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In this paper authors research the problem of traceability of assumptions in logical derivation. The essence of this task is to trace which assumptions from the available knowledge base of assumptions are necessary to derive a certain conclusion. The paper presents a new derivation procedure for propositional logic, which ensures traceability feature. For the derivable conclusion formula derivation procedure also returns the smallest set of assumptions those are enough to get derivation of the conclusion formula. Verification of the procedure were performed using authors implementation.

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 branching-time temporal logic with until operator. This sequent calculus uses efficient loop-checktechinque. 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.

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.

This paper defines decision algorithm for subclass of BKD45DKDIKD logic which is based on known algorithm for temporal BKD45DKDIKD logic [2]. BDI logics are widely used in agent based systems. Such usage of BDI logic can be found in [1]. The original decision algorithm uses loop-check technique for BEL and temporal operators. Applied loop-check technique is not optimized and therefore loop-check takes most of the time used in decision algorithm. Some examples of efficient loop-check applications for logic KT, S4 and some subclasses of intuitionistic logic can be found in [4]. Another efficient loop-check can be found in work [3]. We concentrate on our attitude on loop-check optimization for BEL operator. This paper defines decision algorithm modification, which uses efficient loop-check for BEL operator, but do not effect performance of other parts of algorithm. We define optimization only for BEL operator and therefore we omit temporal operators in this paper.

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BDI logic is widely used to describe agent based systems, since it can express a lot of different real world domains with three main operators: belief, desire and intention. There are lots of works where BDI logic is used as descriptive language, but authors do not talk about implementation issues [5,2,4,8,1]. There is even known general agent implementation scheme, which do not describe howto use formal logic for such scheme implementation [12]. The main aim of this paper is to show the bridge, which connects formal and practical parts of logic. We describe how sequent calculus can be used for implementing some parts of general scheme. In this paper, we show how flexible constraints can be implemented. Sequent calculus usage is illustrated with simple examples.