The paper introduces recurrent Graph Neural Networks (GNNs) that use set-based aggregation and establishes conditions that can be verified directly from the network weights. It proves a two‑directional equivalence between these networks and the Boolean closure of reachability and safety properties, corresponding to the fragment BΣ◦₁ of the modal μ‑calculus. This equivalence allows for verifiable symbolic explanations of networks that satisfy the identified conditions, without relying on counting logic or external halting signals.
By Blai Bonet
arXiv:2606. 17882v1 Announce Type: new Abstract: Bridges between graph neural networks (GNNs) and logical formalisms have been established by fixing architectural choices, such as the types of aggregation, combination, and activation functions.
By Przemys{\l}aw Andrzej Wa{\l}\k{e}ga, Bernardo Cuenca Grau
arXiv:2603. 14846v3 Announce Type: replace Abstract: We define an information-complexity property for aggregation functions, capturing a vast range of practical aggregations, and prove that any Message-Passing Graph Neural Network (MP-GNN) model with such aggregations induces only a polynomial number of equivalence classes on all graphs - while the number of non-isomorphic graphs is super-exponential (in number of vertices).
By Eran Rosenbluth
The paper introduces a general theoretical framework for fibrations on graphs labeled by a commutative monoid, extending the classic theory of graph fibrations to weighted and algebraically labeled graphs. It also accommodates approximate fibrations and demonstrates how this framework can be used to compress arbitrary neural networks, including CNNs, providing a solid theoretical basis for recent findings on fibration symmetries in geometric deep learning.
By Paolo Boldi
The paper presents a complete characterization of when two deep ReLU networks realize the same function, showing that this occurs iff one can be transformed into the other using a set of axioms from many‑valued logic. It introduces a symbolic calculus that maps networks to substitution graphs, proves a completeness theorem linking equivalent formulas, and provides an algorithm to reconstruct networks from these graphs. The framework yields a new compositional normal form for MV logic that preserves the algebraic structure of deep ReLU networks.
By Yani Zhang, Helmut B\"olcskei
arXiv:2601. 23207v2 Announce Type: replace-cross Abstract: Understanding what graph neural networks can learn, especially their ability to learn to execute algorithms, remains a central theoretical challenge.
By Muhammad Fetrat Qharabagh, Artur Back de Luca, George Giapitzakis, Kimon Fountoulakis
arXiv:2608. 11136v1 Announce Type: new Abstract: Logic Tensor Networks (LTN) provide a neurosymbolic framework in which first-order logic is interpreted through tensor operations, enabling logical constraints to be integrated with differentiable learning.
By Davide Rinaldi, Luciano Serafini
arXiv:2609.08961v1 Announce Type: cross
Abstract: For a finite set $O$ of Boolean functions, we consider the class of propositional formulas built using the functions in $O$ as connectives. We determ...
By Balder ten Cate
The paper applies parameterised graph theory to tensor networks, showing that cutwidth and tree‑cutwidth bound the bond‑dimension overhead needed to represent a tensor‑network state as a matrix product state or tree tensor network. It derives graph‑dependent upper bounds on the sample and computational complexity of tensor‑network tomography, introducing a new graph parameter called learning complexity. Finally, it extends the framework to an agnostic learner that approximates any state with a tensor‑network state of given bond dimension, providing explicit graph‑dependent complexity bounds.
By Matthias C. Caro, Natalie McHugh, Sergii Strelchuk
The paper introduces GUT, a method that uses directed acyclic graphs to represent all possible reasoning branches of Large Language Models (LLMs). It comprises two modules: GUT-Q, which quantifies reasoning uncertainty by approximating graph complexity, and GUT-O, which reduces uncertainty through reinforcement learning that rewards lower uncertainty. Experiments on four LLMs across five datasets demonstrate GUT’s effectiveness in measuring and mitigating reasoning uncertainty.
By Shuang Liang, Xin-Yu Hu, Xiang-Jun Ou, Shao-Qun Zhang
arXiv:2608.25598v2 Announce Type: replace
Abstract: The purpose of this paper is to provide a general, comprehensive, theoretical framework that allows one to deal with fibrations on graphs labelled...
By Paolo Boldi, Osvaldo M. Velarde, Hernan A. Makse
arXiv:2608. 11716v1 Announce Type: new Abstract: Chain of Thought (CoT) lifts the expressive ceiling of bounded-depth Transformers, with characterizations tying the number of CoT steps to circuit complexity classes.
By Debanjan Dutta, Anish Chakrabarty, Swagatam Das