Lower Bounds for Existential Pebble Games and k-Consistency Tests

The existential k-pebble game characterizes the expressive power of the existential-positive k-variable fragment of first-order logic on finite structures. The winner of the existential k-pebble game on two given finite structures can be determined in time O(n2k) by dynamic programming on the graph of game configurations. We show that there is no O(n(k-3)/12)-time algorithm that decides which player can win the existential k-pebble game on two given structures. This lower bound is unconditional and does not rely on any complexity-theoretic assumptions. Establishing strong k-consistency is a well-known heuristic for solving the constraint satisfaction problem (CSP). By the game characterization of Kolaitis and Vardi our result implies that there is no O(n(k-3)/12)-time algorithm that decides if strong k-consistency can be established for a given CSP-instance

Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width

Marx (STOC 2010, J. ACM 2013) introduced the notion of submodular width of a conjunctive query (CQ) and showed that for any class Phi of Boolean CQs of bounded submodular width, the model-checking problem for Phi on the class of all finite structures is fixed-parameter tractable (FPT). Note that for non-Boolean queries, the size of the query result may be far too large to be computed entirely within FPT time. We investigate the free-connex variant of submodular width and generalise Marx\u27s result to non-Boolean queries as follows: For every class Phi of CQs of bounded free-connex submodular width, within FPT-preprocessing time we can build a data structure that allows to enumerate, without repetition and with constant delay, all tuples of the query result. Our proof builds upon Marx\u27s splitting routine to decompose the query result into a union of results; but we have to tackle the additional technical difficulty to ensure that these can be enumerated efficiently

Answering UCQs under Updates and in the Presence of Integrity Constraints

We investigate the query evaluation problem for fixed queries over fully dynamic databases where tuples can be inserted or deleted. The task is to design a dynamic data structure that can immediately report the new result of a fixed query after every database update. We consider unions of conjunctive queries (UCQs) and focus on the query evaluation tasks testing (decide whether an input tuple belongs to the query result), enumeration (enumerate, without repetition, all tuples in the query result), and counting (output the number of tuples in the query result). We identify three increasingly restrictive classes of UCQs which we call t-hierarchical, q-hierarchical, and exhaustively q-hierarchical UCQs. Our main results provide the following dichotomies: If the query\u27s homomorphic core is t-hierarchical (q-hierarchical, exhaustively q-hierarchical), then the testing (enumeration, counting) problem can be solved with constant update time and constant testing time (delay, counting time). Otherwise, it cannot be solved with sublinear update time and sublinear testing time (delay, counting time), unless the OV-conjecture and/or the OMv-conjecture fails. We also study the complexity of query evaluation in the dynamic setting in the presence of integrity constraints, and we obtain similar dichotomy results for the special case of small domain constraints (i.e., constraints which state that all values in a particular column of a relation belong to a fixed domain of constant size)

Answering FO+MOD Queries Under Updates on Bounded Degree Databases

We investigate the query evaluation problem for fixed queries over fully dynamic databases, where tuples can be inserted or deleted. The task is to design a dynamic algorithm that immediately reports the new result of a fixed query after every database update. We consider queries in first-order logic (FO) and its extension with modulo-counting quantifiers (FO+MOD), and show that they can be efficiently evaluated under updates, provided that the dynamic database does not exceed a certain degree bound. In particular, we construct a data structure that allows to answer a Boolean FO+MOD query and to compute the size of the query result within constant time after every database update. Furthermore, after every update we are able to immediately enumerate the new query result with constant delay between the output tuples. The time needed to build the data structure is linear in the size of the database. Our results extend earlier work on the evaluation of first-order queries on static databases of bounded degree and rely on an effective Hanf normal form for FO+MOD recently obtained by [Heimberg, Kuske, and Schweikardt, LICS, 2016]

Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy

Given two structures G and H distinguishable in FO^k (first-order logic with k variables), let A^k(G,H) denote the minimum alternation depth of a FO^k formula distinguishing G from H. Let A^k(n) be the maximum value of A^k(G,H) over n-element structures. We prove the strictness of the quantifier alternation hierarchy of FO^2 in a strong quantitative form, namely A^2(n) >= n/8-2, which is tight up to a constant factor. For each k >= 2, it holds that A^k(n) > log_(k+1) n-2 even over colored trees, which is also tight up to a constant factor if k >= 3. For k >= 3 the last lower bound holds also over uncolored trees, while the alternation hierarchy of FO^2 collapses even over all uncolored graphs. We also show examples of colored graphs G and H on n vertices that can be distinguished in FO^2 much more succinctly if the alternation number is increased just by one: while in Sigma_i it is possible to distinguish G from H with bounded quantifier depth, in Pi_i this requires quantifier depth Omega(n2). The quadratic lower bound is best possible here because, if G and H can be distinguished in FO^k with i quantifier alternations, this can be done with quantifier depth n^(2k-2)

Die Rolle der Gene CXCL9 und NR1I2 bei chronisch entzündlichen Darmerkrankungen im Kindesalter

Chronisch entzündliche Darmerkrankungen haben einen multifaktoriellen Ursprung. Dabei scheinen genetische Faktoren eine entscheidende Rolle zu spielen und Einfluss auf den Krankheitsverlauf und unterschiedliche Erkrankungsmuster zu haben. Insbesondere bei Kindern ist eine genetische Disposition in der Pathogenese von chronisch entzündlichen Darmerkrankungen anzunehmen. Die vorliegende Dissertation ist die erste Untersuchung zur Expression der Gene CXCL9 und NR1I2 im Kolongewebe bei M. Crohn und C. ulcerosa im Kindesalter. Auch die Allelvarianten rs2276886 und rs3814055 in beiden Genen wurden erstmals bei Kindern mit chronisch entzündlichen Darmerkrankungen untersucht. Ein Zusammenhang zwischen M. Crohn und C. ulcerosa im Kindesalter und der Expression des Gens NR1I2 im Kolongewebe und dem SNP rs3814055 konnte in dieser Studie nicht gefunden werden. Eine deutliche Überexpression des Gens CXCL9 und die signifikante Assoziation des SNP rs2276886 mit M. Crohn lassen jedoch eine Rolle dieses Gens in der Pathogenese chronisch entzündlicher Darmerkrankungen im Kindesalter vermuten