Evaluating Datalog via Tree Automata and Cycluits

We investigate parameterizations of both database instances and queries that make query evaluation fixed-parameter tractable in combined complexity. We show that clique-frontier-guarded Datalog with stratified negation (CFG-Datalog) enjoys bilinear-time evaluation on structures of bounded treewidth for programs of bounded rule size. Such programs capture in particular conjunctive queries with simplicial decompositions of bounded width, guarded negation fragment queries of bounded CQ-rank, or two-way regular path queries. Our result is shown by translating to alternating two-way automata, whose semantics is defined via cyclic provenance circuits (cycluits) that can be tractably evaluated.Comment: 56 pages, 63 references. Journal version of "Combined Tractability of Query Evaluation via Tree Automata and Cycluits (Extended Version)" at arXiv:1612.04203. Up to the stylesheet, page/environment numbering, and possible minor publisher-induced changes, this is the exact content of the journal paper that will appear in Theory of Computing Systems. Update wrt version 1: latest reviewer feedbac

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

Eliminating Recursion from Monadic Datalog Programs on Trees

We study the problem of eliminating recursion from monadic datalog programs on trees with an infinite set of labels. We show that the boundedness problem, i.e., determining whether a datalog program is equivalent to some nonrecursive one is undecidable but the decidability is regained if the descendant relation is disallowed. Under similar restrictions we obtain decidability of the problem of equivalence to a given nonrecursive program. We investigate the connection between these two problems in more detail

Finite-Cliquewidth Sets of Existential Rules: Toward a General Criterion for Decidable yet Highly Expressive Querying

In our pursuit of generic criteria for decidable ontology-based querying, we introduce finite-cliquewidth sets (fcs) of existential rules, a model-theoretically defined class of rule sets, inspired by the cliquewidth measure from graph theory. By a generic argument, we show that fcs ensures decidability of entailment for a sizable class of queries (dubbed DaMSOQs) subsuming conjunctive queries (CQs). The fcs class properly generalizes the class of finite-expansion sets (fes), and for signatures of arity ? 2, the class of bounded-treewidth sets (bts). For higher arities, bts is only indirectly subsumed by fcs by means of reification. Despite the generality of fcs, we provide a rule set with decidable CQ entailment (by virtue of first-order-rewritability) that falls outside fcs, thus demonstrating the incomparability of fcs and the class of finite-unification sets (fus). In spite of this, we show that if we restrict ourselves to single-headed rule sets over signatures of arity ? 2, then fcs subsumes fus

Linear Datalog and Bounded Path Duality of Relational Structures

In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem \csp(\best) with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL

Tree-width for first order formulae

We introduce tree-width for first order formulae \phi, fotw(\phi). We show that computing fotw is fixed-parameter tractable with parameter fotw. Moreover, we show that on classes of formulae of bounded fotw, model checking is fixed parameter tractable, with parameter the length of the formula. This is done by translating a formula \phi\ with fotw(\phi)<k into a formula of the k-variable fragment L^k of first order logic. For fixed k, the question whether a given first order formula is equivalent to an L^k formula is undecidable. In contrast, the classes of first order formulae with bounded fotw are fragments of first order logic for which the equivalence is decidable. Our notion of tree-width generalises tree-width of conjunctive queries to arbitrary formulae of first order logic by taking into account the quantifier interaction in a formula. Moreover, it is more powerful than the notion of elimination-width of quantified constraint formulae, defined by Chen and Dalmau (CSL 2005): for quantified constraint formulae, both bounded elimination-width and bounded fotw allow for model checking in polynomial time. We prove that fotw of a quantified constraint formula \phi\ is bounded by the elimination-width of \phi, and we exhibit a class of quantified constraint formulae with bounded fotw, that has unbounded elimination-width. A similar comparison holds for strict tree-width of non-recursive stratified datalog as defined by Flum, Frick, and Grohe (JACM 49, 2002). Finally, we show that fotw has a characterization in terms of a cops and robbers game without monotonicity cost

Decidability of Querying First-Order Theories via Countermodels of Finite Width

We propose a generic framework for establishing the decidability of a wide range of logical entailment problems (briefly called querying), based on the existence of countermodels that are structurally simple, gauged by certain types of width measures (with treewidth and cliquewidth as popular examples). As an important special case of our framework, we identify logics exhibiting width-finite finitely universal model sets, warranting decidable entailment for a wide range of homomorphism-closed queries, subsuming a diverse set of practically relevant query languages. As a particularly powerful width measure, we propose Blumensath's partitionwidth, which subsumes various other commonly considered width measures and exhibits highly favorable computational and structural properties. Focusing on the formalism of existential rules as a popular showcase, we explain how finite partitionwidth sets of rules subsume other known abstract decidable classes but -- leveraging existing notions of stratification -- also cover a wide range of new rulesets. We expose natural limitations for fitting the class of finite unification sets into our picture and provide several options for remedy

Decomposing quantified conjunctive (or disjunctive) formulas

Model checking---deciding if a logical sentence holds on a structure---is a basic computational task that is well known to be intractable in general. For first-order logic on finite structures, it is PSPACE-complete, and the natural evaluation algorithm exhibits exponential dependence on the formula. We study model checking on the quantified conjunctive fragment of first-order logic, namely, prenex sentences having a purely conjunctive quantifier-free part. Following a number of works, we associate a graph to the quantifier-free part; each sentence then induces a prefixed graph, a quantifier prefix paired with a graph on its variables. We give a comprehensive classification of the sets of prefixed graphs on which model checking is tractable based on a novel generalization of treewidth that generalizes and places into a unified framework a number of existing results