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Proving a conjecture of Andreka on temporal logic

In 3], a large number of completeness results about variants of discrete linear-time temporal logic are obtained. One of them is left as an open problem: the completeness of the logic of initially and next, for which a deductive system is proposed. This simple logic is of practical importance, since the proof of program invariants only require these modalities. We show here that the conjectured medium completeness of this system indeed holds; further, we show that this decision problem is PSPACE-complete, while deciding the problem of validity is only NP-complete.