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We introduce sequent calculus for multi-modal logic KD45n which uses efficient loop-check. Efficiency of the used loop-check is obtained by using marked modal operator squarei which is used as an alternative to sequent with histories ([2,3]).We use inference rules with or branches to make all rules invertible or semi-invertible. We showthe maximum height of the constructed derivation tree. Also polynomial space complexity is proved.

In this paper, we present sequent calculus for linear temporal logic. This sequent calculus uses efficient loop-check techinque. We prove that we can use not all but only several special sequents from the derivation tree for the loop-check. We use indexes to discover these special sequents in the sequent calculus. These restrictions let us to get efficient decision procedure based on introduced sequent calculus.