1,480 research outputs found

    The MacWilliams Identity for Krawtchouk Association Schemes

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    The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of its dual is the MacWilliams Identity, first developed for the Hamming association scheme. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via eigenvalues of the association scheme. The functional transformation form can, in particular, be used to derive important moment identities for the weight distribution of codes. In this thesis, we focus initially on extending the functional transformation to codes based on skew-symmetric and Hermitian matrices. A generalised b-algebra and new fundamental homogeneous polynomials are then identified and proven to generate the eigenvalues of a specific subclass of association schemes, Krawtchouk association schemes. Based on the new set of MacWilliams Identities as a functional transform, we derive several moments of the weight distribution for all of these codes

    Fragments and frame classes:Towards a uniform proof theory for modal fixed point logics

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    This thesis studies the proof theory of modal fixed point logics. In particular, we construct proof systems for various fragments of the modal mu-calculus, interpreted over various classes of frames. With an emphasis on uniform constructions and general results, we aim to bring the relatively underdeveloped proof theory of modal fixed point logics closer to the well-established proof theory of basic modal logic. We employ two main approaches. First, we seek to generalise existing methods for basic modal logic to accommodate fragments of the modal mu-calculus. We use this approach for obtaining Hilbert-style proof systems. Secondly, we adapt existing proof systems for the modal mu-calculus to various classes of frames. This approach yields proof systems which are non-well-founded, or cyclic.The thesis starts with an introduction and some mathematical preliminaries. In Chapter 3 we give hypersequent calculi for modal logic with the master modality, building on work by Ori Lahav. This is followed by an Intermezzo, where we present an abstract framework for cyclic proofs, in which we give sufficient conditions for establishing the bounded proof property. In Chapter 4 we generalise existing work on Hilbert-style proof systems for PDL to the level of the continuous modal mu-calculus. Chapter 5 contains a novel cyclic proof system for the alternation-free two-way modal mu-calculus. Finally, in Chapter 6, we present a cyclic proof system for Guarded Kleene Algebra with Tests and take a first step towards using it to establish the completeness of an algebraic counterpart

    The Infimum Problem as a Generalization of the Inclusion Problem for Automata

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    This thesis is concerned with automata over infinite trees. They are given a labeled infinite tree and accept or reject this tree based on its labels. A generalization of these automata with binary decisions are weighted automata. They do not just decide 'yes' or 'no', but rather compute an arbitrary value from a given algebraic structure, e.g., a semiring or a lattice. When passing from unweighted to weighted formalisms, many problems can be translated accordingly. The purpose of this work is to determine the feasibility of solving the inclusion problem for automata on infinite trees and its generalization to weighted automata, the infimum aggregation problem

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    A computation of the ninth Dedekind Number

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    In this article, we present an algorithm to compute the 9th Dedekind Number. The key aspects are the use of matrix multiplication and symmetries in the free distributive lattice, which are detected with techniques from Formal Concept Analysis.Comment: - 11 pages - isomorphic interval definition fixe

    Investigating the learning potential of the Second Quantum Revolution: development of an approach for secondary school students

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    In recent years we have witnessed important changes: the Second Quantum Revolution is in the spotlight of many countries, and it is creating a new generation of technologies. To unlock the potential of the Second Quantum Revolution, several countries have launched strategic plans and research programs that finance and set the pace of research and development of these new technologies (like the Quantum Flagship, the National Quantum Initiative Act and so on). The increasing pace of technological changes is also challenging science education and institutional systems, requiring them to help to prepare new generations of experts. This work is placed within physics education research and contributes to the challenge by developing an approach and a course about the Second Quantum Revolution. The aims are to promote quantum literacy and, in particular, to value from a cultural and educational perspective the Second Revolution. The dissertation is articulated in two parts. In the first, we unpack the Second Quantum Revolution from a cultural perspective and shed light on the main revolutionary aspects that are elevated to the rank of principles implemented in the design of a course for secondary school students, prospective and in-service teachers. The design process and the educational reconstruction of the activities are presented as well as the results of a pilot study conducted to investigate the impact of the approach on students' understanding and to gather feedback to refine and improve the instructional materials. The second part consists of the exploration of the Second Quantum Revolution as a context to introduce some basic concepts of quantum physics. We present the results of an implementation with secondary school students to investigate if and to what extent external representations could play any role to promote students’ understanding and acceptance of quantum physics as a personal reliable description of the world

    Advances and Applications of DSmT for Information Fusion. Collected Works, Volume 5

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    This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered. First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes. Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification. Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well

    Parallel Algorithms for Equilevel Predicates

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    We define a new class of predicates called equilevel predicates on a distributive lattice which eases the analysis of parallel algorithms. Many combinatorial problems such as the vertex cover problem, the bipartite matching problem, and the minimum spanning tree problem can be modeled as detecting an equilevel predicate. The problem of detecting an equilevel problem is NP-complete, but equilevel predicates with the helpful property can be detected in polynomial time in an online manner. An equilevel predicate has the helpful property with a polynomial time algorithm if the algorithm can return a nonempty set of indices such that advancing on any of them can be used to detect the predicate. Furthermore, the refined independently helpful property allows online parallel detection of such predicates in NC. When the independently helpful property holds, advancing on all the specified indices in parallel can be used to detect the predicate in polylogarithmic time. We also define a special class of equilevel predicates called solitary predicates. Unless NP = RP, this class of predicate also does not admit efficient algorithms. Earlier work has shown that solitary predicates with the efficient advancement can be detected in polynomial time. We introduce two properties called the antimonotone advancement and the efficient rejection which yield the detection of solitary predicates in NC. Finally, we identify the minimum spanning tree, the shortest path, and the conjunctive predicate detection as problems satisfying such properties, giving alternative certifications of their NC memberships as a result.Comment: To appear in ICDCN 202
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