We study Linear Temporal Logic Modulo Theories over Finite Traces (LTLfMT), a
recently introduced extension of LTL over finite traces (LTLf) where
propositions are replaced by first-order formulas and where first-order
variables referring to different time points can be compared. In general,
LTLfMT was shown to be semi-decidable for any decidable first-order theory
(e.g., linear arithmetics), with a tableau-based semi-decision procedure.
In this paper we present a sound and complete pruning rule for the LTLfMT
tableau. We show that for any LTLfMT formula that satisfies an abstract,
semantic condition, that we call finite memory, the tableau augmented with the
new rule is also guaranteed to terminate. Last but not least, this technique
allows us to establish novel decidability results for the satisfiability of
several fragments of LTLfMT, as well as to give new decidability proofs for
classes that are already known.

本文研究了线性时间逻辑模块理论在有限轨迹上的应用，其中命题使用一阶公式替代，并且可以比较不同时间点引用的一阶变量。我们提出了一种用于 LTLfMT 表格的完备且有效的修剪规则，该规则在满足有限内存的抽象语义条件的任何 LTLfMT 公式下，通过增加新规则可以保证表格也能终止。最重要的是，这种技术使我们能够建立关于 LTLfMT 的多个片段可满足性的新的可判定性结果，并对已知类别给出新的可判定性证明。