Smooth Approximations and Relational Width Collapses

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ≥ 3) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al.. In particular, the bounded width hierarchy collapses in those cases as well.

[1]  Manuel Bodirsky,et al.  Equivalence Constraint Satisfaction Problems , 2012, CSL.

[2]  Michael Pinsker,et al.  Topological Birkhoff , 2012, ArXiv.

[3]  Michael Pinsker,et al.  Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems) , 2019, 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[4]  Pengming Wang,et al.  The pebbling comonad in Finite Model Theory , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[5]  Manuel Bodirsky,et al.  Datalog and constraint satisfaction with infinite templates , 2006, J. Comput. Syst. Sci..

[6]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[7]  B. Larose,et al.  Bounded width problems and algebras , 2007 .

[8]  Michael Pinsker,et al.  Pseudo‐loop conditions , 2018, The bulletin of the London Mathematical Society.

[9]  Libor Barto,et al.  The collapse of the bounded width hierarchy , 2016, J. Log. Comput..

[10]  Bernhard Nebel,et al.  Reasoning about temporal relations: a maximal tractable subclass of Allen's interval algebra , 1994, JACM.

[11]  Manuel Bodirsky,et al.  A Dichotomy for First-Order Reducts of Unary Structures , 2016, Log. Methods Comput. Sci..

[12]  Michal Wrona.  Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width , 2020, STACS.

[13]  Libor Barto,et al.  Constraint Satisfaction Problems Solvable by Local Consistency Methods , 2014, JACM.

[14]  Carsten Lutz,et al.  Ontology-Based Data Access , 2014, ACM Trans. Database Syst..

[15]  Barnaby Martin,et al.  Constraint satisfaction problems for reducts of homogeneous graphs , 2016, ICALP.

[16]  Hubie Chen,et al.  Asking the Metaquestions in Constraint Tractability , 2016, TOCT.

[17]  László Babai,et al.  Graph isomorphism in quasipolynomial time [extended abstract] , 2015, STOC.

[18]  Michael Pinsker,et al.  Hrushovski's Encoding and ω-Categorical CSP Monsters , 2020, ICALP.

[19]  Dmitriy Zhuk,et al.  A Proof of CSP Dichotomy Conjecture , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[20]  Manuel Bodirsky,et al.  The Complexity of Equality Constraint Languages , 2006, Theory of Computing Systems.

[21]  Iain A. Stewart,et al.  Constraint Satisfaction, Logic and Forbidden Patterns , 2007, SIAM J. Comput..

[22]  Rewritability in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics , 2017, Log. Methods Comput. Sci..

[23]  Michael Pinsker,et al.  PROJECTIVE CLONE HOMOMORPHISMS , 2014, The Journal of Symbolic Logic.

[24]  M. Maróti,et al.  Existence theorems for weakly symmetric operations , 2008 .

[25]  Manuel Bodirsky,et al.  Ramsey classes: examples and constructions , 2015, Surveys in Combinatorics.

[26]  Martin Grohe,et al.  Fixed-Point Definability and Polynomial Time on Graphs with Excluded Minors , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[27]  Michal Wrona.  On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width , 2020, LICS.

[28]  Michael Pinsker,et al.  Canonical Functions: a proof via topological dynamics , 2016, Contributions Discret. Math..

[29]  Libor Barto,et al.  The wonderland of reflections , 2015, Israel Journal of Mathematics.

[30]  Libor Barto,et al.  The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[31]  Justin Pearson,et al.  Closure Functions and Width 1 Problems , 1999, CP.

[32]  László Babai,et al.  Graph isomorphism in quasipolynomial time [extended abstract] , 2016, STOC.

[33]  Michael Pinsker,et al.  Smooth approximations and CSPs over finitely bounded homogeneous structures , 2020, LICS.

[34]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[35]  M. Valeriote.  A Subalgebra Intersection Property for Congruence Distributive Varieties , 2009, Canadian Journal of Mathematics.

[36]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[37]  Michael Pinsker,et al.  Schaefer's Theorem for Graphs , 2015, J. ACM.

[38]  Michael Pinsker,et al.  Decidability of Definability , 2010, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[39]  Manuel Bodirsky,et al.  A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP , 2018, LICS.

[40]  Barnaby Martin,et al.  Discrete Temporal Constraint Satisfaction Problems , 2018, J. ACM.

[41]  Libor Barto,et al.  Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem , 2012, Log. Methods Comput. Sci..

[42]  Andrei A. Bulatov,et al.  On the Power of k -Consistency , 2007, ICALP.

[43]  M. Bodirsky,et al.  Temporal Constraint Satisfaction Problems in Fixed-Point Logic , 2020, LICS.

[44]  Andrei A. Bulatov,et al.  Bounded relational width , 2009 .

[45]  Carsten Lutz,et al.  Containment in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics , 2016, KR.

[46]  V. Pestov,et al.  Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups , 2003 .

[47]  Libor Barto,et al.  Congruence Distributivity Implies Bounded Width , 2009, SIAM J. Comput..