Hardness of Network Satisfaction for Relation Algebras with Normal Representations

We study the computational complexity of the general network satisfaction problem for a finite relation algebra $A$ with a normal representation $B$. If $B$ contains a non-trivial equivalence relation with a finite number of equivalence classes, then the network satisfaction problem for $A$ is NP-hard. As a second result, we prove hardness if $B$ has domain size at least three and contains no non-trivial equivalence relations but a symmetric atom $a$ with a forbidden triple $(a,a,a)$, that is, $a \not\leq a \circ a$. We illustrate how to apply our conditions on two small relation algebras.

[1]  Peter B. Ladkin,et al.  On binary constraint problems , 1994, JACM.

[2]  P. Cameron,et al.  PERMUTATION GROUPS , 2019, Group Theory for Physicists.

[3]  Manuel Bodirsky,et al.  Determining the consistency of partial tree descriptions , 2007, Artif. Intell..

[4]  Matteo Cristani,et al.  The complexity of constraint satisfaction problems for small relation algebras , 2004, Artif. Intell..

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

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

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

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

[9]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[10]  Libor Barto,et al.  The equivalence of two dichotomy conjectures for infinite domain constraint satisfaction problems , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

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

[12]  Andrei A. Bulatov,et al.  Tractable conservative constraint satisfaction problems , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[13]  Henry A. Kautz,et al.  Constraint propagation algorithms for temporal reasoning: a revised report , 1989 .

[14]  Peter J. Cameron,et al.  Permutation Groups: Frontmatter , 1999 .

[15]  Robin Hirsch,et al.  A Finite Relation Algebra with Undecidable Network Satisfaction Problem , 1999, Log. J. IGPL.

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

[17]  Peter Jonsson,et al.  A Model-Theoretic View on Qualitative Constraint Reasoning , 2017, J. Artif. Intell. Res..

[18]  Ivo Düntsch,et al.  Relation Algebras and their Application in Temporal and Spatial Reasoning , 2005, Artificial Intelligence Review.

[19]  Manuel Bodirsky,et al.  Finite Relation Algebras with Normal Representations , 2018, RAMiCS.

[20]  Wilfrid Hodges,et al.  A Shorter Model Theory , 1997 .

[21]  Libor Barto,et al.  Equations in oligomorphic clones and the Constraint Satisfaction Problem for $ω$-categorical structures , 2016, J. Math. Log..

[22]  Bernhard Nebel,et al.  On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus , 1999, Artif. Intell..

[23]  Robin Hirsch,et al.  Relation Algebras of Intervals , 1996, Artif. Intell..

[24]  I. Hodkinson,et al.  Relation Algebras by Games , 2002 .

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

[26]  Roger D. Maddux,et al.  Relation Algebras , 1997, Relational Methods in Computer Science.

[27]  Jaroslav Nesetril,et al.  Constraint Satisfaction with Countable Homogeneous Templates , 2003, J. Log. Comput..

[28]  Andrei A. Bulatov,et al.  Conservative constraint satisfaction re-revisited , 2014, J. Comput. Syst. Sci..

[29]  Bernhard Nebel,et al.  Qualitative Spatial Reasoning Using Constraint Calculi , 2007, Handbook of Spatial Logics.

[30]  Libor Barto,et al.  The Dichotomy for Conservative Constraint Satisfaction Problems Revisited , 2011, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[31]  Robin Hirsch,et al.  Expressive Power and Complexity in Algebraic Logic , 1997, J. Log. Comput..

[32]  Roger D. Maddux,et al.  Representations for Small Relation Algebras , 1994, Notre Dame J. Formal Log..