We characterise the computational power of recurrent graph neural networks (GNNs) in terms of arithmetic circuits over the real numbers. Our networks are not restricted to aggregate-combine GNNs or other particular types. Generalising similar notions from the literature, we introduce the model of recurrent arithmetic circuits, which can be seen as arithmetic analogues of sequential or logical circuits. These circuits utilise so-called memory gates which are used to store data between iterations of the recurrent circuit. While (recurrent) GNNs work on labelled graphs, we construct arithmetic circuits that obtain encoded labelled graphs as real valued tuples and then compute the same function. For the other direction we construct recurrent GNNs which are able to simulate the computations of recurrent circuits. These GNNs are given the circuit-input as initial feature vectors and then, after the GNN-computation, have the circuit-output among the feature vectors of its nodes. In this way we establish an exact correspondence between the expressivity of recurrent GNNs and recurrent arithmetic circuits operating over real numbers. Our results both deepen our understanding of the capabilities of trained neural networks and open new approaches to study recurrent neural networks using the lens of circuit complexity theory.

The expressive power of Graph Neural Networks (GNNs) is often analysed via correspondence to the Weisfeiler-Leman (WL) algorithm and fragments of first-order logic. Standard GNNs are limited to performing aggregation over immediate neighbourhoods or over global read-outs. To increase their expressivity, recent attempts have been made to incorporate substructural information (e.g. cycle counts and subgraph properties). In this paper, we formalise this architectural trend by introducing Template GNNs (T-GNNs), a generalised framework where node features are updated by aggregating over valid template embeddings from a specified set of graph templates. We propose a corresponding logic, Graded template-modal logic (GML(T)), and generalised notions of template-based bisimulation and WL algorithm. We establish an equivalence between the expressive power of T-GNNs and GML(T), and provide a unifying approach for analysing GNN expressivity: we show how standard AC-GNNs and its recent variants such as AC+-GNNs can be interpreted as instantiations of T-GNNs.

Abstract Linear temporal logic (LTL) is used in system verification to write formal specifications for reactive systems. However, some relevant properties, e.g. non-inference in information flow security, cannot be expressed in LTL. A class of such properties that has recently received ample attention is known as hyperproperties. There are two major streams in the research regarding capturing hyperproperties, namely hyperlogics, which extend LTL with trace quantifiers (HyperLTL), and logics that employ team semantics, extending truth to sets of traces. In this article we explore the relation between asynchronous LTL under set-based team semantics (TeamLTL) and HyperLTL. In particular we consider the extensions of TeamLTL with the Boolean disjunction and a fragment of the extension of TeamLTL with the Boolean negation, where the negation cannot occur in the right-hand side of the strong release operator or within the global operator. We show that TeamLTL extended with the Boolean disjunction is equi-expressive with the positive Boolean closure of HyperLTL restricted to one universal quantifier, while the right-downward closed fragment of TeamLTL extended with the Boolean negation is expressively equivalent with the Boolean closure of HyperLTL restricted to one universal quantifier. Furthermore, we show that formulae of TeamLTL extended with the Boolean negation are equivalent with sentences of first-order logic interpreted over grid structures.

We introduce a general abstract framework for database repairs, where the repair notions are defined using formal logic. We distinguish between integrity constraints and so-called query constraints. The former are used to model consistency and desirable properties of the data (such as functional dependencies and independencies), while the latter relate two database instances according to their answers to the query constraints. The framework allows for a distinction between hard and soft queries, allowing the answers to a core set of queries to be preserved, as well as defining a distance between instances based on query answers. We illustrate how different repair notions from the literature can be modelled in our framework. The framework generalises both set-based and cardinality based repairs to semiring annotated databases. Finally, we initiate a complexity-theoretic analysis of consistent query answering and checking existence of a repair in our setting.

Graph Neural Networks (GNNs) are a class of machine-learning models that operate on graph-structured data. Their expressive power is intimately related to logics that are invariant under graded bisimilarity. Current proposals for recurrent GNNs either assume that the graph size is given to the model, or suffer from a lack of termination guarantees. In this paper, we propose a halting mechanism for recurrent GNNs. We prove that our halting model can express all node classifiers definable in graded modal mu-calculus, even for the standard GNN variant that is oblivious to the graph size. To prove our main result, we develop a new approximate semantics for graded mu-calculus, which we believe to be of independent interest. We leverage this new semantics into a new model-checking algorithm, called the counting algorithm, which is oblivious to the graph size. In a final step we show that the counting algorithm can be implemented on a halting recurrent GNN.

We introduce and develop a set-based semantics for asynchronous TeamLTL. We consider two canonical logics in this setting: the extensions of TeamLTL by the Boolean disjunction and by the Boolean negation. We relate the new semantics with the original semantics based on multisets and establish one of the first positive complexity theoretic results in the temporal team semantics setting. In particular we show that both logics enjoy normal forms that can be utilised to obtain results related to expressivity and complexity (decidability) of the new logics.

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.

We embark on a study of the consistent answers of queries over databases annotated with values from a naturally ordered positive semiring. In this setting, the consistent answers of a query are defined as the minimum of the semiring values that the query takes over all repairs of an inconsistent database. The main focus is on self-join free conjunctive queries and key constraints, which is the most extensively studied case of consistent query answering over standard databases. We introduce a variant of first-order logic with a limited form of negation, define suitable semiring semantics, and then establish the main result of the paper: the consistent query answers of a self-join free conjunctive query under key constraints are rewritable in this logic if and only if the attack graph of the query contains no cycles. This result generalizes an analogous result of Koutris and Wijsen for ordinary databases, but also yields new results for a multitude of semirings, including the bag semiring, the tropical semiring, and the fuzzy semiring. Further, for the bag semiring, we show that computing the consistent answers of any self-join free conjunctive query whose attack graph has a strong cycle is not only NP-hard but also it is NP-hard to even approximate the consistent answers with a constant relative approximation guarantee.

