arXiv subjects

Christoph Standke

Publications and source records attributed to Christoph Standke.

10 recordsLinked to original sources

arXiv2026

The Importance of Parameters in Ranking Functions

How important is the weight of a given column in determining the ranking of tuples in a table? To address such an explanation question about a ranking function, we investigate the computation of SHAP scores for column weights, adopting a recent framework by Grohe et al.[ICDT'24]. The exact definition of this score depends on three key components: (1) the ranking function in use, (2) an effect function that quantifies the impact of using alternative weights on the ranking, and (3) an underlying weight distribution. We analyze the computational complexity of different instantiations of this framework for a range of fundamental ranking and effect functions, focusing on probabilistically independent finite distributions for individual columns. For the ranking functions, we examine lexicographic orders and score-based orders defined by the summation, minimum, and maximum functions. For the effect functions, we consider global, top-k, and local perspectives: global measures quantify the divergence between the perturbed and original rankings, top-k measures inspect the change in the set of top-k answers, and local measures capture the impact on an individual tuple of interest. Although all cases admit an additive fully polynomial-time randomized approximation scheme (FPRAS), we establish the complexity of exact computation, identifying which cases are solvable in polynomial time and which are #P-hard. We further show that all complexity results, lower bounds and upper bounds, extend to a related task of computing the Shapley value of whole columns (regardless of their weight).

cs.DB↗

arXiv2026

Recursive querying of neural networks via weighted structures

Expressive querying of machine learning models - viewed as a form of intentional data - enables their verification and interpretation using declarative languages, thereby making learned representations of data more accessible. Motivated by the querying of feedforward neural networks, we investigate logics for weighted structures. In the absence of a bound on neural network depth, such logics must incorporate recursion; thereto we revisit the functional fixpoint mechanism proposed by Gr\"adel and Gurevich. We adopt it in a Datalog-like syntax; we extend normal forms for fixpoint logics to weighted structures; and show an equivalent "loose" fixpoint mechanism that allows values of inductively defined weight functions to be overwritten. We propose a "scalar" restriction of functional fixpoint logic, of polynomial-time data complexity, and show it can express all PTIME model-agnostic queries over reduced networks with polynomially bounded weights. In contrast, we show that very simple model-agnostic queries are already NP-complete. Finally, we consider transformations of weighted structures by iterated transductions.

cs.LO↗

arXiv2025

Tractability Frontiers of the Shapley Value for Aggregate Conjunctive Queries

In recent years, the Shapley value has emerged as a general game-theoretic measure for assessing the contribution of a tuple to the result of a database query. We study the complexity of calculating the Shapley value of a tuple for an aggregate conjunctive query, which applies an aggregation function to the result of a conjunctive query (CQ) based on a value function that assigns a number to each query answer. Prior work by Livshits, Bertossi, Kimelfeld, and Sebag (2020) established that this task is #P-hard for every nontrivial aggregation function when the query is non-hierarchical with respect to its existential variables, assuming the absence of self-joins. They further showed that this condition precisely characterizes the class of intractable CQs when the aggregate function is sum or count. In addition, they posed as open problems the complexity of other common aggregate functions such as min, max, count-distinct, average, and quantile (including median). Towards the resolution of these problems, we identify for each aggregate function a class of hierarchical CQs where the Shapley value is tractable with every value function, as long as it is local (i.e., determined by the tuples of one relation). We further show that each such class is maximal: for every CQ outside of this class, there is a local (easy-to-compute) value function that makes the Shapley value #P-hard. Interestingly, our results reveal that each aggregate function corresponds to a different generalization of the class of hierarchical CQs from Boolean to non-Boolean queries. In particular, max, min, and count-distinct match the class of CQs that are all-hierarchical (i.e., hierarchical with respect to all variables), and average and quantile match the narrower class of q-hierarchical CQs introduced by Berkholz, Keppeler, and Schweikardt (2017) in the context of the fine-grained complexity of query answering.

cs.DB↗

arXiv2024

Query languages for neural networks

We lay the foundations for a database-inspired approach to interpreting and understanding neural network models by querying them using declarative languages. Towards this end we study different query languages, based on first-order logic, that mainly differ in their access to the neural network model. First-order logic over the reals naturally yields a language which views the network as a black box; only the input--output function defined by the network can be queried. This is essentially the approach of constraint query languages. On the other hand, a white-box language can be obtained by viewing the network as a weighted graph, and extending first-order logic with summation over weight terms. The latter approach is essentially an abstraction of SQL. In general, the two approaches are incomparable in expressive power, as we will show. Under natural circumstances, however, the white-box approach can subsume the black-box approach; this is our main result. We prove the result concretely for linear constraint queries over real functions definable by feedforward neural networks with a fixed number of hidden layers and piecewise linear activation functions.

cs.AI↗

arXiv2024

Untangling Gaussian Mixtures

Tangles were originally introduced as a concept to formalize regions of high connectivity in graphs. In recent years, they have also been discovered as a link between structural graph theory and data science: when interpreting similarity in data sets as connectivity between points, finding clusters in the data essentially amounts to finding tangles in the underlying graphs. This paper further explores the potential of tangles in data sets as a means for a formal study of clusters. Real-world data often follow a normal distribution. Accounting for this, we develop a quantitative theory of tangles in data sets drawn from Gaussian mixtures. To this end, we equip the data with a graph structure that models similarity between the points and allows us to apply tangle theory to the data. We provide explicit conditions under which tangles associated with the marginal Gaussian distributions exist asymptotically almost surely. This can be considered as a sufficient formal criterion for the separabability of clusters in the data.

