Query evaluation in monadic second-order logic (MSO) is tractable on trees and treelike instances, even though it is hard for arbitrary instances. This tractability result has been extended to several tasks related to query evaluation, such as counting query results [3] or performing query evaluation on probabilistic trees [10]. These are two examples of the more general problem of computing augmented query output, that is referred to as provenance. This article presents a provenance framework for trees and treelike instances, by describing a linear-time construction of a circuit provenance representation for MSO queries. We show how this provenance can be connected to the usual definitions of semiring provenance on relational instances [20], even though we compute it in an unusual way, using tree automata; we do so via intrinsic definitions of provenance for general semirings, independent of the operational details of query evaluation. We show applications of this provenance to capture existing counting and probabilistic results on trees and treelike instances, and give novel consequences for probability evaluation.

Knowledge bases such as Wikidata, DBpedia, or YAGO contain millions of entities and facts. In some knowledge bases, the correctness of these facts has been evaluated. However, much less is known about their completeness, i.e., the proportion of real facts that the knowledge bases cover. In this work, we investigate different signals to identify the areas where a knowledge base is complete. We show that we can combine these signals in a rule mining approach, which allows us to predict where facts may be missing. We also show that completeness predictions can help other applications such as fact prediction.

We consider the information extraction framework known as document spanners and study the problem of efficiently computing the results of the extraction from an input document, where the extraction task is described as a sequential variable-set automaton (VA). We pose this problem in the setting of enumeration algorithms, where we can first run a preprocessing phase and must then produce the results with a small delay between any two consecutive results. Our goal is to have an algorithm that is tractable in combined complexity, i.e., in the sizes of the input document and the VA, while ensuring the best possible data complexity bounds in the input document size, i.e., constant delay in the document size. Several recent works at PODS’18 proposed such algorithms but with linear delay in the document size or with an exponential dependency in size of the (generally nondeterministic) input VA. In particular, Florenzano et al. suggest that our desired runtime guarantees cannot be met for general sequential VAs. We refute this and show that, given a nondeterministic sequential VA and an input document, we can enumerate the mappings of the VA on the document with the following bounds: the preprocessing is linear in the document size and polynomial in the size of the VA, and the delay is independent of the document and polynomial in the size of the VA. The resulting algorithm thus achieves tractability in combined complexity and the best possible data complexity bounds. Moreover, it is rather easy to describe, particularly for the restricted case of so-called extended VAs. Finally, we evaluate our algorithm empirically using a prototype implementation.

We consider entailment problems involving powerful constraint languages such as guarded existential rules, in which additional semantic restrictions are put on a set of distinguished relations. We consider restricting a relation to be transitive, restricting a relation to be the transitive closure of another relation, and restricting a relation to be a linear order. We give some natural generalizations of guardedness that allow inference to be decidable in each case, and isolate the complexity of the corresponding decision problems. Finally we show that slight changes in our conditions lead to undecidability. arXiv:1607.00813v1 [cs.DB] 4 Jul 2016The size, width, and CQ-rank bounds after performing this rewriting are straightforward to check.The rewrite rules preserve the polarity of subformulas, which helps ensure that coveredness is preserved during this conversion.

Query evaluation on probabilistic databases is generally intractable (#P-hard).Existing dichotomy results [19,18,23] have identified which queries are tractable (or safe), and connected them to tractable lineages [36]. In our previous work [2], using different tools, we showed that query evaluation is linear-time on probabilistic databases for arbitrary monadic second-order queries, if we bound the treewidth of the instance.In this paper, we study limitations and extensions of this result. First, for probabilistic query evaluation, we show that MSO tractability cannot extend beyond bounded treewidth: there are even FO queries that are hard on any efficiently constructible unbounded-treewidth class of graphs. This dichotomy relies on recent polynomial bounds on the extraction of planar graphs as minors [10], and implies lower bounds in non-probabilistic settings, for query evaluation and match counting in subinstance-closed families. Second, we show how to explain our tractability result in terms of lineage: the lineage of MSO queries on bounded-treewidth instances can be represented as bounded-treewidth circuits, polynomial-size OBDDs, and linear-size d-DNNFs. By contrast, we can strengthen the previous dichotomy to lineages, and show that there are even UCQs with disequalities that have superpolynomial OBDDs on all unbounded-treewidth graph classes; we give a characterization of such queries. Last, we show how bounded-treewidth tractability explains the tractability of the inversion-free safe queries: we can rewrite their input instances to have bounded-treewidth.

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that boundedtreewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs). ACM Subject Classification H.2 DATABASE MANAGEMENTThis landscape is summarized in Fig. 1, and we review known translations and separation results that describe the relative expressive power of these features.The second contribution of this paper (in Sections 4 and 5) is to show an upper bound on the compilation of bounded-treewidth classes to d-SDNNFs, and of bounded-pathwidth classes to OBDD variants. For pathwidth, existing work had already studied the compilation of bounded-pathwidth circuits to OBDDs [34, Corollary 2.13], which can be made constructive [4, Lemma 6.9]. Specifically, they show that a circuit of pathwidth k can be converted in polynomial time into an OBDD of width 2 (k+2)2 k+2 . Our first contribution is to show that, by using unambiguous OBDDs (uOBDDs), we can do the same but with linear time complexity, and with the size of the uOBDD as well as its width (in the classical knowledge compilation sense) being singly exponential in the pathwidth. Specifically:For treewidth, we show that bounded-treewidth circuits can be compiled to the class of d-SDNNF circuits:The proof of Result 2, and its variant that shows Result 1, is quite challenging: we transform the input circuit bottom-up by considering all possible valuations of th...

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We give an algorithm to enumerate the results on trees of monadic second-order (MSO) queries represented by nondeterministic tree automata. After linear time preprocessing (in the input tree), we can enumerate answers with linear delay (in each answer). We allow updates on the tree to take place at any time, and we can then restart the enumeration after logarithmic time in the tree. Further, all our combined complexities are polynomial in the automaton.Our result follows our previous circuit-based enumeration algorithms based on deterministic tree automata, and is also inspired by our earlier result on words and nondeterministic sequential extended variable-set automata in the context of document spanners. We extend these results and combine them with a recent tree balancing scheme by Niewerth, so that our enumeration structure supports updates to the underlying tree in logarithmic time (with leaf insertions, leaf deletions, and node relabelings). Our result implies that, for MSO queries with free first-order variables, we can enumerate the results with linear preprocessing and constantdelay and update the underlying tree in logarithmic time, which improves on several known results for words and trees.Building on lower bounds from data structure research, we also show unconditionally that up to a doubly logarithmic factor the update time of our algorithm is optimal. Thus, unlike other settings, there can be no algorithm with constant update time.