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 probabilistic Bu ̈chi automata for this logic that are exponential in the size of the formula. One consequence of our construction is a new, improved EXPTIME model checking algorithm (as opposed to the previously known doubly exponential time) for Markov Decision Processes and LTL\GU formulae.
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