math.ST↗

arXiv2024

The Importance of Parameters in Database Queries

We propose and study a framework for quantifying the importance of the choices of parameter values to the result of a query over a database. These parameters occur as constants in logical queries, such as conjunctive queries. In our framework, the importance of a parameter is its SHAP score - a popular instantiation of the game-theoretic Shapley value to measure the importance of feature values in machine learning models. We make the case for the rationale of using this score by explaining the intuition behind SHAP, and by showing that we arrive at this score in two different, apparently opposing, approaches to quantifying the contribution of a parameter. The application SHAP requires two components in addition to the query and the database: (a) a probability distribution over the combinations of parameter values, and (b) a utility function that measures the similarity between the result for the original parameters and the result for hypothetical parameters. The main question addressed in the paper is the complexity of calculating the SHAP score for different distributions and similarity measures. In particular, we devise polynomial-time algorithms for the case of full acyclic conjunctive queries for certain (natural) similarity functions. We extend our results to conjunctive queries with parameterized filters (e.g., inequalities between variables and parameters). We also illustrate the application of our results to "why-not" explanations (aiming to explain the absence of a query answer), where we consider the task of quantifying the contribution of query components to the elimination of a non-answer in consideration. Finally, we discuss a simple approximation technique for the case of correlated parameters.

cs.DB↗

arXiv2023

The Complexity of Homomorphism Reconstructibility

Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph $G$ as a finite vector of homomorphism counts from some fixed finite set of graphs to $G$. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructability problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph $G$ that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an $\mathsf{NP}^{#\mathsf{P}}$-hard problem, which still can be $\mathsf{NP}$-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph $G$ as additional input, the problem cannot be $\mathsf{NP}$-hard unless $\mathsf{P} = \mathsf{NP}$. For this regime, we obtain partial positive results. We also investigate the problem's parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given.

cs.DS↗

arXiv2023

Probabilistic Query Evaluation with Bag Semantics

We study the complexity of evaluating queries on probabilistic databases under bag semantics. We focus on self-join free conjunctive queries, and probabilistic databases where occurrences of different facts are independent, which is the natural generalization of tuple-independent probabilistic databases to the bag semantics setting. For set semantics, the data complexity of this problem is well understood, even for the more general class of unions of conjunctive queries: it is either in polynomial time, or #P-hard, depending on the query (Dalvi & Suciu, JACM 2012). A reasonably general model of bag probabilistic databases may have unbounded multiplicities. In this case, the probabilistic database is no longer finite, and a careful treatment of representation mechanisms is required. Moreover, the answer to a Boolean query is a probability distribution over (possibly all) non-negative integers, rather than a probability distribution over { true, false }. Therefore, we discuss two flavors of probabilistic query evaluation: computing expectations of answer tuple multiplicities, and computing the probability that a tuple is contained in the answer at most k times for some parameter k. Subject to mild technical assumptions on the representation systems, it turns out that expectations are easy to compute, even for unions of conjunctive queries. For query answer probabilities, we obtain a dichotomy between solvability in polynomial time and #P-hardness for self-join free conjunctive queries.

cs.DB↗

arXiv2022

Tuple-Independent Representations of Infinite Probabilistic Databases

Probabilistic databases (PDBs) are probability spaces over database instances. They provide a framework for handling uncertainty in databases, as occurs due to data integration, noisy data, data from unreliable sources or randomized processes. Most of the existing theory literature investigated finite, tuple-independent PDBs (TI-PDBs) where the occurrences of tuples are independent events. Only recently, Grohe and Lindner (PODS '19) introduced independence assumptions for PDBs beyond the finite domain assumption. In the finite, a major argument for discussing the theoretical properties of TI-PDBs is that they can be used to represent any finite PDB via views. This is no longer the case once the number of tuples is countably infinite. In this paper, we systematically study the representability of infinite PDBs in terms of TI-PDBs and the related block-independent disjoint PDBs. The central question is which infinite PDBs are representable as first-order views over tuple-independent PDBs. We give a necessary condition for the representability of PDBs and provide a sufficient criterion for representability in terms of the probability distribution of a PDB. With various examples, we explore the limits of our criteria. We show that conditioning on first order properties yields no additional power in terms of expressivity. Finally, we discuss the relation between purely logical and arithmetic reasons for (non-)representability.

cs.DB↗

arXiv2017

A minimum-change version of the Chung-Feller theorem for Dyck paths

A Dyck path with $2k$ steps and $e$ flaws is a path in the integer lattice that starts at the origin and consists of $k$ many $\nearrow$-steps and $k$ many $\searrow$-steps that change the current coordinate by $(1,1)$ or $(1,-1)$, respectively, and that has exactly $e$ many $\searrow$-steps below the line $y=0$. Denoting by $D_{2k}^e$ the set of Dyck paths with $2k$ steps and $e$ flaws, the Chung-Feller theorem asserts that the sets $D_{2k}^0,D_{2k}^1,\ldots,D_{2k}^k$ all have the same cardinality $\frac{1}{k+1}\binom{2k}{k}=C_k$, the $k$-th Catalan number. The standard combinatorial proof of this classical result establishes a bijection $f'$ between $D_{2k}^e$ and $D_{2k}^{e+1}$ that swaps certain parts of the given Dyck path $x$, with the effect that $x$ and $f'(x)$ may differ in many positions. In this paper we strengthen the Chung-Feller theorem by presenting a simple bijection $f$ between $D_{2k}^e$ and $D_{2k}^{e+1}$ which has the additional feature that $x$ and $f(x)$ differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of $f$ in constant time per generated Dyck path. As an application, we use our minimum-change bijection $f$ to construct cycle-factors in the odd graph $O_{2k+1}$ and the middle levels graph $M_{2k+1}$ --- two intensively studied families of vertex-transitive graphs --- that consist of $C_k$ many cycles of the same length.

math.CO↗