Graph Transformers are popular neural networks that extend the well-known Transformer architecture to the graph domain. These architectures operate by applying self-attention on graph nodes and incorporating graph structure through the use of positional encodings (e.g., Laplacian positional encoding) or structural encodings (e.g., random-walk structural encoding). The quality of such encodings is critical, since they provide the necessary \emph{graph inductive biases} to condition the model on graph structure. In this work, we propose \emph{motif structural encoding} (MoSE) as a flexible and powerful structural encoding framework based on counting graph homomorphisms. Theoretically, we compare the expressive power of MoSE to random-walk structural encoding and relate both encodings to the expressive power of standard message passing neural networks. Empirically, we observe that MoSE outperforms other well-known positional and structural encodings across a range of architectures, and it achieves state-of-the-art performance on a widely studied molecular property prediction dataset.

Knowledge Graph Foundation Models (KGFMs) are at the frontier for deep learning on knowledge graphs (KGs), as they can generalize to completely novel knowledge graphs with different relational vocabularies. Despite their empirical success, our theoretical understanding of KGFMs remains very limited. In this paper, we conduct a rigorous study of the expressive power of KGFMs. Specifically, we show that the expressive power of KGFMs directly depends on the *motifs* that are used to learn the relation representations. We then observe that the most typical motifs used in the existing literature are *binary*, as the representations are learned based on how pairs of relations interact, which limits the model's expressiveness. As part of our study, we design more expressive KGFMs using richer motifs, which necessitate learning relation representations based on, e.g., how triples of relations interact with each other. Finally, we empirically validate our theoretical findings, showing that the use of richer motifs results in better performance on a wide range of datasets drawn from different domains.

Graph machine learning architectures are typically tailored to specific tasks on specific datasets, which hinders their broader applicability. This has led to a new quest in graph machine learning: \emph{how to build graph foundation models (GFMs)} capable of generalizing across arbitrary graphs and features? In this work, we present a recipe for designing GFMs for node-level tasks from first principles. The key ingredient underpinning our study is a systematic investigation of the symmetries that a graph foundation model must respect. In a nutshell, we argue that label permutation-equivariance alongside feature permutation-invariance are necessary in addition to the common node permutation-equivariance on each local neighborhood of the graph. To this end, we first characterize the space of linear transformations that are equivariant to permutations of nodes and labels, and invariant to permutations of features. We then prove that the resulting network is a universal approximator on multisets that respect the aforementioned symmetries. Our recipe uses such layers on the multiset of features induced by the local neighborhood of the graph to obtain a class of graph foundation models for node property prediction. We validate our approach through extensive experiments on 29 real-world node classification datasets, demonstrating both strong zero-shot empirical performance and consistent improvement as the number of training graphs increases.

Modeling the transport dynamics of natural processes from population-level observations is a ubiquitous problem in the natural sciences. Such models rely on key assumptions about the underlying process in order to enable faithful learning of governing dynamics that mimic the actual system behavior. The de facto assumption in current approaches relies on the principle of least action that results in gradient field dynamics and leads to trajectories minimizing an energy functional between two probability measures. However, many real-world systems, such as cell cycles in single-cell RNA, are known to exhibit non-gradient, periodic behavior, which fundamentally cannot be captured by current state-of-the-art methods such as flow and bridge matching. In this paper, we introduce Curly Flow Matching (Curly-FM), a novel approach that is capable of learning non-gradient field dynamics by designing and solving a Schrödinger bridge problem with a non-zero drift reference process---in stark contrast to typical zero-drift reference processes---which is constructed using inferred velocities in addition to population snapshot data. We showcase Curly-FM by solving the trajectory inference problems for single cells, computational fluid dynamics, and ocean currents with approximate velocities. We demonstrate that Curly-FM can learn trajectories that better match both the reference process and population marginals. Curly-FM expands flow matching models beyond the modeling of populations and towards the modeling of known periodic behavior in physical systems. Our code repository is accessible at: https://github.com/kpetrovicc/curly-flow-matching.git

