We study the expressive power and computational properties of first-order logic and its extensions under the semiring semantics originating from the seminal work of Green, Karvounarakis, and Tannen. While semiring semantics is currently extensively used, e.g., in the study of provenance in database theory and description logic, a comprehensive computational analysis of these logics acting over general semirings is still lacking. We analyse expressivity, and complexity of model-checking of first-order formulas in this framework, providing characterizations in terms of generalized Blum–Shub–Smale machines over semirings. We also show a variant of Fagin's theorem, i.e., a logical characterization of nondeterministic polynomial time over semirings using a version of existential second-order logic. We further generalize Cook's theorem for the semiring framework and show that propositional satisfiability in the semiring semantics is complete for this notion of NP, and that the true existential first-order theory of the semiring is complete for its Boolean fragment.

Abstract We study the expressivity and the complexity of various logics in probabilistic team semantics with the Boolean negation. In particular, we study the extension of probabilistic independence logic with the Boolean negation, and a recently introduced logic first-order theory of random variables with probabilistic independence. We give several results that compare the expressivity of these logics with the most studied logics in probabilistic team semantics setting, as well as relating their expressivity to a numerical variant of second-order logic. In addition, we introduce novel entropy atoms and show that the extension of first-order logic by entropy atoms subsumes probabilistic independence logic. Finally, we obtain some results on the complexity of model checking, validity and satisfiability of our logics.

The training problem of neural networks (NNs) is known to be ER-complete with respect to ReLU and linear activation functions. We show that the training problem for NNs equipped with arbitrary activation functions is polynomial-time bireducible to the existential theory of the reals extended with the corresponding activation functions. For effectively continuous activation functions (e.g., the sigmoid function), we obtain an inclusion to low levels of the arithmetical hierarchy. Consequently, the sigmoid activation function leads to the existential theory of the reals with the exponential function, and hence the decidability of training NNs using the sigmoid activation function is equivalent to the decidability of the existential theory of the reals with the exponential function, a long-standing open problem. In contrast, we obtain that the training problem is undecidable if sinusoidal activation functions are considered.

Information inequalities appear in many database applications such as query output size bounds, query containment, and implication between data dependencies. Recently Khamis et al. proposed to study the algorithmic aspects of information inequalities, including the information inequality problem: decide whether a linear inequality over entropies of random variables is valid. While the decidability of this problem is a major open question, applications often involve only inequalities that adhere to specific syntactic forms linked to useful semantic invariance properties. This paper studies the information inequality problem in different syntactic and semantic scenarios that arise from database applications. Focusing on the boundary between tractability and intractability, we show that the information inequality problem is coNP-complete if restricted to normal polymatroids, and in polynomial time if relaxed to monotone functions. We also examine syntactic restrictions related to query output size bounds, and provide an alternative proof, through monotone functions, for the polynomial-time computability of the entropic bound over simple sets of degree constraints.

Conditional independence plays a foundational role in database theory, probability theory, information theory, and graphical models. In databases, conditional independence appears in database normalization and is known as the (embedded) multivalued dependency. Many properties of conditional independence are shared across various domains, and to some extent these commonalities can be studied through a measure-theoretic approach. The present paper proposes an alternative approach via semiring relations, defined by extending database relations with tuple annotations from some commutative semiring. Integrating various interpretations of conditional independence in this context, we investigate how the choice of the underlying semiring impacts the corresponding axiomatic and decomposition properties. We specifically identify positivity and multiplicative cancellativity as the key semiring properties that enable extending results from the relational context to the broader semiring framework. Additionally, we explore the relationships between different conditional independence notions through model theory, and consider how methods to test logical consequence and validity generalize from database theory and information theory to semiring relations.

We present exact complexity bounds on the discovery of cross joins from database relations, and algorithms that work evidently well on real-world data sets within those bounds.

Semiring semantics for first-order logic provides a way to trace how facts represented by a model are used to deduce satisfaction of a formula. Team semantics is a framework for studying logics of dependence and independence in diverse contexts such as databases, quantum mechanics, and statistics by extending first-order logic with atoms that describe dependencies between variables. Combining these two, we propose a unifying approach for analysing the concepts of dependence and independence via a novel semiring team semantics, which subsumes all the previously considered variants for first-order team semantics. In particular, we study the preservation of satisfaction of dependencies and formulae between different semirings. In addition we create links to reasoning tasks such as provenance, counting, and repairs.

