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A Theory of Well-Connected Relations 99
INFORMA TION SCIENCES 19, 97- 134 (1979) 97 A Theory of Well-conwcted Relations SUDHIR K. ARORA and K. C. SMITH Lkpartment of Conputer Science, Uniwrsity of Toronto, Toronto, Cuna& MSS IA7 Communicated by John M. Richardson ABSTRACT The simple concept of a well-connected relation (WCR) is defined and then developed into a theory for studying the structure of binary relations. Several interesting partitions of a binary relation are identified. It is shown that binary relations can be separated into families which form a hierarchy. Finally, concepts of universal algebra are extended to the family of “prime” relations. One application for the theory is in the logical design of data bases. INTRODUCTION Data in an information system have been studied from several points of view [2, 4, 9, 10, Ill. With the growing interest in data base systems, the possibility for structuring the data in various ways has been explored by several authors. For example, Delobel [9] defines “elementary functional relations” and proposes a theory based on this building block; Zaniolo [lo] defines “atomic relations” as a basic building block for data; Furtado and Kerschberg [ 1 l] have “quotient relations.” Further, many authors have noted and studied specific data structures, such as “functional dependencies” [4], “multivalued dependencies” [5], “contextual dependencies” [6], “mutual de- pendencies” [12] (which happen to be same as contextual dependencies), and “hierarchical dependencies” [ 131. In this paper we define a new building block, “well-connected relations,” and propose a theory to study binary relations. Elsewhere [6] we have applied this theory to the study of dependency structures of data. -
Sets, Relations, Pos and Lattices
Contents 1 Discrete structures F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 1 / 23 Discrete structures Section outline Lattices (contd.) Boolean lattice 1 Discrete structures Boolean lattice structure Sets Boolean algebra Relations Additional Boolean algebra Lattices properties F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 2 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg Discrete structures Sets Sets A set A of elements: A = fa; b; cg F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 S = fXjX -
Simple Laws About Nonprominent Properties of Binary Relations
Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation arXiv:1806.05036v2 [math.LO] 20 Nov 2018 Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . 24 4.4 Asymmetry . 24 4.5 Symmetry . 25 4.6 Quasi-transitivity . 26 4.7 Anti-transitivity . 28 4.8 Incomparability-transitivity . 28 4.9 Euclideanness . 33 4.10 Density . 38 4.11 Connex and semi-connex relations . 39 4.12 Seriality . 40 4.13 Uniqueness . 42 4.14 Semi-order property 1 . 43 4.15 Semi-order property 2 . 45 5 Examples 48 6 Implementation issues 62 6.1 Improved relation enumeration . 62 6.2 Quine-McCluskey implementation . 64 6.3 On finding \nice" laws . 66 7 References 69 List of Figures 1 Source code for transitivity check . .5 2 Source code to search for right Euclidean non-transitive relations . .5 3 Timing vs. universe cardinality . -
Representing and Aggregating Conflicting Beliefs
Journal of Artificial Intelligence Research 19 (2003) 155-203 Submitted 1/03; published 9/03 Representing and Aggregating Conflicting Beliefs Pedrito Maynard-Zhang [email protected] Department of Computer Science and Systems Analysis Miami University Oxford, Ohio 45056, USA Daniel Lehmann [email protected] School of Computer Science and Engineering Hebrew University Jerusalem 91904, Israel Abstract We consider the two-fold problem of representing collective beliefs and aggregating these beliefs. We propose a novel representation for collective beliefs that uses modular, transitive relations over possible worlds. They allow us to represent conflicting opinions and they have a clear semantics, thus improving upon the quasi-transitive relations often used in social choice. We then describe a way to construct the belief state of an agent informed by a set of sources of varying degrees of reliability. This construction circumvents Arrow’s Impossibility Theorem in a satisfactory manner by accounting for the explicitly encoded conflicts. We give a simple set-theory-based operator for combining the informa- tion of multiple agents. We show that this operator satisfies the desirable invariants of idempotence, commutativity, and associativity, and, thus, is well-behaved when iterated, and we describe a computationally effective way of computing the resulting belief state. Finally, we extend our framework to incorporate voting. 1. Introduction We are interested in the multi-agent setting where agents are informed by sources of varying levels of reliability, and where agents can iteratively combine their belief states. This setting introduces three problems: (1) Finding an appropriate representation for collective beliefs; (2) Constructing an agent’s belief state by aggregating the information from informant sources, accounting for the relative reliability of these sources; and, (3) Combining the information of multiple agents in a manner that is well-behaved under iteration. -
Simple Laws About Nonprominent Properties of Binary Relations Jochen Burghardt
Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt To cite this version: Jochen Burghardt. Simple Laws about Nonprominent Properties of Binary Relations. 2018. hal- 02948115 HAL Id: hal-02948115 https://hal.archives-ouvertes.fr/hal-02948115 Preprint submitted on 24 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . -
PDF Version 4