Ontology-based data access is an extensively studied paradigm aiming at improving query answers with the use of an "ontology". An ontology is a specification of a domain of interest, which, in this context, is described via a logical theory. As a form of logical entailment, ontology-mediated query answering is fully interpretable, which makes it possible to derive explanations for ontological query answers. This is a quite important aspect, as the fact that many recent AI systems mostly operating as black boxes has led to some serious concerns. In the literature, various works on explanations in the context of description logics (DLs) have appeared, mostly focusing on explaining concept subsumption and concept unsatisfiability in the ontologies. Some works on explaining query entailment in DLs have appeared as well, however, mainly dealing with inconsistency-tolerant semantics and, actually, non-entailment of the queries. Surprisingly, explaining ontological query entailment has received little attention for ontology languages based on existential rules. In fact, although DLs are popular formalisms to model ontologies, it is generally agreed that rule-based ontologies are well-suited for data-intensive applications, as they allow us to conveniently deal with higher-arity relations, which naturally occur in standard relational databases. The goal of this work is to close this gap, and study the problem of explaining query entailment in the context of existential rules ontologies in terms of minimal subsets of database facts. We provide a thorough complexity analysis for several decision problems associated with minimal explanations for various classes of existential rules, and for different complexity measures.

Machine learning on graphs, especially using graph neural networks (GNNs), has seen a surge in interest due to the wide availability of graph data across a broad spectrum of disciplines, from life to social and engineering sciences. Despite their practical success, our theoretical understanding of the properties of GNNs remains highly incomplete. Recent theoretical advancements primarily focus on elucidating the coarse-grained expressive power of GNNs, predominantly employing combinatorial techniques. However, these studies do not perfectly align with practice, particularly in understanding the generalization behavior of GNNs when trained with stochastic first-order optimization techniques. In this position paper, we argue that the graph machine learning community needs to shift its attention to developing a balanced theory of graph machine learning, focusing on a more thorough understanding of the interplay of expressive power, generalization, and optimization.

Graph neural networks are popular architectures for graph machine learning, based on iterative computation of node representations of an input graph through a series of invariant transformations. A large class of graph neural networks follow a standard message-passing paradigm: at every layer, each node state is updated based on an aggregate of messages from its neighborhood. In this work, we propose a novel framework for training graph neural networks, where every node is viewed as a player that can choose to either listen, broadcast, listen and broadcast, or to isolate. The standard message propagation scheme can then be viewed as a special case of this framework where every node listens and broadcasts to all neighbors. Our approach offers a more flexible and dynamic message-passing paradigm, where each node can determine its own strategy based on their state, effectively exploring the graph topology while learning. We provide a theoretical analysis of the new message-passing scheme which is further supported by an extensive empirical analysis on a synthetic and real-world datasets.

A large body of work has investigated the properties of graph neural networks and identified several limitations, particularly pertaining to their expressive power. Their inability to count certain patterns (e.g., cycles) in a graph lies at the heart of such limitations, since many functions to be learned rely on the ability of counting such patterns. Two prominent paradigms aim to address this limitation by enriching the graph features with subgraph or homomorphism pattern counts. In this work, we show that both of these approaches are sub-optimal in a certain sense and argue for a more fine-grained approach, which incorporates the homomorphism counts of all structures in the “basis” of the target pattern. This yields strictly more expressive architectures without incurring any additional overhead in terms of computational complexity compared to existing approaches. We prove a series of theoretical results on node-level and graph-level motif parameters and empirically validate them on standard benchmark datasets.