A cross join between two attribute sets holds on a relation whenever its projection onto the union of the attribute sets is the cross join between its projections on the first and second attribute set. Hence, the cross join is a fundamental operator on database relations. For example, it can rewrite the division operator into a simple projection, or measure the independence of tuple values between two attribute sets during cardinality estimation. It is therefore surprising that we present the first research on the discovery problem of cross joins. We show that the problem of deciding whether there is a cross join that holds on a given relation is not only NP-complete but W[3]-complete in its arguably most natural parameter, namely its arity. We establish the first algorithms that discover all cross joins that hold on a given relation. We illustrate in experiments with benchmark data that our algorithms perform well within the limits established by our hardness results. Our treatment of cross joins and the design of our algorithms enables us to extend our findings to the discovery of cross joins that meet a given approximation ratio. Our experiments quantify the trade-off between discovery time and targeted ratio.

Since 2005, significant progress has been made in the problem of Consistent Query Answering (CQA) with respect to primary keys. In this problem, the input is a database instance that may violate one or more primary key constraints. A repair is defined as a maximal subinstance that satisfies all primary keys. Given a Boolean query q, the question then is whether q holds true in every repair.

We define and study logics in the framework of probabilistic team semantics and over metafinite structures. Our work is paralleled by the recent development of novel axiomatizable and tractable logics in team semantics that are closed under the Boolean negation. Our logics employ new probabilistic atoms that resemble so-called extended atoms from the team semantics literature. We also define counterparts of our logics over metafinite structures and show that all of our logics can be translated into functional fixed point logic implying a polynomial time upper bound for data complexity with respect to BSS-computations.

Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additive real arithmetic can only express NP properties. Lastly, we present a sound and complete axiomatization for probabilistic inclusion logic at the atomic level.

Inclusion logic differs from many other logics of dependence and independence in that it can only describe polynomial-time properties. In this article we examine more closely connections between syntactic fragments of inclusion logic and different complexity classes. Our focus is on two computational problems: maximal subteam membership and the model checking problem for a fixed inclusion logic formula. We show that very simple quantifier-free formulae with one or two inclusion atoms generate instances of these problems that are complete for (non-deterministic) logarithmic space and polynomial time. We also present a safety game for the maximal subteam membership problem and use it to investigate this problem over teams in which one variable is a key. Furthermore, we relate our findings to consistent query answering over inclusion dependencies, and present a fragment of inclusion logic that captures non-deterministic logarithmic space in ordered models.

We introduce a novel variant of BSS machines called Separate Branching BSS machines (S-BSS in short) and develop a Fagin-type logical characterisation for languages decidable in nondeterministic polynomial time by S-BSS machines. We show that NP on S-BSS machines is strictly included in NP on BSS machines and that every NP language on S-BSS machines is a countable disjoint union of closed sets in the usual topology of Rn. Moreover, we establish that on Boolean inputs NP on S-BSS machines without real constants characterises a natural fragment of the complexity class ∃R (a class of problems polynomial time reducible to the true existential theory of the reals) and hence lies between NP and PSPACE. Finally we apply our results to determine the data complexity of probabilistic independence logic.

Abstract Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the fruitful interplay between team semantics and database dependency theory, we define Polyteam Semantics in which formulae are evaluated over a family of teams. We begin by defining a novel polyteam variant of dependence atoms and give a finite axiomatization for the associated implication problem. We relate polyteam semantics to team semantics and investigate in which cases logics over the former can be simulated by logics over the latter. We also characterize the expressive power of poly-dependence logic by properties of polyteams that are downwards closed and definable in existential second-order logic ($\textsf{ESO}$). The analogous result is shown to hold for poly-independence logic and all $\textsf{ESO}$-definable properties. We also relate poly-inclusion logic to greatest fixed point logic.

The computational properties of modal and propositional dependence logics have been extensively studied over the past few years, starting from a result by Sevenster showing NEXPTIME-completeness of the satisfiability problem for modal dependence logic. Thus far, however, the validity and entailment properties of these logics have remained mostly unaddressed. This paper provides a comprehensive classification of the complexity of validity and entailment in various modal and propositional dependence logics. The logics examined are obtained by extending the standard modal and propositional logics with notions of dependence, independence, and inclusion in the team semantics context. In particular, we address the question of the complexity of validity in modal dependence logic. By showing that it is NEXPTIME-complete we refute an earlier conjecture proposing a higher complexity for the problem.

