On the Computational Complexity of Decidable Fragments of First-Order Linear Temporal Logics Ian Hodkinson, Roman Kontchakov, Agi Kurucz, Frank Wolter, and Michael Zakharyaschev Department of Computing, Imperial College, London SW7 2AZ, U.K. Department of Computer Science, King’s College, Strand, London WC2R 2LS, U.K. Department of Computer Science, University of Liverpool, Liverpool L69 7ZF, U.K. (e-mails: imh@doc.ic.ac.uk, romanvk,kuag,mz @dcs.kcl.ac.uk, frank@csc.liv.ac.uk) Abstract We study the complexity of some fragments of first- order temporal logic over natural numbers time. The one- variable fragment of linear first-order temporal logic even with sole temporal operator is EXPSPACE-complete (this solves an open problem of [10]). So are the one-variable, two-variable and monadic monodic fragments with Until and Since. If we add the operators , with given in binary, the fragments become 2EXPSPACE-complete. The packed monodic fragment has the same complexity as its pure first-order part — 2EXPTIME-complete. Over any class of flows of time containing one with an infinite as- cending sequence — e.g., rationals and real numbers time, and arbitrary strict linear orders — we obtain EXPSPACE lower bounds (which solves an open problem of [16]). Our results continue to hold if we restrict to models with finite first-order domains. 1. Introduction What is known about the computational complexity of linear time temporal logics? Everything seems to be clear in the propositional case. The logics with only one tempo- ral operator (‘always in the future’) are known to be co- NP-complete for linear time, for the flows of time and [15] as well as for [22]. The complexity remains the same if one adds the corresponding past oper- ator [15, 22, 25]. The addition of the ‘next-time’ operator and/or the ‘until’ operator to this primitive language makes the logic PSPACE-complete over [22], and , , and the class of arbitrary strict linear or- ders [17, 18]. The succinctness of the operators (‘in moments of time’), where is given in binary, in- creases the complexity to EXPSPACE (over ) [1], but, of course, does not change the expressive power of the language. Compared to this ‘well cultivated garden’, the complex- ity of first-order temporal logics and their fragments is still terra incognita. There are well known ‘negative’ results: for example, -completeness of the two-variable monadic temporal logic of the flow of time ; see, e.g., [11] and references therein. But we could find only one ‘positive’ result: Halpern and Vardi [10] and, independently, Sistla and German [23] showed that the one-variable fragment of the logic with , , and/or over is EXPSPACE- complete. 1 Halpern and Vardi considered this fragment as a propositional epistemic temporal logic with one agent mod- elled by the propositional modal system . They conjec- tured that, as in the propositional case, “even with knowl- edge operators in the language, the complexity still becomes much simpler without and ” [10, page 231]. We take up this conjecture as a starting point of our in- vestigation of the computational complexity of decidable fragments of first-order linear temporal logic. The main technical result of this paper is that over a wide range of flows of time, the one-variable fragment of linear temporal logic even with sole operator is EXPSPACE-hard. We also establish matching EXPSPACE upper bounds for the one-variable, two-variable and monadic monodic fragments of the first-order temporal logic based on the flow of time and having , , , and (since) as their temporal operators. The fragments are EXPSPACE- complete even if we restrict to models with finite first-order domains. If we add the operators , with given in bi- nary, the fragments become 2EXPSPACE-complete. Fi- nally, the packed monodic fragment turns out to be as com- plex as its pure first-order part, i.e., 2EXPTIME-complete 1 [21] and [2] determined the complexity of certain temporalised de- scription logics, which can be regarded as fragments of first-order temporal logics.