We present a new angle on the expressive power of graph neural networks (GNNs) by studying how the predictions of real-valued GNN classifiers, such as those classifying graphs probabilistically, evolve as we apply them on larger graphs drawn from some random graph model. We show that the output converges to a constant function, which upper-bounds what these classifiers can uniformly express. This strong convergence phenomenon applies to a very wide class of GNNs, including state of the art models, with aggregates including mean and the attention-based mechanism of graph transformers. Our results apply to a broad class of random graph models, including sparse and dense variants of the Erdős-Rényi model, the stochastic block model, and the Barabási-Albert model. We empirically validate these findings, observing that the convergence phenomenon appears not only on random graphs but also on some real-world graphs.

Generative modeling over discrete data has recently seen numerous success stories, with applications spanning language modeling, biological sequence design, and graph-structured molecular data. The predominant generative modeling paradigm for discrete data is still autoregressive, with more recent alternatives based on diffusion or flow-matching falling short of their impressive performance in continuous data settings, such as image or video generation. In this work, we introduce Fisher-Flow, a novel flow-matching model for discrete data. Fisher-Flow takes a manifestly geometric perspective by considering categorical distributions over discrete data as points residing on a statistical manifold equipped with its natural Riemannian metric: the \emph{Fisher-Rao metric}. As a result, we demonstrate discrete data itself can be continuously reparameterised to points on the positive orthant of the $d$-hypersphere $\mathbb{S}^d_+$, which allows us to define flows that map any source distribution to target in a principled manner by transporting mass along (closed-form) geodesics of $\mathbb{S}^d_+$. Furthermore, the learned flows in Fisher-Flow can be further bootstrapped by leveraging Riemannian optimal transport leading to improved training dynamics. We prove that the gradient flow induced by Fisher-FLow is optimal in reducing the forward KL divergence. We evaluate Fisher-Flow on an array of synthetic and diverse real-world benchmarks, including designing DNA Promoter, and DNA Enhancer sequences. Empirically, we find that Fisher-Flow improves over prior diffusion and flow-matching models on these benchmarks.

Message Passing Neural Networks (MPNNs) are a staple of graph machine learning. MPNNs iteratively update each node’s representation in an input graph by aggregating messages from the node’s neighbors, which necessitates a memory complexity of the order of the __number of graph edges__. This complexity might quickly become prohibitive for large graphs provided they are not very sparse. In this paper, we propose a novel approach to alleviate this problem by approximating the input graph as an intersecting community graph (ICG) -- a combination of intersecting cliques. The key insight is that the number of communities required to approximate a graph __does not depend on the graph size__. We develop a new constructive version of the Weak Graph Regularity Lemma to efficiently construct an approximating ICG for any input graph. We then devise an efficient graph learning algorithm operating directly on ICG in linear memory and time with respect to the __number of nodes__ (rather than edges). This offers a new and fundamentally different pipeline for learning on very large non-sparse graphs, whose applicability is demonstrated empirically on node classification tasks and spatio-temporal data processing.

Graph neural networks (GNNs) are the de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erdős–Rényi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or one. This class includes the popular graph convolutional network architecture. The result establishes `zero-one laws' for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.

Graph neural networks are prominent models for representation learning over graphs, where the idea is to iteratively compute representations of nodes of an input graph through a series of transformations in such a way that the learned graph function is isomorphism-invariant on graphs, which makes the learned representations graph invariants. On the other hand, it is well-known that graph invariants learned by these class of models are incomplete: there are pairs of non-isomorphic graphs which cannot be distinguished by standard graph neural networks. This is unsurprising given the computational difficulty of graph isomorphism testing on general graphs, but the situation begs to differ for special graph classes, for which efficient graph isomorphism testing algorithms are known, such as planar graphs. The goal of this work is to design architectures for efficiently learning complete invariants of planar graphs. Inspired by the classical planar graph isomorphism algorithm of Hopcroft and Tarjan, we propose PlanE as a framework for planar representation learning. PlanE includes architectures which can learn complete invariants over planar graphs while remaining practically scalable. We empirically validate the strong performance of the resulting model architectures on well-known planar graph benchmarks, achieving multiple state-of-the-art results.

