  Formalisms based on temporal logics interpreted over finite strict linear
orders, known in the literature as finite traces, have been used for temporal
specification in automated planning, process modelling, (runtime) verification
and synthesis of programs, as well as in knowledge representation and
reasoning. In this paper, we focus on first-order temporal logic on finite
traces. We first investigate preservation of equivalences and satisfiability of
formulas between finite and infinite traces, by providing a set of semantic and
syntactic conditions to guarantee when the distinction between reasoning in the
two cases can be blurred. Moreover, we show that the satisfiability problem on
finite traces for several decidable fragments of first-order temporal logic is
ExpSpace-complete, as in the infinite trace case, while it decreases to
NExpTime when finite traces bounded in the number of instants are considered.
This leads also to new complexity results for temporal description logics over
finite traces. Finally, we investigate applications to planning and
verification, in particular by establishing connections with the notions of
insensitivity to infiniteness and safety from the literature.
