On the Relationship between Consistent Query Answering and Constraint Satisfaction Problems

Recently, Fontaine has pointed out a connection between consistent query answering (CQA) and constraint satisfaction problems (CSP) [Fontaine, LICS 2013]. We investigate this connection more closely, identifying classes of CQA problems based on denial constraints and GAV constraints that correspond exactly to CSPs in the sense that a complexity classification of the CQA problems in each class is equivalent (up to FO-reductions) to classifying the complexity of all CSPs. We obtain these classes by admitting only monadic relations and only a single variable in denial constraints/GAVs and restricting queries to hypertree UCQs. We also observe that dropping the requirement of UCQs to be hypertrees corresponds to transitioning from CSP to its logical generalization MMSNP and identify a further relaxation that corresponds to transitioning from MMSNP to GMSNP (also know as MMSNP_2). Moreover, we use the CSP connection to carry over decidability of FO-rewritability and Datalog-rewritability to some of the identified classes of CQA problems

Rewritability in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics

We study rewritability of monadic disjunctive Datalog programs, (the complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and on conjunctive queries. We show that rewritability into FO and into monadic Datalog (MDLog) are decidable, and that rewritability into Datalog is decidable when the original query satisfies a certain condition related to equality. We establish 2NExpTime-completeness for all studied problems except rewritability into MDLog for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze the shape of rewritings, which in the MMSNP case correspond to obstructions, and give a new construction of canonical Datalog programs that is more elementary than existing ones and also applies to formulas with free variables

Universality for and in induced-hereditary graph properties

The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 20 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(20 ) properties in the lattice K< of induced-hereditary properties of which only at most 20 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.http://www.discuss.wmie.uz.zgora.pl/gt/am201

Strict width for Constraint Satisfaction Problems over homogeneous strucures of finite duality

We investigate the `local consistency implies global consistency' principle of strict width among structures within the scope of the Bodirsky-Pinsker dichotomy conjecture for infinite-domain Constraint Satisfaction Problems (CSPs). Our main result implies that for certain CSP templates within the scope of that conjecture, having bounded strict width has a concrete consequence on the expressive power of the template called implicational simplicity. This in turn yields an explicit bound on the relational width of the CSP, i.e., the amount of local consistency needed to ensure the satisfiability of any instance. Our result applies to first-order expansions of any homogeneous $k$-uniform hypergraph, but more generally to any CSP template under the assumption of finite duality and general abstract conditions mainly on its automorphism group. In particular, it overcomes the restriction to binary signatures in the pioneering work of Wrona.Comment: 22 page

Homogeneity and Homogenizability: Hard Problems for the Logic SNP

We show that the question whether a given SNP sentence defines a homogenizable class of finite structures is undecidable, even if the sentence comes from the connected Datalog fragment and uses at most binary relation symbols. As a byproduct of our proof, we also get the undecidability of some other properties for Datalog programs, e.g., whether they can be rewritten in MMSNP, whether they solve some finite-domain CSP, or whether they define the age of a reduct of a homogeneous Ramsey structure in a finite relational signature. We subsequently show that the closely related problem of testing the amalgamation property for finitely bounded classes is EXPSPACE-hard or PSPACE-hard, depending on whether the input is specified by a universal sentence or a set of forbidden substructures.Comment: 34 pages, 3 figure

UNIVERSAL STRUCTURES AND THE LOGIC OF FORBIDDEN PATTERNS

We show that forbidden patterns problems, when restricted to some classes of input structures, are in fact constraint satisfaction problems. This contrasts with the case of unrestricted input structures, for which it is known that there are forbidden patterns problems that are not constraint satisfaction problems. We show that if the input comes from a class of connected structures with low tree-depth decomposition then every forbidden patterns problem is in fact a constraint satisfaction problem. In particular, our result covers input restrictions such as: structures of bounded degree, planar graphs, structures of bounded tree-width and, more generally, classes definable by at least one forbidden minor. This result can also be rephrased in terms of expressiveness of the logic MMSNP, introduced by Feder and Vardi in relation with constraint satisfaction problems. Our approach follows and generalises that of Neˇsetˇril and Ossona de Mendez’s, who investigated restricted dualities, which corresponds in our setting to investigating the restricted case when the considered forbidden patterns problems are captured by a first-order sentence. Note also that our result holds in the general setting of problems over arbitrary relational structures (not just for graphs)

Universal Structures and the logic of Forbidden Patterns

Forbidden Patterns Problems (FPPs) are a proper generalisation of Constraint Satisfaction Problems (CSPs). However, we show that when the input is connected and belongs to a class which has low tree-depth decomposition (e.g. structure of bounded degree, proper minor closed class and more generally class of bounded expansion) any FPP becomes a CSP. This result can also be rephrased in terms of expressiveness of the logic MMSNP, introduced by Feder and Vardi in relation with CSPs. Our proof generalises that of a recent paper by Nesetril and Ossona de Mendez. Note that our result holds in the general setting of problems over arbitrary relational structures (not just for graphs)