The design of voting rules is traditionally guided by desirable axioms.

Gradients play a pivotal role in neural networks explanation. The inherent high dimensionality and structural complexity of neural networks result in the original gradients containing a significant amount of noise. While several approaches were proposed to reduce noise with smoothing, there is little discussion of the rationale behind smoothing gradients in neural networks. In this work, we proposed a gradient smooth theoretical framework for neural networks based on the function mollification and Monte Carlo integration. The framework intrinsically axiomatized gradient smoothing and reveals the rationale of existing methods. Furthermore, we provided an approach to design new smooth methods derived from the framework. By experimental measurement of several newly designed smooth methods, we demonstrated the research potential of our framework.

Ibeling et al. (2023). axiomatize increasingly expressive languages of causation and probability, and Mosse et al. (2024) show that reasoning (specifically the satisfiability problem) in each causal language is as difficult, from a computational complexity perspective, as reasoning in its merely probabilistic or "correlational" counterpart. Introducing a summation operator to capture common devices that appear in applications -- such as the $do$-calculus of Pearl (2009) for causal inference, which makes ample use of marginalization -- van der Zander et al. (2023) partially extend these earlier complexity results to causal and probabilistic languages with marginalization. We complete this extension, fully characterizing the complexity of probabilistic and causal reasoning with summation, demonstrating that these again remain equally difficult. Surprisingly, allowing free variables for random variable values results in a system that is undecidable, so long as the ranges of these random variables are unrestricted. We finally axiomatize these languages featuring marginalization (or more generally summation), resolving open questions posed by Ibeling et al. (2023).

Natural language explanations have become a proxy for evaluating explainable and multi-step Natural Language Inference (NLI) models. However, assessing the validity of explanations for NLI is challenging as it typically involves the crowd-sourcing of apposite datasets, a process that is time-consuming and prone to logical errors. To address existing limitations, this paper investigates the verification and refinement of natural language explanations through the integration of Large Language Models (LLMs) and Theorem Provers (TPs). Specifically, we present a neuro-symbolic framework, named Explanation-Refiner, that augments a TP with LLMs to generate and formalise explanatory sentences and suggest potential inference strategies for NLI. In turn, the TP is employed to provide formal guarantees on the logical validity of the explanations and to generate feedback for subsequent improvements. We demonstrate how Explanation-Refiner can be jointly used to evaluate explanatory reasoning, autoformalisation, and error correction mechanisms of state-of-the-art LLMs as well as to automatically enhance the quality of human-annotated explanations of variable complexity in different domains.

Causal intervention is an essential tool in causal inference. It is axiomatized under the rules of do-calculus in the case of structure causal models. We provide simple axiomatizations for families of probability distributions to be different types of interventional distributions. Our axiomatizations neatly lead to a simple and clear theory of causality that has several advantages: it does not need to make use of any modeling assumptions such as those imposed by structural causal models; it only relies on interventions on single variables; it includes most cases with latent variables and causal cycles; and more importantly, it does not assume the existence of an underlying true causal graph as we do not take it as the primitive object--in fact, a causal graph is derived as a by-product of our theory. We show that, under our axiomatizations, the intervened distributions are Markovian to the defined intervened causal graphs, and an observed joint probability distribution is Markovian to the obtained causal graph; these results are consistent with the case of structural causal models, and as a result, the existing theory of causal inference applies. We also show that a large class of natural structural causal models satisfy the theory presented here. We note that the aim of this paper is axiomatization of interventional families, which is subtly different from "causal modeling."

We propose communication pattern logic. A communication pattern describes how processes or agents inform each other, independently of the information content. The full-information protocol in distributed computing is the special case wherein all agents inform each other. We study this protocol in distributed computing models where communication might fail: an agent is certain about the messages it receives, but it may be uncertain about the messages other agents have received. In a dynamic epistemic logic with distributed knowledge and with modalities for communication patterns, the latter are interpreted by updating Kripke models. We propose an axiomatization of communication pattern logic, and we show that collective bisimilarity (comparing models on their distributed knowledge) is preserved when updating models with communication patterns. We can also interpret communication patterns by updating simplicial complexes, a well-known topological framework for distributed computing. We show that the different semantics correspond, and propose collective bisimulation between simplicial complexes.

The Metadata Object Description Schema (MODS) was developed to describe bibliographic concepts and metadata and is maintained by the Library of Congress. Its authoritative version is given as an XML schema based on an XML mindset which means that it has significant limitations for use in a knowledge graphs context. We have therefore developed the Modular MODS Ontology (MMODS-O) which incorporates all elements and attributes of the MODS XML schema. In designing the ontology, we adopt the recent Modular Ontology Design Methodology (MOMo) with the intention to strike a balance between modularity and quality ontology design on the one hand, and conservative backward compatibility with MODS on the other.

Causality is crucial in human reasoning and knowledge. Defining and formalizing causality has been a significant area of research in philosophy and formal methods [12, 21, 24, 11]. In recent years, with the rise of machine learning and AI, there has been growing interest in formalizing causal reasoning. One of the key areas of AI research is designing algorithms capable of comprehending causal information and performing causal reasoning [5, 29, 30]. Causal reasoning can be instrumental in formally modeling notions such as responsibility, blame, harm, and explanation, which are important aspects in designing ethical and responsible AI systems [3]. In this article we focus on the kind of causality known as "actual causality" (a.k.a.

Epistemic logics model how agents reason about their beliefs and the beliefs of other agents. Existing logics typically assume the ability of agents to reason perfectly about propositions of unbounded modal depth. We present DBEL, an extension of S5 that models agents that can reason about epistemic formulas only up to a specific modal depth. To support explicit reasoning about agent depths, DBEL includes depth atoms Ead (agent a has depth exactly d) and Pad (agent a has depth at least d). We provide a sound and complete axiomatization of DBEL. We extend DBEL to support public announcements for bounded depth agents and show how the resulting DPAL logic generalizes standard axioms from public announcement logic. We present two alternate extensions and identify two undesirable properties, amnesia and knowledge leakage, that these extensions have but DPAL does not. We provide axiomatizations of these logics as well as complexity results for satisfiability and model checking. Finally, we use these logics to illustrate how agents with bounded modal depth reason in the classical muddy children problem, including upper and lower bounds on the depth knowledge necessary for agents to successfully solve the problem.