We consider the task of enumerating and counting answers to k-ary conjunctive queries against relational databases that may be updated by inserting or deleting tuples.We exhibit a new notion of q-hierarchical conjunctive queries and show that these can be maintained efficiently in the following sense. During a linear time preprocessing phase, we can build a data structure that enables constant delay enumeration of the query results; and when the database is updated, we can update the data structure and restart the enumeration phase within constant time. For the special case of self-join free conjunctive queries we obtain a dichotomy: if a query is not q-hierarchical, then query enumeration with sublinear * delay and sublinear update time (and arbitrary preprocessing time) is impossible.For answering Boolean conjunctive queries and for the more general problem of counting the number of solutions of k-ary queries we obtain complete dichotomies: if the query's homomorphic core is q-hierarchical, then size of the the query result can be computed in linear time and maintained with constant update time. Otherwise, the size of the query result cannot be maintained with sublinear update time.All our lower bounds rely on the OMv-conjecture, a conjecture on the hardness of online matrix-vector multiplication that has recently emerged in the field of fine-grained complexity to characterise the hardness of dynamic problems. The lower bound for the counting problem additionally relies on the orthogonal vectors conjecture, which in turn is implied by the strong exponential time hypothesis. * ) By sublinear we mean O(n 1−ε ) for some ε > 0, where n is the size of the active domain of the current database. * This is the full version of the conference contribution [6].

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We consider a scenario where we want to query a large dataset that is stored in external memory and does not fit into main memory. The most constrained resources in such a situation are the size of the main memory and the number of random accesses to external memory. We note that sequentially streaming data from external memory through main memory is much less prohibitive.We propose an abstract model of this scenario in which we restrict the size of the main memory and the number of random accesses to external memory, but do not restrict sequential reads. A distinguishing feature of our model is that it admits the usage of unlimited external memory for storing intermediate results, such as several hard disks that can be accessed in parallel. In practice, such auxiliary external memory can be crucial. For example, in a first sequential pass the data can be annotated, and in a second pass this annotation can be used to answer the query. Koch's [9] ARB system for answering XPath queries is based on such a strategy.In this model, we prove lower bounds for sorting the input data. As opposed to related results for models without auxiliary external memory for intermediate results, we cannot rely on communication complexity to establish these lower bounds. Instead, we simulate our model by a non-uniform computation model for which we can establish the lower bounds by combinatorial means.

This paper is the tutorial we wish we had had available when starting our own research on constant delay enumeration for conjunctive queries. It provides precise statements and detailed, self-contained proofs of the fundamental results in this area.

This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every first-order formula ϕ(y, z) over the signature {<, +} there is a first-order formula ψ(x, z) which expresses over the structure N, <, + (respectively, over initial segments of this structure) that the variable x is interpreted exactly by the number of possible interpretations of the variable y for which the formula ϕ(y, z) is satisfied. Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in finite graphs is not expressible in first-order logic with unary counting quantifiers and addition. Furthermore, the above result on Presburger arithmetic helps to show the failure of a particular version of the Crane Beach conjecture.