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**1 - 4**of**4**### Weak completeness theorem for . . .

, 2012

"... We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in [20] which roughly follows the idea of t ..."

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We prove weak (finite set of premises) completeness theorem for extended propositional linear time temporal logic with irreflexive version of until-operator. We base it on the proof of completeness for basic propositional linear time temporal logic given in [20] which roughly follows the idea of the Henkin-Hasenjaeger method for classical logic. We show that a temporal model exists for every formula which negation is not derivable (Satisfiability Theorem). The contrapositive of that theorem leads to derivability of every valid formula. We build a tree of consistent and complete PNPs which is used to construct the model.

### The Properties of Sets of Temporal Logic

, 2012

"... This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [17]. We introduce two modified definitions of a subformula. In the former one we trea ..."

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This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [17]. We introduce two modified definitions of a subformula. In the former one we treat until-formula as indivisible. In the latter one, we extend the set of subformulas of until-formulas by a special disjunctive formula. This is needed to construct a temporal model. We also define an ordered positive-negative pair of finite sequences of formulas (PNP). PNPs represent states of a temporal model.

### DOI: 10.2478/v10037-012-0025-x versita.com/fm/ The Derivations of Temporal Logic

"... Summary. This is a preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [12]. We introduce n-ary connectives and prove their properties. We derive temporal ..."

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Summary. This is a preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [12]. We introduce n-ary connectives and prove their properties. We derive temporal logic formulas.

### Propositional Linear Temporal Logic equipped with Initial Validity Semantics

"... Summary. In article [9] a formal system for Propositional Linear Tempo-ral Logic (in short LTLB) with normal semantics was introduced. The language of this logic consists of “until ” operator in very strict version. The very strict “until ” operator enables to express all other temporal operators. I ..."

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Summary. In article [9] a formal system for Propositional Linear Tempo-ral Logic (in short LTLB) with normal semantics was introduced. The language of this logic consists of “until ” operator in very strict version. The very strict “until ” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [11]. Initial semantics means that we define the validity of formula in model as satisfaction of formula in the initial state of model only instead of satisfaction in all states of model. We prove the Deduction Theorem, and soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [9] and the one introduced in this article. Formal systems for temporal logics are applied in verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within selected formal system. The formal systems introduced in [9] and in this article can be used to carry out such verifications in Mizar.