Graph neural networks are prominent models for representation learning over graph-structured data. While the capabilities and limitations of these models are well-understood for simple graphs, our understanding remains incomplete in the context of knowledge graphs. Our goal is to provide a systematic understanding of the landscape of graph neural networks for knowledge graphs pertaining to the prominent task of link prediction. Our analysis entails a unifying perspective on seemingly unrelated models and unlocks a series of other models. The expressive power of various models is characterized via a corresponding relational Weisfeiler-Leman algorithm. This analysis is extended to provide a precise logical characterization of the class of functions captured by a class of graph neural networks. The theoretical findings presented in this paper explain the benefits of some widely employed practical design choices, which are validated empirically.

Knowledge graph completion is the task of inferring missing facts based on existing data in a knowledge graph. Temporal knowledge graph completion (TKGC) is an extension of this task to temporal knowledge graphs, where each fact is additionally associated with a time stamp. Current approaches for TKGC primarily build on existing embedding models which are developed for static knowledge graph completion, and extend these models to incorporate time, where the idea is to learn latent representations for entities, relations, and timestamps and then use the learned representations to predict missing facts at various time steps. In this paper, we propose BoxTE, a box embedding model for TKGC, building on the static knowledge graph embedding model BoxE. We show that BoxTE is fully expressive, and possesses strong inductive capacity in the temporal setting. We then empirically evaluate our model and show that it achieves state-of-the-art results on several TKGC benchmarks

We investigate quantum circuits for graph representation learning, and propose equivariant quantum graph circuits (EQGCs), as a class of parameterized quantum circuits with strong relational inductive bias for learning over graph-structured data. Conceptually, EQGCs serve as a unifying framework for quantum graph representation learning, allowing us to define several interesting subclasses which subsume existing proposals. In terms of the representation power, we prove that the studied subclasses of EQGCs are universal approximators for functions over the bounded graph domain. This theoretical perspective on quantum graph machine learning methods opens many directions for further work, and could lead to models with capabilities beyond those of classical approaches. We empirically verify the expressive power of EQGCs through a dedicated experiment on synthetic data, and additionally observe that the performance of EQGCs scales well with the depth of the model and does not suffer from barren plateu issues.

Weighted model counting consists of computing the weighted sum of all satisfying assignments of a propositional formula. Weighted model counting is well-known to be #P-hard for exact solving, but admits a fully polynomial randomized approximation scheme when restricted to DNF structures. In this work, we study weighted model integration, a generalization of weighted model counting which involves real variables in addition to propositional variables, and pose the following question: Does weighted model integration on DNF structures admit a fully polynomial randomized approximation scheme? Building on classical results from approximate weighted model counting and approximate volume computation, we show that weighted model integration on DNF structures can indeed be approximated for a class of weight functions. Our approximation algorithm is based on three subroutines, each of which can be a weak (i.e., approximate), or a strong (i.e., exact) oracle, and in all cases, comes along with accuracy guarantees. We experimentally verify our approach over randomly generated DNF instances of varying sizes, and show that our algorithm scales to large problem instances, involving up to 1K variables, which are currently out of reach for existing, general-purpose weighted model integration solvers.

We study the problem of query evaluation on probabilistic graphs, namely, tuple-independent probabilistic databases over signatures of arity two. We focus on the class of queries closed under homomorphisms, or, equivalently, the infinite unions of conjunctive queries. Our main result states that the probabilistic query evaluation problem is #P-hard for all unbounded queries from this class. As bounded queries from this class are equivalent to a union of conjunctive queries, they are already classified by the dichotomy of Dalvi and Suciu (2012). Hence, our result and theirs imply a complete data complexity dichotomy, between polynomial time and #P-hardness, on evaluating homomorphism-closed queries over probabilistic graphs. This dichotomy covers in particular all fragments of infinite unions of conjunctive queries over arity-two signatures, such as negation-free (disjunctive) Datalog, regular path queries, and a large class of ontology-mediated queries. The dichotomy also applies to a restricted case of probabilistic query evaluation called generalized model counting, where fact probabilities must be 0, 0.5, or 1. We show the main result by reducing from the problem of counting the valuations of positive partitioned 2-DNF formulae, or from the source-to-target reliability problem in an undirected graph, depending on properties of minimal models for the query.

