This paper discusses and evaluates ideas of data balancing and data augmentation in the context of mathematical objects: an important topic for both the symbolic computation and satisfiability checking communities, when they are making use of machine learning techniques to optimise their tools. We consider a dataset of non-linear polynomial problems and the problem of selecting a variable ordering for cylindrical algebraic decomposition to tackle these with. By swapping the variable names in already labelled problems, we generate new problem instances that do not require any further labelling when viewing the selection as a classification problem. We find this augmentation increases the accuracy of ML models by 63% on average. We study what part of this improvement is due to the balancing of the dataset and what is achieved thanks to further increasing the size of the dataset, concluding that both have a very significant effect. We finish the paper by reflecting on how this idea could be applied in other uses of machine learning in mathematics.

Knowledge graph (KG) reasoning is an important problem for knowledge graphs. In this paper, we propose a novel and principled framework called \textbf{RulE} (stands for {Rul}e {E}mbedding) to effectively leverage logical rules to enhance KG reasoning. Unlike knowledge graph embedding (KGE) methods, RulE learns rule embeddings from existing triplets and first-order {rules} by jointly representing \textbf{entities}, \textbf{relations} and \textbf{logical rules} in a unified embedding space. Based on the learned rule embeddings, a confidence score can be calculated for each rule, reflecting its consistency with the observed triplets. This allows us to perform logical rule inference in a soft way, thus alleviating the brittleness of logic. On the other hand, RulE injects prior logical rule information into the embedding space, enriching and regularizing the entity/relation embeddings. This makes KGE alone perform better too. RulE is conceptually simple and empirically effective. We conduct extensive experiments to verify each component of RulE. Results on multiple benchmarks reveal that our model outperforms the majority of existing embedding-based and rule-based approaches.

A common feature of non-monotonic logics is that the classical notion of equivalence does not preserve the intended meaning in light of additional information. Consequently, the term strong equivalence was coined in the literature and thoroughly investigated. In the present paper, the knowledge representation formalism under consideration is claim-augmented argumentation frameworks (CAFs) which provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective. CAFs extend Dung AFs by associating a claim to each argument representing its conclusion. In this paper, we investigate both ordinary and strong equivalence in CAFs. Thereby, we take the fact into account that one might either be interested in the actual arguments or their claims only. The former point of view naturally yields an extension of strong equivalence for AFs to the claim-based setting while the latter gives rise to a novel equivalence notion which is genuine for CAFs. We tailor, examine and compare these notions and obtain a comprehensive study of this matter for CAFs. We conclude by investigating the computational complexity of naturally arising decision problems.

In imperative programming, the Domain-Driven Design methodology helps in coping with the complexity of software development by materializing in code the invariants of a domain of interest. Code is cleaner and more secure because any implicit assumption is removed in favor of invariants, thus enabling a fail fast mindset and the immediate reporting of unexpected conditions. This article introduces a notion of template for Answer Set Programming that, in addition to the don't repeat yourself principle, enforces locality of some predicates by means of a simple naming convention. Local predicates are mapped to the usual global namespace adopted by mainstream engines, using universally unique identifiers to avoid name clashes. This way, local predicates can be used to enforce invariants on the expected outcome of a template in a possibly empty context of application, independently by other rules that can be added to such a context. Template applications transpiled this way can be processed by mainstream engines and safely shared with other knowledge designers, even when they have zero knowledge of templates.

In this work, following the intuition that adverbs describing scene-sequences are best identified by reasoning over high-level concepts of object-behavior, we propose the design of a new framework that reasons over object-behaviours extracted from raw-video-clips to recognize the clip's corresponding adverb-types. Importantly, while previous works for general scene adverb-recognition assume knowledge of the clips underlying action-types, our method is directly applicable in the more general problem setting where the action-type of a video-clip is unknown. Specifically, we propose a novel pipeline that extracts human-interpretable object-behaviour-facts from raw video clips and propose novel symbolic and transformer based reasoning methods that operate over these extracted facts to identify adverb-types. Experiment results demonstrate that our proposed methods perform favourably against the previous state-of-the-art. Additionally, to support efforts in symbolic video-processing, we release two new datasets of object-behaviour-facts extracted from raw video clips - the MSR-VTT-ASP and ActivityNet-ASP datasets.

An autonomous system is constructed by a manufacturer, operates in a society subject to norms and laws, and interacts with end users. All of these actors are stakeholders affected by the behavior of the autonomous system. We address the challenge of how the ethical views of such stakeholders can be integrated in the behavior of an autonomous system. We propose an ethical recommendation component called Jiminy which uses techniques from normative systems and formal argumentation to reach moral agreements among stakeholders. A Jiminy represents the ethical views of each stakeholder by using normative systems, and has three ways of resolving moral dilemmas that involve the opinions of the stakeholders. First, the Jiminy considers how the arguments of the stakeholders relate to one another, which may already resolve the dilemma. Secondly, the Jiminy combines the normative systems of the stakeholders such that the combined expertise of the stakeholders may resolve the dilemma. Thirdly, and only if these two other methods have failed, the Jiminy uses context-sensitive rules to decide which of the stakeholders take preference over the others. At the abstract level, these three methods are characterized by adding arguments, adding attacks between arguments, and revising attacks between arguments. We show how a Jiminy can be used not only for ethical reasoning and collaborative decision-making, but also to provide explanations about ethical behavior.

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.

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.

In this paper, we propose a modal logic in which counting modalities appear in linear inequalities. We show that each formula can be transformed into an equivalent graph neural network (GNN). We also show that each GNN can be transformed into a formula. We show that the satisfiability problem is decidable. We also discuss some variants that are in PSPACE.

Epistemic logic proposed by von Wright and Hintikka (see [24, 11]) is a logical formalism for reasoning about knowledge of agents. It deals with propositional knowledge, that is, the knowledge expressed as knowing that ϕ is true. In recent years, other patterns of knowledge besides knowing that are attracting increasing attention in logic community, such as knowing whether [8, 4], knowing who [3], knowing the value [2, 6], and knowing why [28] (see a survey in [27]). Motivated by different scenarios in philosophy and AI, reasoning about knowing how assertions are particularly interesting [23]. The discussion about formalizing the notion of knowing how can date back to [16, 17]. Currently, there are two main approaches of formalizing knowing how. One of them is connecting knowing how with logics of knowing that and ability (see e.g.