On the complexity of linear temporal logic with team semantics

A specification given as a formula in linear temporal logic (LTL) defines a system by its set of traces. However, certain features such as information flow security constraints are rather modeled as so-called hyperproperties, which are sets of sets of traces. One logical approach to this is team logic, which is a logical framework for the specification of dependence and independence of information. LTL with team semantics has recently been discovered as a logic for hyperproperties. We study the complexity theoretic aspects of LTL with so-called synchronous team semantics and Boolean negation, and prove that both its model checking and satisfiability problems are highly undecidable, and equivalent to the decision problem of third-order arithmetic. Furthermore, we prove that this complexity already appears at small temporal depth and with only the “future” modality F. Finally, we also introduce a team-semantical generalization of stutter-invariance.

[1]  J. R. Büchi.  Symposium on Decision Problems: On a Decision Method in Restricted Second Order Arithmetic , 1966 .

[2]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[3]  Martin Zimmermann,et al.  The Keys to Decidable HyperLTL Satisfiability: Small Models or Very Simple Formulas , 2019, CSL.

[4]  Bernd Finkbeiner,et al.  The First-Order Logic of Hyperproperties , 2016, STACS.

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

[6]  Juha Kontinen.  Dependence Logic: A survey of some recent work , 2013 .

[7]  Jouko A. Väänänen,et al.  Dependence Logic - A New Approach to Independence Friendly Logic , 2007, London Mathematical Society student texts.

[8]  Fan Yang,et al.  Propositional team logics , 2016, Ann. Pure Appl. Log..

[9]  Martin Lück,et al.  On the Succinctness of Atoms of Dependency , 2019, Log. Methods Comput. Sci..

[10]  Fan Yang.  Modal dependence logics: axiomatizations and model-theoretic properties , 2017, Log. J. IGPL.

[11]  Christel Baier,et al.  Principles of model checking , 2008 .

[12]  Doron A. Peled,et al.  Stutter-Invariant Temporal Properties are Expressible Without the Next-Time Operator , 1997, Inf. Process. Lett..

[13]  Julian-Steffen Müller,et al.  A Van Benthem Theorem for Modal Team Semantics , 2015, CSL.

[14]  Fan Yang,et al.  Structural completeness in propositional logics of dependence , 2015, Arch. Math. Log..

[15]  Jouko Vnnen,et al.  Dependence Logic: A New Approach to Independence Friendly Logic (London Mathematical Society Student Texts) , 2007 .

[16]  Arne Meier,et al.  A Team Based Variant of CTL , 2015, 2015 22nd International Symposium on Temporal Representation and Reasoning (TIME).

[17]  Arne Meier,et al.  Team Semantics for the Specification and Verification of Hyperproperties , 2018, MFCS.

[18]  Philippe Schnoebelen,et al.  The Complexity of Propositional Linear Temporal Logics in Simple Cases , 1998, Inf. Comput..

[19]  Bernd Finkbeiner,et al.  Deciding Hyperproperties , 2016, CONCUR.

[20]  Bernd Finkbeiner,et al.  Temporal Logics for Hyperproperties , 2013, POST.

[21]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[22]  Fred Kröger,et al.  Temporal Logic of Programs , 1987, EATCS Monographs on Theoretical Computer Science.

[23]  Fan Yang,et al.  Propositional logics of dependence , 2014, Ann. Pure Appl. Log..

[24]  Martin Lück,et al.  Team logic: axioms, expressiveness, complexity , 2020 .

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

[26]  Philippe Schnoebelen,et al.  The Complexity of Temporal Logic Model Checking , 2002, Advances in Modal Logic.

[27]  Antti Kuusisto.  A Double Team Semantics for Generalized Quantifiers , 2015, J. Log. Lang. Inf..

[28]  Klaus Schneider,et al.  Verification of Reactive Systems: Formal Methods and Algorithms , 2003 .

[29]  Pierre Wolper,et al.  An Automata-Theoretic Approach to Automatic Program Verification (Preliminary Report) , 1986, LICS.

[30]  Martin Luck.  Canonical Models and the Complexity of Modal Team Logic , 2017 .

[31]  de Boudewijn Bruin.  New perspectives on games and interaction , 2008 .