We study the expressive power of fragments of inclusion logic under the so-called lax team semantics. The fragments are defined either by restricting the number of universal quantifiers, the number of inclusion atoms, or the arity of inclusion atoms. We show that the whole expressive power of inclusion logic can be captured using only five inclusion atoms in finite ordered models or, alternatively, only one universal quantifier in general. The arity hierarchy is shown to be strict by relating the question to the study of arity hierarchies in fixed point logics.

We classify the computational complexity of the satisfiability, validity, and model-checking problems for propositional independence, inclusion, and team logic. Our main result shows that the satisfiability and validity problems for propositional team logic are complete for alternating exponential-time with polynomially many alternations.

Large complexity classes, like the exponential time hierarchy, received little attention in terms of finding complete problems. In this work a generalization of propositional logic is investigated which fills this gap with the introduction of Boolean higher-order quantifiers or equivalently Boolean Skolem functions. This builds on the important results of Wrathall and Stockmeyer regarding complete problems, namely QBF and QBF-k, for the polynomial hierarchy. Furthermore it generalizes the Dependency QBF problem introduced by Peterson, Reif and Azhar which is complete for NEXP, the first level of the exponential hierarchy. Also it turns out that the hardness results do not collapse at the consideration of conjunctive and disjunctive normal forms, in contrast to plain QBF.

Inclusion dependencies are one of the most important database constraints. In isolation their finite and unrestricted implication problems coincide, are finitely axiomatizable, PSPACE-complete, and fixed-parameter tractable in their arity. In contrast, finite and unrestricted implication problems for the combined class of functional and inclusion de- pendencies deviate from one another and are each undecidable. The same holds true for the class of embedded multivalued dependencies. An important embedded tractable fragment of embedded multivalued dependencies are independence atoms. These stipulate independence between two attribute sets in the sense that for every two tuples there is a third tuple that agrees with the first tuple on the first attribute set and with the second tuple on the second attribute set. For independence atoms, their finite and unrestricted implication problems coincide, are finitely axiomatizable, and decidable in cubic time. In this article, we study the implication problems of the combined class of independence atoms and inclusion dependencies. We show that their finite and unrestricted implication problems coincide, are finitely axiomatizable, PSPACE-complete, and fixed-parameter tractable in their arity. Hence, significant expressivity is gained without sacrificing any of the desirable properties that inclusion dependencies have in isolation. Finally, we establish an efficient condition that is sufficient for independence atoms and inclusion dependencies not to inter- act. The condition ensures that we can apply known algorithms for deciding implication of the individual classes of independence atoms and inclusion dependencies, respectively, to decide implication for an input that combines both individual classes.

We study the expressive power of fragments of inclusion and independence logic defined by restricting the number k of universal quantifiers in formulas. Assuming the so-called strict semantics for these logics, we relate these fragments of inclusion and independence logic to sublogics ESO_f(k\forall) of existential second-order logic, which in turn are known to capture the complexity classes NTIME_{RAM}(n^k).

Uniqueness and independence are two fundamental properties of data. Their enforcement in knowledge systems can lead to higher quality data, faster data service response time, better data-driven decision making and knowledge discovery from data. The applications can be effectively unlocked by providing efficient solutions to the underlying implication problems of keys and independence atoms. Indeed, for the sole class of keys and the sole class of independence atoms the associated finite and general implication problems coincide and enjoy simple axiomatizations. However, the situation changes drastically when keys and independence atoms are combined. We show that the finite and the general implication problems are already different for keys and unary independence atoms. Furthermore, we establish a finite axiomatization for the general implication problem, and show that the finite implication problem does not enjoy a k-ary axiomatization for any k.

We study the expressive power of fragments of inclusion and independence logic defined either by restricting the number of universal quantifiers or the arity of inclusion and independence atoms in formulas. Assuming the so-called lax semantics for these logics, we relate these fragments of inclusion and independence logic to familiar sublogics of existential second-order logic. We also show that, with respect to the stronger strict semantics, inclusion logic is equivalent to existential second-order logic.