  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.
