A Model-Theoretic View on Qualitative Constraint Reasoning

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Madelaine

Mottet

2018

Self Cite

The logic MMSNP is a restricted fragment of existential second-order logic which allows to express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finitedomain CSPs which does not rely on the results of Kun. This new proof allows us to obtain a stronger statement and to verify the more general Bodirsky-Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite ω-categorical structures; moreover, by a recent result of Hubička and Nešetřil, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property. This allows us to use the universal-algebraic approach to study the computational complexity of MMSNP.

Section: Conclusion and Open Problemsmentioning

confidence: 99%

A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Madelaine

Mottet

2018

Self Cite

“…The class of CSPs for ω-categorical theories arguably coincides with the class of CSPs for qualitative formalisms studied e.g. in temporal and spatial reasoning; see [9].…”

Section: Introductionmentioning

confidence: 84%

“…about space and time) is not performed with absolute numerical values, but rather with qualitative predicates (such as within, before, etc.). There is no universally accepted definition in the literature that defines what a qualitative CSP is, but a proposal has been made in [9]; the central mathematical property for this proposal is ω-categoricity. A theory is called ω-categorical if it has up to isomorphism only one countable model.…”

Section: Introductionmentioning

confidence: 99%

Complexity of Combinations of Qualitative Constraint Satisfaction Problems

Greiner

2018

Automated Reasoning

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The CSP of a first-order theory T is the problem of deciding for a given finite set S of atomic formulas whether T ∪ S is satisfiable. Let T 1 and T 2 be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomialtime decidability) of CSP(T 1 ∪ T 2 ) under the assumption that CSP(T 1 ) and CSP(T 2 ) are decidable (or polynomial-time decidable). We show that for a large class of ω-categorical theories T 1 , T 2 the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of CSP(T 1 ∪ T 2 ) (unless P=NP).

“…The so-called network satisfaction problem for a finite relation algebra can be used to model many computational problems in temporal and spatial reasoning [8,24,40]. In 1996, Robin Hirsch [26] asked the Really Big Complexity Problem (RBCP): can we classify the computational complexity of the network satisfaction problem for every finite relation algebra?…”

Section: Introductionmentioning

confidence: 99%

Hardness of Network Satisfaction for Relation Algebras with Normal Representations

Relational and Algebraic Methods in Computer Science

Knäuer

2020

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Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation B. We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type result of Nešetřil and Rdl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.