On Finite Domains in First-Order Linear Temporal Logic

We consider First-Order Linear Temporal Logic (FO-LTL) over linear time N. We focus on finding models with finite first-order domains for FO-LTL formulas. More precisely, we investigate the complexity of the following problem: given a formula φ and an integer n, is there a model of φ with domain of cardinality at most n? We show that depending on the logic considered (FO or FO-LTL) and on the precise encoding of the problem, the problem is either NP-complete, NEXPTIMEcomplete, PSPACE-complete or NEXPSPACE-complete. In a second part, we exhibit cases where the Finite Model Property can be lifted from fragments of FO to their FO-LTL extension.