Recently, explanations for query answers under existential rules have been investigated, where an explanation is an inclusion-minimal subset of a given database that, together with the ontology, entails the query. In this paper, we take a step further and study explanations under different minimality criteria. In particular, we first study cardinality-minimal explanations and hence focus on deriving explanations of minimum size. We then study a more general preference order induced by a weight distribution. We assume that every database fact is annotated with a (penalization) weight, and we are interested in explanations with minimum overall weight. For both preference orders, we study a variety of explanation problems, such as recognizing a preferred explanation, all preferred explanations, a relevant or necessary fact, and the existence of a preferred explanation not containing forbidden sets of facts. We provide a detailed complexity analysis for all the aforementioned problems, thereby providing a more complete picture for explaining query answers under existential rules.

Graph neural networks (GNNs) are effective models for representation learning on relational data. However, standard GNNs are limited in their expressive power, as they cannot distinguish graphs beyond the capability of the Weisfeiler-Leman graph isomorphism heuristic. In order to break this expressiveness barrier, GNNs have been enhanced with random node initialization (RNI), where the idea is to train and run the models with randomized initial node features. In this work, we analyze the expressive power of GNNs with RNI, and prove that these models are universal, a first such result for GNNs not relying on computationally demanding higher-order properties. This universality result holds even with partially randomized initial node features, and preserves the invariance properties of GNNs in expectation. We then empirically analyze the effect of RNI on GNNs, based on carefully constructed datasets. Our empirical findings support the superior performance of GNNs with RNI over standard GNNs.

Weighted model counting (WMC) has emerged as a prevalent approach for probabilistic inference. In its most general form, WMC is #P-hard. Weighted DNF counting (weighted #DNF) is a special case, where approximations with probabilistic guarantees are obtained in O(nm), where n denotes the number of variables, and m the number of clauses of the input DNF, but this is not scalable in practice. In this paper, we propose a neural model counting approach for weighted #DNF that combines approximate model counting with deep learning, and accurately approximates model counts in linear time when width is bounded. We conduct experiments to validate our method, and show that our model learns and generalizes very well to large-scale #DNF instances.

Ontology-mediated query answering is a paradigm that seeks to exploit the semantic knowledge expressed in terms of ontologies to improve query answers over incomplete data sources. In this paper, we focus on description logic ontologies, and study the problem of explaining why an ontology-mediated query is entailed from a given data source. Specifically, we view explanations as minimal sets of assertions from an ABox, which satisfy the ontologymediated query. Based on such explanations, we study a variety of problems taken from the recent literature on explanations (studied for existential rules), such as recognizing all minimal explanations. Our results establish tight connections between intractable explanation problems and variants of propositional satisfiability problems. We provide insights on the inherent computational difficulty of deriving explanations for ontology-mediated queries

We study the problem of probabilistic query evaluation (PQE) over probabilistic graphs, namely, tuple-independent probabilistic databases (TIDs) on signatures of arity two. Our focus is the class of queries that is closed under homomorphisms, or equivalently, the infinite unions of conjunctive queries, denoted UCQ∞. Our main result states that all unbounded queries in UCQ∞ are #P-hard for PQE. As bounded queries in UCQ∞ are already classified by the dichotomy of Dalvi and Suciu [Dalvi and Suciu, 2012], our results and theirs imply a complete dichotomy on PQE for UCQ∞ queries over probabilistic graphs. This dichotomy covers in particular all fragments in UCQ∞ such as negation-free (disjunctive) Datalog, regular path queries, and a large class of ontology-mediated queries on arity-two signatures. Our result is shown by reducing from counting the valuations of positive partitioned 2-DNF formulae (#PP2DNF) for some queries, or from the source-to-target reliability problem in an undirected graph (#U-ST-CON) for other queries, depending on properties of minimal models.