Introduction to Mathematical Philosophy by Bertrand Russell Originally published by George Allen & Unwin, Ltd., London. May . Online Corrected Edition version . (February , ), based on the “second edition” (second printing) of April , incorporating additional corrections, marked in green. ii Introduction to Mathematical Philosophy [Russell’s blurb from the original dustcover:] CONTENTS This book is intended for those who have no previous acquain- tance with the topics of which it treats, and no more knowledge of mathematics than can be acquired at a primary school or even at Eton. It sets forth in elementary form the logical definition of number, the analysis of the notion of order, the modern doctrine Contents.............................. iii of the infinite, and the theory of descriptions and classes as sym- Preface............................... iv bolic fictions. The more controversial and uncertain aspects of the Editor’s Note............................ vi subject are subordinated to those which can by now be regarded I. The Series of Natural Numbers............ as acquired scientific knowledge. These are explained without the II. Definition of Number.................. use of symbols, but in such a way as to give readers a general un- III. Finitude and Mathematical Induction........ derstanding of the methods and purposes of mathematical logic, IV. The Definition of Order................ which, it is hoped, will be of interest not only to those who wish V. Kinds of Relations.................... to proceed to a more serious study of the subject, but also to that VI. Similarity of Relations................. wider circle who feel a desire to know the bearings of this impor- VII. Rational, Real, and Complex Numbers........ tant modern science. VIII. -
Relations and Partial Orders
“mcs-ftl” — 2010/9/8 — 0:40 — page 213 — #219 7 Relations and Partial Orders A relation is a mathematical tool for describing associations between elements of sets. Relations are widely used in computer science, especially in databases and scheduling applications. A relation can be defined across many items in many sets, but in this text, we will focus on binary relations, which represent an association between two items in one or two sets. 7.1 Binary Relations 7.1.1 Definitions and Examples Definition 7.1.1. Given sets A and B, a binary relation R A B from1 A W ! to B is a subset of A B. The sets A and B are called the domain and codomain of R, respectively. We commonly use the notation aRb or a R b to denote that .a; b/ R. 2 A relation is similar to a function. In fact, every function f A B is a rela- W ! tion. In general, the difference between a function and a relation is that a relation might associate multiple elements ofB with a single element ofA, whereas a func- tion can only associate at most one element of B (namely, f .a/) with each element a A. 2 We have already encountered examples of relations in earlier chapters. For ex- ample, in Section 5.2, we talked about a relation between the set of men and the set of women where mRw if man m likes woman w. In Section 5.3, we talked about a relation on the set of MIT courses where c1Rc2 if the exams for c1 and c2 cannot be given at the same time. -
Three Axiom Systems for Additive Semi-Ordered Structures
SIAM J. APPL.MATH. Vol. 25, No. 1, July 1973 THREE AXIOM SYSTEMS FOR ADDITIVE SEMIORDERED STRUCTURES* R. DUNCAN LUCEi Abstract. Axioms are provided for extensive, probability, and (two-component, additive) conjoint structures which are semiordered, rather than weakly ordered. These are sufficient to construct the usual representation for the natural, induced weak order and a tight, threshold representation for the semiorder itself. The fundamental difficulty is in finding axioms that are simple in the primitive semi- order, rather than simple in its induced weak order; from this point of view, the results are only partially successful. 1. Introduction. A theory of measurement is any axiomatization of an ordered algebraic structure that, on the one hand, has at least one model with a plausible empirical interpretation and, on the other hand, each model is homomorphic to some numerical structure. The most familiar examples are from physics and are of three types. First are the extensive structures, with interpretations as length, mass, and the like, in which a set of objects is ordered by 2 (a 2 b means a has at least as much length, or mass, etc. as b) and has a binary operation o (a o b means some relevant combination of a and b). In the numerical representation, 2 maps to 2 and 0 to +. Second are simple conjoint structures, with interpretations as momentum, kinetic energy, and the like, in which a Cartesian product is ordered by 2. In the numerical representation, 2 is mapped into 2 and the product structure into products of numbers. Third are probabilistic structures in which an algebra of sets (events) is ordered by 2. -
Topological Representation of Precontact Algebras and A
Topological Representation of Precontact Algebras and a Connected Version of the Stone Duality Theorem – I Georgi Dimov∗ and Dimiter Vakarelov Department of Mathematics and Informatics, University of Sofia, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria Abstract The notions of a 2-precontact space and a 2-contact space are introduced. Using them, new representation theorems for precontact and contact algebras are proved. They incorporate and strengthen both the discrete and topological representation theorems from [8, 5, 6, 9, 24]. It is shown that there are bijec- tive correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems [22, 19] for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem (in the sense of [20]) for a kind of arXiv:1508.02220v4 [math.GN] 20 Nov 2015 compact T0-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion of a Stone adjacency space and using it, we prove another representation theorem for precontact algebras. We even obtain a bi- jective correspondence between the class of all, up to isomorphism, precontact algebras and the class of all, up to isomorphism, Stone adjacency spaces. ∗The first (resp., the second) author of this paper was supported by the contract no. 7/2015 “Contact algebras and extensions of topological spaces” (resp., contract no. 5/2015) of the Sofia University Science Fund. 1Keywords: (pre)contact algebra, 2-(pre)contact space, (extremally disconnected) Stone space, Stone 2-space, (extremally) connected space, (Stone) duality, C-semiregular spaces (extensions), (complete) Boolean algebra, Stone adjacency space, (closed) relations, continuous extension of maps. -
