Measuring Inconsistency in Argument Graphs

There have been a number of developments in measuring inconsistency in logic-based representations of knowledge. In contrast, the development of inconsistency measures for computational models of argument has been limited. To address this shortcoming, this paper provides a general framework for measuring inconsistency in abstract argumentation, together with some proposals for specific measures, and a consideration of measuring inconsistency in logic-based instantiations of argument graphs, including a review of some existing proposals and a consideration of how existing logic-based measures of inconsistency can be applied.Comment: 29 page

The core Hopf algebra

We study the core Hopf algebra underlying the renormalization Hopf algebra.Comment: 9p, contributed to the Proceedings for Alain Connes' 60th birthda

Connectivity for matroids based on rough sets

In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems. The connectivity of a matroid is an important measure of its robustness as a network. Therefore, it is very necessary to investigate the conditions under which a matroid is connected. In this paper, the connectivity for matroids is studied through relation-based rough sets. First, a symmetric and transitive relation is introduced from a general matroid and its properties are explored from the viewpoint of matroids. Moreover, through the relation introduced by a general matroid, an undirected graph is generalized. Specifically, the connection of the graph can be investigated by the relation-based rough sets. Second, we study the connectivity for matroids by means of relation-based rough sets and some conditions under which a general matroid is connected are presented. Finally, it is easy to prove that the connectivity for a general matroid with some special properties and its induced undirected graph is equivalent. These results show an important application of relation-based rough sets to matroids.Comment: 16 pages, 8figure

Knowledge Extraction and Knowledge Integration governed by {\L}ukasiewicz Logics

The development of machine learning in particular and artificial intelligent in general has been strongly conditioned by the lack of an appropriate interface layer between deduction, abduction and induction. In this work we extend traditional algebraic specification methods in this direction. Here we assume that such interface for AI emerges from an adequate Neural-Symbolic integration. This integration is made for universe of discourse described on a Topos governed by a many-valued {\L}ukasiewicz logic. Sentences are integrated in a symbolic knowledge base describing the problem domain, codified using a graphic-based language, wherein every logic connective is defined by a neuron in an artificial network. This allows the integration of first-order formulas into a network architecture as background knowledge, and simplifies symbolic rule extraction from trained networks. For the train of such neural networks we changed the Levenderg-Marquardt algorithm, restricting the knowledge dissemination in the network structure using soft crystallization. This procedure reduces neural network plasticity without drastically damaging the learning performance, allowing the emergence of symbolic patterns. This makes the descriptive power of produced neural networks similar to the descriptive power of {\L}ukasiewicz logic language, reducing the information lost on translation between symbolic and connectionist structures. We tested this method on the extraction of knowledge from specified structures. For it, we present the notion of fuzzy state automata, and we use automata behaviour to infer its structure. We use this type of automata on the generation of models for relations specified as symbolic background knowledge.Comment: 38 page

On the maximum number of integer colourings with forbidden monochromatic sums

Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{\lceil n/2\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leq 5$. Similar results were obtained by H\`an and Jim\'enez in the setting of finite abelian groups.Comment: 24 pages + 3 page appendi

Diamond Twin

As noticed in 2006 by the author of the present article, the hypothetical crystal---described by crystallographer F. Laves (1932) for the first time and designated ``Laves' graph of girth ten" by geometer H. S. M. Coxeter (1955)---is a unique crystal net sharing a remarkable symmetric property with the diamond crystal, thus deserving to be called the diamond twin although their shapes look quite a bit different at first sight. In this short note, we shall provide an interesting mutual relationship between them, expressed in terms of ``building blocks" and ``period lattices." This may give further justification to employ the word ``twin." What is more, our discussion brings us to the notion of ``orthogonally symmetric lattice," a generalization of irreducible root lattices, which makes the diamond and its twin very distinct among all crystal structures.Comment: 22 pages, 17 figure

Physically-interpretable classification of biological network dynamics for complex collective motions

Understanding biological network dynamics is a fundamental issue in various scientific and engineering fields. Network theory is capable of revealing the relationship between elements and their propagation; however, for complex collective motions, the network properties often transiently and complexly change. A fundamental question addressed here pertains to the classification of collective motion network based on physically-interpretable dynamical properties. Here we apply a data-driven spectral analysis called graph dynamic mode decomposition, which obtains the dynamical properties for collective motion classification. Using a ballgame as an example, we classified the strategic collective motions in different global behaviours and discovered that, in addition to the physical properties, the contextual node information was critical for classification. Furthermore, we discovered the label-specific stronger spectra in the relationship among the nearest agents, providing physical and semantic interpretations. Our approach contributes to the understanding of principles of biological complex network dynamics from the perspective of nonlinear dynamical systems.Comment: 42 pages with 7 figures and 3 tables. The latest version is published in Scientific Reports, 202

A local characterization of B2B_2 regular crystals

Stembridge characterized regular crystals associated with a simply-laced generalized Cartan matrix (GCM) in terms of local graph-theoretic quantities. We give a similar axiomatization for $B_2$ regular crystals and thus for regular crystals associated with a finite GCM except $G_2$ and an affine GCM except $A^{(1)}_{1},G^{(1)}_{2},A^{(2)}_{2},D^{(3)}_4$.Comment: 15 pages, submitted to a journal as a revision in October 201

Causal Discovery in the Presence of Measurement Error: Identifiability Conditions

Measurement error in the observed values of the variables can greatly change the output of various causal discovery methods. This problem has received much attention in multiple fields, but it is not clear to what extent the causal model for the measurement-error-free variables can be identified in the presence of measurement error with unknown variance. In this paper, we study precise sufficient identifiability conditions for the measurement-error-free causal model and show what information of the causal model can be recovered from observed data. In particular, we present two different sets of identifiability conditions, based on the second-order statistics and higher-order statistics of the data, respectively. The former was inspired by the relationship between the generating model of the measurement-error-contaminated data and the factor analysis model, and the latter makes use of the identifiability result of the over-complete independent component analysis problem.Comment: 15 pages, 5 figures, 1 tabl

Higher melonic theories

We classify a large set of melonic theories with arbitrary $q$-fold interactions, demonstrating that the interaction vertices exhibit a range of symmetries, always of the form $\mathbb{Z}_2^n$ for some $n$, which may be $0$. The number of different theories proliferates quickly as $q$ increases above $8$ and is related to the problem of counting one-factorizations of complete graphs. The symmetries of the interaction vertex lead to an effective interaction strength that enters into the Schwinger-Dyson equation for the two-point function as well as the kernel used for constructing higher-point functions.Comment: 43 pages, 12 figure