Sentential Calculus with Identity (SCI) is an extension of classical propositional logic, featuring a new connective of identity between formulas. In SCI two formulas are said to be identical if they share the same denotation. In the semantics of the logic, truth values are distinguished from denotations, hence the identity connective is strictly stronger than classical equivalence. In this paper we present a sound, complete, and terminating algorithm deciding the satisfiability of SCI-formulas, based on labelled tableaux. To the best of our knowledge, it is the first implemented decision procedure for SCI which runs in NP, i.e., is complexity-optimal. The obtained complexity bound is a result of dividing derivation rules in the algorithm into two sets: decomposition and equality rules, whose interplay yields derivation trees with branches of polynomial length with respect to the size of the investigated formula. We describe an implementation of the procedure and compare its performance with implementations of other calculi for SCI (for which, however, the termination results were not established). We show possible refinements of our algorithm and discuss the possibility of extending it to other non-Fregean logics.

中文翻译：

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非Tablet身份逻辑的基于Tableau的决策程序

具有身份的语句演算（SCI）是经典命题逻辑的扩展，具有公式之间的身份的新连接。在SCI中，如果两个公式具有相同的符号，则称它们是相同的。在逻辑的语义中，真值与表示法有所区别，因此，标识连接词严格比经典对等词强。在本文中，我们提出了一种可靠，完整和终止的算法，该算法基于标记的tableaux决定SCI公式的可满足性。据我们所知，这是在NP中运行的SCI的第一个已实现的决策程序，即复杂度最优。所获得的复杂度界限是将算法中的推导规则分为两组的结果：分解规则和等式规则，其相互作用产生了相对于所研究公式的大小具有多项式长度分支的导数树。我们描述了该程序的实现，并将其性能与SCI其他计算的实现进行比较（但是，尚未确定终止结果）。我们展示了算法的可能改进，并讨论了将其扩展到其他非弗拉芒逻辑的可能性。