined in  , shows that model checking MDPs against LTL\GU is in EXPTIME; we prove a matching lower bound in this paper as well. Thus, for a large, expressively rich subclass of LTL specifications, our results provide an exponential improvement to the complexity of model checking MDPs. Another consequence of our main result is that it gives us a way to translate LTL\GU formulae to exponential sized probabilistic Büchi automata (PBA)  . Probabilistic Büchi automata are like Büchi automata, except that they probabilistically choose the next state on reading an input symbol. On input w, such a machine can be seen as defining a probability measure on the space of all runs on w. A PBA is said to accept a (infinite length) string w iff the set of all accepting runs (i.e., runs that visit some final state infinitely often) have measure > 0. We use the observation that any assignment of non-zero probabilities to the nondeterministic choices of a limit deterministic NBA, results in a PBA that accepts the same language  . This result also generalizes some of the results in  where exponential sized weak probabilistic monitors 1 are constructed for the LTL fragment with just the temporal operators X and G.