Weighted model counting (WMC) consists of computing the weighted sum of all satisfying assignments of a propositional formula. WMC is well-known to be #P-hard for exact solving, but admits a fully polynomial randomised approximation scheme (FPRAS) when restricted to DNF structures. In this work, we study weighted model integration, a generalization of weighted model counting which involves real variables in addition to propositional variables, and pose the following question: Does weighted model integration on DNF structures admit an FPRAS? Building on classical results from approximate volume computation and approximate weighted model counting, we show that weighted model integration on DNF structures can indeed be approximated for a class of weight functions. Our approximation algorithm is based on three subroutines, each of which can be a weak (i.e., approximate), or a strong (i.e., exact) oracle, and in all cases, comes along with accuracy guarantees. We experimentally verify our approach over randomly generated DNF instances of varying sizes, and show that our algorithm scales to large problem instances, involving up to 1K variables, which are currently out of reach for existing, general-purpose weighted model integration solvers.

Ontology-mediated query answering is an extensively studied paradigm, where the conceptual knowledge provided by an ontology is leveraged towards more enhanced querying of data sources. A major advantage of ontological reasoning is its interpretability, which allows one to derive explanations for query answers. Indeed, explanations have a long history in knowledge representation, and have also been investigated for ontology languages based on description logics and existential rules. Existing works on existential rules, however, merely focus on understanding why a query is entailed, i.e., explaining positive query answers. In this paper, we continue this line of research and address another important problem, namely, explaining why a query is not entailed under existential rules, i.e., explaining negative query answers. We consider various problems related to explaining non-entailments from the abduction literature, and also introduce new problems. For all considered problems, we give a detailed complexity analysis for a wide range of existential rule languages and complexity measures.

Knowledge base completion (KBC) aims to automatically infer missing facts by exploiting information already present in a knowledge base (KB). A promising approach for KBC is to embed knowledge into latent spaces and make predictions from learned embeddings. However, existing embedding models are subject to at least one of the following limitations: (1) theoretical inexpressivity, (2) lack of support for prominent inference patterns (e.g., hierarchies), (3) lack of support for KBC over higher-arity relations, and (4) lack of support for incorporating logical rules. Here, we propose a spatio-translational embedding model, called BoxE, that simultaneously addresses all these limitations. BoxE embeds entities as points, and relations as a set of hyper-rectangles (or boxes), which spatially characterize basic logical properties. This seemingly simple abstraction yields a fully expressive model offering a natural encoding for many desired logical properties. BoxE can both capture and inject rules from rich classes of rule languages, going well beyond individual inference patterns. By design, BoxE naturally applies to higher-arity KBs. We conduct a detailed experimental analysis, and show that BoxE achieves state-of-the-art performance, both on benchmark knowledge graphs and on more general KBs, and we empirically show the power of integrating logical rules.

Large-scale knowledge bases are at the heart of modern information systems. Their knowledge is inherently uncertain, and hence they are often materialized as probabilistic databases. However, probabilistic database management systems typically lack the capability to incorporate implicit background knowledge and, consequently, fail to capture some intuitive query answers. Ontology-mediated query answering is a popular paradigm for encoding commonsense knowledge, which can provide more complete answers to user queries. We propose a new data model that integrates the paradigm of ontology-mediated query answering with probabilistic databases, employing a log-linear probability model. We compare our approach to existing proposals, and provide supporting computational results.

Ontology-mediated query answering is an extensively studied paradigm, which aims at improving query answers with the use of a logical theory. As a form of logical entailment, ontology-mediated query answering is fully interpretable, which makes it possible to derive explanations for query answers. Surprisingly, however, explaining answers for ontology-mediated queries has received little attention for ontology languages based on existential rules. In this paper, we close this gap, and study the problem of explaining query answers in terms of minimal subsets of database facts. We provide a thorough complexity analysis for several decision problems associated with minimal explanations under existential rules.

