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Register a new userDatalog and Constraint Satisfaction with Infinite Templates
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Manuel Bodirsky
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template Gamma is omega-categorical, we present various equivalent characterizations of those Gamma such that the constraint satisfaction problem (CSP) for Gamma can be solved by a Datalog program. We also show that CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical structures Gamma if the input is restricted to instances of bounded treewidth. Finally, we characterize those omega-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width k.
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We characterise the sentences in Monadic Second-order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class C of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all integers l,k, there exists a *canonical* Datalog program Pi of width (l,k), that is, a Datalog program of width (l,k) which is sound for C (i.e., Pi only derives the goal predicate on a finite structure A if A is in C) and with the property that Pi derives the goal predicate whenever *some* Datalog program of width (l,k) which is sound for C derives the goal predicate. The same characterisations also hold for Guarded Second-order Logic (GSO), which properly extends MSO. To prove our results, we show that every class C in GSO whose complement is closed under homomorphisms is a finite union of constraint satisfaction problems (CSPs) of countably categorical structures.
We investigate the impact of modifying the constraining relations of a Constraint Satisfaction Problem (CSP) instance, with a fixed template, on the set of solutions of the instance. More precisely we investigate sensitive instances: an instance of the CSP is called sensitive, if removing any tuple from any constraining relation invalidates some solution of the instance. Equivalently, one could require that every tuple from any one of its constraints extends to a solution of the instance. Clearly, any non-trivial template has instances which are not sensitive. Therefore we follow the direction proposed (in the context of strict width) by Feder and Vardi (SICOMP 1999) and require that only the instances produced by a local consistency checking algorithm are sensitive. In the language of the algebraic approach to the CSP we show that a finite idempotent algebra $mathbf{A}$ has a $k+2$ variable near unanimity term operation if and only if any instance that results from running the $(k, k+1)$-consistency algorithm on an instance over $mathbf{A}^2$ is sensitive. A version of our result, without idempotency but with the sensitivity condition holding in a variety of algebras, settles a question posed by G. Bergman about systems of projections of algebras that arise from some subalgebra of a finite product of algebras. Our results hold for infinite (albeit in the case of $mathbf{A}$ idempotent) algebras as well and exhibit a surprising similarity to the strict width $k$ condition proposed by Feder and Vardi. Both conditions can be characterized by the existence of a near unanimity operation, but the arities of the operations differ by 1.
We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander. We consider the spectral definition of high-dimensional expansion used by Dinur and Kaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure the expansion in terms of a parameter $gamma$ which is the analogue of the second singular value for expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS 2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet $[q]$ is a high-dimensional expander with parameter $gamma$, then it is possible to approximate the maximum fraction of satisfiable constraints up to an additive error $epsilon$ using $q^{O(k)} cdot (k/epsilon)^{O(1)}$ levels of the sum-of-squares SDP hierarchy, provided $gamma leq epsilon^{O(1)} cdot (1/(kq))^{O(k)}$. Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs, which can be used to extend the results for approximating 2-CSPs on low threshold-rank graphs. We show that if an instance of MAX k-CSP has threshold rank $r$ for a threshold $tau = (epsilon/k)^{O(1)} cdot (1/q)^{O(k)}$, then it is possible to approximately solve the instance up to additive error $epsilon$, using $r cdot q^{O(k)} cdot (k/epsilon)^{O(1)}$ levels of the sum-of-squares hierarchy. As in the case of graphs, high-dimensional expanders (with sufficiently small $gamma$) have threshold rank 1 according to our definition.
Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes coincides with an important dichotomy in universal algebra; in particular, the border can be described by a strong height-one Maltsev condition. For infinite-domain CSPs, the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the Boolean rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.
The universal-algebraic approach has proved a powerful tool in the study of the complexity of CSPs. This approach has previously been applied to the study of CSPs with finite or (infinite) omega-categorical templates, and relies on two facts. The first is that in finite or omega-categorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. The second is that every finite or omega-categorical structure is homomorphically equivalent to a core structure. In this paper, we present generalizations of these facts to infinite structures that are not necessarily omega-categorical. (This abstract has been severely curtailed by the space constraints of arXiv -- please read the full abstract in the article.) Finally, we present applications of our general results to the description and analysis of the complexity of CSPs. In particular, we give general hardness criteria based on the absence of polymorphisms that depend on more than one argument, and we present a polymorphism-based description of those CSPs that are first-order definable (and therefore can be solved in polynomial time).
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