Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.

[1]  Petr Hliněný,et al.  FO Model Checking on Posets of Bounded Width , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[2]  Hannu Niemistö.  On locality and uniform reduction , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[3]  Michael Elberfeld,et al.  Order Invariance on Decomposable Structures , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[4]  Michael Benedikt,et al.  Towards a characterization of order-invariant queries over tame graphs , 2009, The Journal of Symbolic Logic.

[5]  Robin Thomas,et al.  Testing first-order properties for subclasses of sparse graphs , 2011, JACM.

[6]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[7]  Benjamin Rossman.  Successor-invariant first-order logic on finite structures , 2007, J. Symb. Log..

[8]  Robert Ganian,et al.  FO Model Checking of Interval Graphs , 2013, ICALP.

[9]  Stephan Kreutzer,et al.  Deciding first-order properties of nowhere dense graphs , 2013, STOC.

[10]  Martin Otto,et al.  Epsilon-logic is more expressive than first-order logic over finite structures , 2000, Journal of Symbolic Logic.

[11]  Dániel Marx,et al.  Structure theorem and isomorphism test for graphs with excluded topological subgraphs , 2011, STOC '12.

[12]  M. S. Ramanujan,et al.  A New Perspective on FO Model Checking of Dense Graph Classes , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[13]  Stephan Kreutzer,et al.  The Generalised Colouring Numbers on Classes of Bounded Expansion , 2016, MFCS.

[14]  Xuding Zhu,et al.  Colouring graphs with bounded generalized colouring number , 2009, Discret. Math..

[15]  Jörg Flum,et al.  Query evaluation via tree-decompositions , 2001, JACM.

[16]  Zdenek Dvorak,et al.  Constant-factor approximation of domination number in sparse graphs , 2011, ArXiv.

[17]  Hal A. Kierstead,et al.  Orderings on Graphs and Game Coloring Number , 2003, Order.

[18]  Stephan Kreutzer,et al.  On brambles, grid-like minors, and parameterized intractability of monadic second-order logic , 2010, SODA '10.

[19]  Johann A. Makowsky,et al.  Model Theoretic Methods in Finite Combinatorics , 2011 .

[20]  Stephan Kreutzer,et al.  Lower Bounds for the Complexity of Monadic Second-Order Logic , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[21]  Stephan Kreutzer,et al.  Locally Excluding a Minor , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[22]  Ashok K. Chandra,et al.  Optimal implementation of conjunctive queries in relational data bases , 1977, STOC '77.

[23]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[24]  Andreas Potthoff,et al.  Logische Klassifizierung regulärer Baumsprachen , 1994 .

[25]  Alejandro A. Schäffer,et al.  Optimal Node Ranking of Trees in Linear Time , 1989, Inf. Process. Lett..

[26]  Paul D. Seymour,et al.  Graph Minors: XVII. Taming a Vortex , 1999, J. Comb. Theory, Ser. B.

[27]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[28]  Detlef Seese,et al.  Linear time computable problems and first-order descriptions , 1996, Mathematical Structures in Computer Science.

[29]  Frederik Harwath,et al.  Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures , 2014, MFCS.

[30]  Stephan Kreutzer,et al.  First-Order and Monadic Second-Order Model-Checking on Ordered Structures , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[31]  Stephan Kreutzer,et al.  Methods for Algorithmic Meta Theorems , 2009, AMS-ASL Joint Special Session.

[32]  Robert E. Tarjan,et al.  Efficiency of a Good But Not Linear Set Union Algorithm , 1972, JACM.

[33]  Martin Grohe,et al.  Deciding first-order properties of locally tree-decomposable structures , 2000, JACM.

[34]  Ken-ichi Kawarabayashi,et al.  Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs , 2016, CSL.

[35]  Klaus Jansen,et al.  Rankings of Graphs , 1998, SIAM J. Discret. Math..

[36]  Benjamin Rossman.  Successor-invariance in the finite , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[37]  Ken-ichi Kawarabayashi,et al.  Model Checking for Successor-Invariant First-Order Logic on Minor-Closed Graph Classes , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.