Numerical Representations of Binary Relations with Thresholds: a Brief Survey 1
Numerical representations of binary relations with thresholds: A brief survey 1 Fuad Aleskerov Denis Bouyssou Bernard Monjardet 11 July 2006, Revised 8 January 2007 Typos corrected 1 March 2008 Additional typos corrected 9 October 2008 More typos corrected: 13 February 2009 1 This text is based on a chapter of a book in preparation co-authored with Fuad Aleskerov and Bernard Monjardet (that will be the second edition of Aleskerov and Monjardet, 2002). Abstract This purpose of this text is to present in a self-contained way a number of classical results on the numerical representation of binary relations on arbitrary sets involving a threshold. We tackle the case of linear orders, weak orders, biorders, interval orders, semiorders and acyclic relations. Keywords : utility, threshold, biorder, interval order, semiorder, acyclic relation. Contents 1 Introduction 1 1.1 Overview . 1 1.2 Definitions and notation . 2 2 Linear orders 4 3 Weak orders 9 3.1 Preliminaries . 9 3.2 Numerical representation . 10 4 Biorders 12 4.1 Preliminaries . 12 4.2 The countable case . 15 4.3 The general case . 17 5 Interval orders 22 5.1 The countable case . 22 5.2 The general case . 23 6 Semiorders 24 6.1 Preliminaries . 26 6.2 Representations with no proper nesting . 27 6.3 Representations with no nesting . 30 6.4 Constant threshold representations . 32 7 Acyclic relations 33 7.1 Preliminaries . 33 7.2 The countable case . 35 7.3 The general case . 36 8 Concluding remarks and guide to the literature 37 8.1 Linear orders . 37 8.2 Weak orders . -
Model Based Horn Contraction
Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning Model Based Horn Contraction Zhiqiang Zhuang and Maurice Pagnucco National ICT Australia and ARC Centre of Excellence in Autonomous Systems School of Computer Science and Engineering, The University of New South Wales Sydney, NSW 2052, Australia Email: fzqzhuang—[email protected] Abstract 2010; Delgrande and Wassermann 2010; Zhuang and Pag- nucco 2010a; 2010b; 2011; Delgrande and Peppas 2011). Following the recent trend of adapting the AGM (Alchourron´ The topic is interesting for several reasons. Horn logic is an and Makinson 1985) framework to propositional Horn logic, important subset of propositional logic which has found use Delgrande and Peppas (Delgrande and Peppas 2011) give a in many artificial intelligence and database applications. The model theoretic account for revision in the Horn logic set- ting. The current paper complements their work by studying study of belief change under Horn logic broadens the practi- the model theoretic approach for contraction. A model based cal applicability of the AGM framework and in particular it Horn contraction is constructed and shown to give a model provides a key step towards applying the AGM framework theoretic account to the transitively relational partial meet to non-classical logics with less expressive and reasoning Horn contraction studied in (Zhuang and Pagnucco 2011). power than propositional logic. Significantly however, in contrast to (Delgrande and Pep- The classic construction for AGM contraction is based pas 2011), our model-based characterisation of Horn contrac- on the notion of remainder sets. Remainder sets of a belief tion does not require the property of Horn compliance and set K with respect to a sentence φ are the maximal1 sub- totality over preorders. -
The Mathematical Landscape a Guided Tour
Claremont Colleges Scholarship @ Claremont CMC Senior Theses CMC Student Scholarship 2011 The aM thematical Landscape Antonio Collazo Claremont McKenna College Recommended Citation Collazo, Antonio, "The aM thematical Landscape" (2011). CMC Senior Theses. Paper 116. http://scholarship.claremont.edu/cmc_theses/116 This Open Access Senior Thesis is brought to you by Scholarship@Claremont. It has been accepted for inclusion in this collection by an authorized administrator. For more information, please contact [email protected]. Claremont McKenna College The Mathematical Landscape A guided tour Submitted To Professor Asuman G. Aksoy and Dean Gregory Hess By Antonio Collazo For Senior Thesis Spring 2011 April 25, 2011 Abstract The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject. By no means, is the following a complete formulation of any of the topics covered. I want to give the reader a tour of the mathematical landscape, hoping maybe to see them again. There is plenty of further details to explore in each section, I have just touched the tip the iceberg. The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building block of Math. The first sections are a exposition into the mathematical objects and their algebras. The last section dives into the foundation of math, sets and logic, and develops the \language" of Math. My hope is that after this, the read will have the necessarily (maybe not sufficient) information needed to talk the language of Math. Antonio Collazo What is Mathematics? Take a random sample from our population, and ask them \What is Mathematics?" Their answer, most likely, is that Mathematics is about numbers; some will add that Mathematics is the study of numbers and their arithmetic.