The Complexity of Flat Freeze LTL

We consider the model-checking problem for freeze LTL on one-counter automata (OCAs). Freeze LTL extends LTL with the freeze quantifier, which allows one to store different counter values of a run in registers so that they can be compared with one another. As the model-checking problem is undecidable in general, we focus on the flat fragment of freeze LTL, in which the usage of the freeze quantifier is restricted. Recently, Lechner et al. showed that model checking for flat freeze LTL on OCAs with binary encoding of counter updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in unary or binary. Like Lechner et al., we rely on a reduction to the reachability problem in OCAs with parameterized tests (OCAPs). The new aspect is that we simulate OCAPs by alternating two-way automata over words. This implies an exponential upper bound on the parameter values that we exploit towards an NP algorithm for reachability in OCAPs with unary updates. We obtain our main result as a corollary.

[1]  Joël Ouaknine,et al.  Model Checking Succinct and Parametric One-Counter Automata , 2010, ICALP.

[2]  Olivier Serre,et al.  Parity Games Played on Transition Graphs of One-Counter Processes , 2006, FoSSaCS.

[3]  Orna Kupferman,et al.  An Automata-Theoretic Approach to Reasoning about Infinite-State Systems , 2000, CAV.

[4]  Markus Lohrey,et al.  Path Checking for MTL and TPTL over Data Words , 2015, DLT.

[5]  Moshe Y. Vardi.  Reasoning about The Past with Two-Way Automata , 1998, ICALP.

[6]  Thierry Cachat.  Two-Way Tree Automata Solving Pushdown Games , 2001, Automata, Logics, and Infinite Games.

[7]  Benedikt Bollig,et al.  Distributed Timed Automata with Independently Evolving Clocks , 2008, CONCUR.

[8]  Stéphane Demri,et al.  Model checking memoryful linear-time logics over one-counter automata , 2010, Theor. Comput. Sci..

[9]  Leslie G. Valiant,et al.  Deterministic One-Counter Automata , 1975, J. Comput. Syst. Sci..

[10]  Véronique Cortier,et al.  Flatness Is Not a Weakness , 2000, CSL.

[11]  Zvi Galil,et al.  Hierarchies of complete problems , 1976, Acta Informatica.

[12]  Joël Ouaknine,et al.  Model Checking Flat Freeze LTL on One-Counter Automata , 2016, CONCUR.

[13]  Joël Ouaknine,et al.  Branching-Time Model Checking of Parametric One-Counter Automata , 2012, FoSSaCS.

[14]  Markus Lohrey,et al.  Branching-Time Model Checking of One-Counter Processes and Timed Automata , 2013, SIAM J. Comput..

[15]  Slawomir Lasota,et al.  Simulation Problems Over One-Counter Nets , 2016, Log. Methods Comput. Sci..

[16]  Pascal Lafourcade,et al.  Intruder Deduction for AC-Like Equational Theories with Homomorphisms , 2005, RTA.

[17]  Stéphane Demri,et al.  LTL with the Freeze Quantifier and Register Automata , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[18]  Thomas Schwentick,et al.  Two-variable logic on data words , 2011, TOCL.

[19]  Christoph Haase,et al.  On the complexity of model checking counter automata , 2012 .

[20]  John Fearnley,et al.  Reachability in two-clock timed automata is PSPACE-complete , 2013, Inf. Comput..

[21]  Thomas A. Henzinger,et al.  Parametric real-time reasoning , 1993, STOC.

[22]  Igor Potapov,et al.  Temporal logic with predicate /spl lambda/-abstraction , 2005, 12th International Symposium on Temporal Representation and Reasoning (TIME'05).

[23]  Stéphane Demri,et al.  On the freeze quantifier in constraint LTL: decidability and complexity , 2005, 12th International Symposium on Temporal Representation and Reasoning (TIME'05).

[24]  Tim French.  Quantified propositional temporal logic with repeating states , 2003, 10th International Symposium on Temporal Representation and Reasoning, 2003 and Fourth International Conference on Temporal Logic. Proceedings..

[25]  Joël Ouaknine,et al.  Reachability in Succinct and Parametric One-Counter Automata , 2009, CONCUR.

[26]  Stéphane Demri,et al.  When Model-Checking Freeze LTL over Counter Machines Becomes Decidable , 2010, FoSSaCS.

[27]  Joël Ouaknine,et al.  On parametric timed automata and one-counter machines , 2017, Inf. Comput..

[28]  Joël Ouaknine,et al.  On Expressiveness and Complexity in Real-Time Model Checking , 2008, ICALP.

[29]  Stéphane Demri,et al.  The Effects of Bounding Syntactic Resources on Presburger LTL , 2007, 14th International Symposium on Temporal Representation and Reasoning (TIME'07).