On the computational complexity of decidable fragments of first-order linear temporal logics

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 /spl square/ is EXPSPACE-complete (this solves an open problem of J. Halpern and M. Vardi (1989)). So are the one-variable, two-variable and monadic monodic fragments with Until and Since. If we add the operators O/sup n/, with n given in binary, the fragment becomes 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 ascending sequence - e.g., rationals and real numbers time, and arbitrary strict linear orders - we obtain EXPSPACE lower bounds (which solves an open problem of M. Reynolds (1997)). Our results continue to hold if we restrict to models with finite first-order domains.

[1]  Enrico Franconi,et al.  A Temporal Description Logic for Reasoning over Conceptual Schemas and Queries , 2002, JELIA.

[2]  Michael Zakharyaschev,et al.  On the Products of Linear Modal Logics , 2001, J. Log. Comput..

[3]  Klaus Schild,et al.  Combining Terminological Logics with Tense Logic , 1993, EPIA.

[4]  Joseph Y. Halpern,et al.  The Complexity of Reasoning about Knowledge and Time. I. Lower Bounds , 1989, J. Comput. Syst. Sci..

[5]  Mark Reynolds,et al.  The complexity of the temporal logic with "until" over general linear time , 2003, J. Comput. Syst. Sci..

[6]  Dov M. Gabbay,et al.  Products of Modal Logics, Part 1 , 1998, Log. J. IGPL.

[7]  Frank Wolter,et al.  Properties of Tense Logics , 1996, Math. Log. Q..

[8]  A. Nakamura,et al.  On the size of refutation Kripke models for some linear modal and tense logics , 1980 .

[9]  A. Prasad Sistla,et al.  Reasoning with Many Processes , 1987, LICS.

[10]  Leslie Lamport,et al.  What Good is Temporal Logic? , 1983, IFIP Congress.

[11]  Erich Grädel,et al.  On the Restraining Power of Guards , 1999, Journal of Symbolic Logic.

[12]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[13]  A. Prasad Sistla,et al.  The complexity of propositional linear temporal logics , 1982, STOC '82.

[14]  Maarten Marx,et al.  Undecidability of Compass Logic , 1999, J. Log. Comput..

[15]  D. Gabbay,et al.  Many-Dimensional Modal Logics: Theory and Applications , 2003 .

[16]  Frank Wolter,et al.  Decidable fragment of first-order temporal logics , 2000, Ann. Pure Appl. Log..

[17]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[18]  A. Wilkie.  THE CLASSICAL DECISION PROBLEM (Perspectives in Mathematical Logic) By Egon Börger, Erich Grädel and Yuri Gurevich: 482 pp., DM.158.–, ISBN 3 540 57073 X (Springer, 1997). , 1998 .

[19]  Mark Reynolds,et al.  A Decidable Temporal Logic of Parallelism , 1997, Notre Dame J. Formal Log..

[20]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[21]  Erich Grädel,et al.  Decision procedures for guarded logics , 1999 .

[22]  A. Prasad Sistla,et al.  On the Eventuality Operator in Temporal Logic , 1987, Logic in Computer Science.

[23]  F. Wolter,et al.  Temporalizing Description Logics , 1998, FroCoS.

[24]  Journal of the Association for Computing Machinery , 1961, Nature.

[25]  Thomas A. Henzinger,et al.  A really temporal logic , 1994, JACM.

[26]  Maarten Marx,et al.  Tolerance Logic , 2001, J. Log. Lang. Inf..