We give a survey on recent advances at the forefront of research on probabilistic knowledge bases for representing and querying large-scale automatically extracted data. We concentrate especially on increasing the semantic expressivity of formalisms for representing and querying probabilistic knowledge (i) by giving up the closed-world assumption, (ii) by allowing for commonsense knowledge (and in parallel giving up the tuple-independence assumption), and (iii) by giving up the closed-domain assumption, while preserving some computational properties of query answering in such formalisms.

Probabilistic databases (PDBs) are usually incomplete, e.g., containing only the facts that have been extracted from the Web with high confidence. However, missing facts are often treated as being false, which leads to unintuitive results when querying PDBs. Recently, open-world probabilistic databases (OpenPDBs) were proposed to address this issue by allowing probabilities of unknown facts to take any value from a fixed probability interval. In this paper, we extend OpenPDBs by Datalog+/- ontologies, under which both upper and lower probabilities of queries become even more informative, enabling us to distinguish queries that were indistinguishable before. We show that the dichotomy between P and PP in (Open)PDBs can be lifted to the case of first-order rewritable positive programs (without negative constraints); and that the problem can become NP^PP-complete, once negative constraints are allowed. We also propose an approximating semantics that circumvents the increase in complexity caused by negative constraints.

Forming the foundations of large-scale knowledge bases, probabilistic databases have been widely studied in the literature. In particular, probabilistic query evaluation has been investigated intensively as a central inference mechanism. However, despite its power, query evaluation alone cannot extract all the relevant information encompassed in large-scale knowledge bases. To exploit this potential, we study two inference tasks; namely finding the most probable database and the most probable hypothesis for a given query. As natural counterparts of most probable explanations (MPE) and maximum a posteriori hypotheses (MAP) in probabilistic graphical models, they can be used in a variety of applications that involve prediction or diagnosis tasks. We investigate these problems relative to a variety of query languages, ranging from conjunctive queries to ontology-mediated queries, and provide a detailed complexity analysis.

Large-scale probabilistic knowledge bases are becoming increasingly important in academia and industry alike. They are constantly extended with new data, powered by modern information extraction tools that associate probabilities with database tuples. In this paper, we revisit the semantics underlying such systems. In particular, the closed-world assumption of probabilistic databases, that facts not in the database have probability zero, clearly conflicts with their everyday use. To address this discrepancy, we propose an open-world probabilistic database semantics, which relaxes the probabilities of open facts to default intervals. For this open-world setting, we lift the existing data complexity dichotomy of probabilistic databases, and propose an efficient evaluation algorithm for unions of conjunctive queries. We also show that query evaluation can become harder for non-monotone queries.

We present an ontological framework, based on preference rankings, that allows users to express their preferences between the knowledge explicitly available in the ontology. Using this formalism, the answers for a given query to an ontology can be ranked by preference, allowing users to retrieve the most preferred answers only. We provide a host of complexity results for the main computational tasks in this framework, for the general case, and for EL and DL-Lite_core as underlying ontology languages.

We study the query evaluation problem in probabilistic databases in the presence of probabilistic existential rules. Our focus is on the Datalog±family of languages for which we define the probabilistic counterpart using a flexible and compact encoding of probabilities. This formalism can be viewed as a generalization of probabilistic databases, as it allows to generate new facts from the given ones, using so-called tuple-generating dependencies, or existential rules. We study the computational cost of this additional expressiveness under two different semantics. First, we use a conventional approach and assume that the probabilistic knowledge base is consistent and employ the standard possible world semantics. Thereafter, we introduce a probabilistic inconsistency-tolerant semantics, which we call inconsistency-tolerant possible world semantics. For both of these cases, we provide a thorough complexity analysis relative to different languages, drawing a complete picture of the complexity of probabilistic query answering in this family.