Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic ( PLTL ) have been invented. In this paper, a sound and complete loop-type sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ G L T for PLTL with the temporal operators “next” and “henceforth always” ( $${\mathbf{PLTL}}^{n,a}$$ PLTL n , a ) is considered at first. We prove that all rules of $$\mathbf{G} _\text {L}{} \mathbf{T} $$ G L T are invertible and that the structural rules of weakening and contraction, as well as the rule of cut, are admissible in $$\mathbf{G} _\text {L}{} \mathbf{T} $$ G L T . We describe a decision procedure for $${\mathbf{PLTL}}^{n,a}$$ PLTL n , a based on the introduced calculus $$\mathbf{G} _\text {L}{} \mathbf{T} $$ G L T . Afterwards, we introduce a sound and complete sequent calculus $$\mathbf{G} _\text {L}{} \mathbf{T} ^\mathcal {U}$$ G L T U for PLTL with the temporal operators “next” and “until”.

中文翻译：

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时序逻辑的循环型序列演算

已经发明了用于命题线性时间逻辑 (PLTL) 的各种类型的演算（希尔伯特、根岑序列、分解演算、表格）。在这篇论文中，一个完整的循环型序列演算 $$\mathbf{G} _\text {L}{} \mathbf{T} $$ GLT 用于 PLTL，带有时间运算符“next”和“hereforth always” ( $${\mathbf{PLTL}}^{n,a}$$ PLTL n , a ) 首先被考虑。我们证明 $$\mathbf{G} _\text {L}{} \mathbf{T} $$ GLT 的所有规则都是可逆的，并且弱收缩的结构规则以及切割规则是在 $$\mathbf{G} _\text {L}{} \mathbf{T} $$ GLT 中可接受。我们描述了 $${\mathbf{PLTL}}^{n,a}$$ PLTL n 的决策过程，a 基于引入的微积分 $$\mathbf{G} _\text {L}{} \mathbf{ T} $$ GLT 。然后，