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.

Causal multiteam semantics is a framework where probabilistic dependencies arising from data and causation between variables can be together formalized and studied logically. We discover complete characterizations of expressivity for several logics that can express probabilistic statements, conditioning and interventionist counterfactuals. The results characterize the languages in terms of families of linear equations and closure conditions that define the corresponding classes of causal multiteams. The characterizations yield a strict hierarchy of expressive power. Finally, we present some undefinability results based on the characterizations.

We characterize the computational power of neural networks that follow the graph neural network (GNN) architecture, not restricted to aggregate-combine GNNs or other particular types. We establish an exact correspondence between the expressivity of GNNs using diverse activation functions and arithmetic circuits over real numbers. In our results the activation function of the network becomes a gate type in the circuit. Our result holds for families of constant depth circuits and networks, both uniformly and non-uniformly, for all common activation functions.

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.

We introduce and develop a set-based semantics for asynchronous TeamLTL. We consider two canonical logics in this setting: the extensions of TeamLTL by the Boolean disjunction and by the Boolean negation. We establish fascinating connections between the original semantics based on multisets and the new set-based semantics as well as show one of the first positive complexity theoretic results in the temporal team semantics setting. In particular we show that both logics enjoy normal forms that can be utilised to obtain results related to expressivity and complexity (decidability) of the new logics. We also relate and apply our results to recently defined logics whose asynchronicity is formalized via time evaluation functions.

In this paper, we study a novel approach to asynchronous hyperproperties by reconsidering the foundations of temporal team semantics. We consider three logics: , and , which are obtained by adding quantification over so-called time evaluation functions controlling the asynchronous progress of traces. We then relate synchronous to our new logics and show how it can be embedded into them. We show that the model checking problem for with Boolean disjunctions is highly undecidable by encoding recurrent computations of non-deterministic 2-counter machines. Finally, we present a translation from to Alternating Asynchronous Büchi Automata and obtain decidability results for the path checking problem as well as restricted variants of the model checking and satisfiability problems.

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.

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.

Abstract Propositional and modal inclusion logic are formalisms that belong to the family of logics based on team semantics. This article investigates the model checking and validity problems of these logics. We identify complexity bounds for both problems, covering both lax and strict team semantics. By doing so, we come close to finalizing the programme that aims to completely classify the complexities of the basic reasoning problems for modal and propositional dependence, independence and inclusion logics.

Second-order transitive-closure logic, SO(TC), is an expressive declarative language that captures the complexity class PSPACE. Already its monadic fragment, MSO(TC), allows the expression of various NP-hard and even PSPACE-hard problems in a natural and elegant manner. As SO(TC) offers an attractive framework for expressing properties in terms of declaratively specified computations, it is interesting to understand the expressivity of different features of the language. This paper focuses on the fragment MSO(TC), as well on the purely existential fragment SO(2TC)(exists); in 2TC, the TC operator binds only tuples of relation variables. We establish that, with respect to expressive power, SO(2TC)(exists) collapses to existential first-order logic. In addition we study the relationship of MSO(TC) to an extension of MSO(TC) with counting features (CMSO(TC)) as well as to order-invariant MSO. We show that the expressive powers of CMSO(TC) and MSO(TC) coincide. Moreover we establish that, over unary vocabularies, MSO(TC) strictly subsumes order-invariant MSO.

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.

We study the complexity of predicate logics based on team semantics. We show that the satisfiability problems of two-variable independence logic and inclusion logic are both NEXPTIME-complete. Furthermore, we show that the validity problem of two-variable dependence logic is undecidable, thereby solving an open problem from the team semantics literature. We also briefly analyse the complexity of the Bernays-Schoenfinkel-Ramsey prefix classes of dependence logic.

We give sound and complete Hilbert-style axiomatizations for propositional dependence logic (PD), modal dependence logic (MDL), and extended modal dependence logic (EMDL) by extending existing axiomatizations for propositional logic and modal logic. In addition, we give novel labeled tableau calculi for PD, MDL, and EMDL. We prove soundness, completeness and termination for each of the labeled calculi.

We introduce two variants of computation tree logic CTL based on team semantics: an asynchronous one and a synchronous one. For both variants we investigate the computational complexity of the satisfiability as well as the model checking problem. The satisfiability problem is shown to be EXPTIME-complete. Here it does not matter which of the two semantics are considered. For model checking we prove a PSPACE-completeess for the synchronous case, and show P-completeness for the asynchronous case. Furthermore we prove several interesting fundamental properties of both semantics.

This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree d receives messages through d input ports and it sends messages through d output ports, both numbered with 1,2,...,d. In this work, VVc is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of VVc:

We study the two-variable fragments D2and IF2of dependence logic and independence-friendly logic. We consider the satisfiability and finite satisfiability problems of these logics and show that for D2, both problems are NEXPTIME-complete, whereas for IF2, the problems are undecidable. We also show that D2is strictly less expressive than IF2and that already in D2, equicardinality of two unary predicates and infinity can be expressed (the latter in the presence of a constant symbol).An extended version of this publication can be found at arxiv.org.