DOCSLIB.ORG

Chain Partitions in Ordered Sets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chain Partitions in Ordered Sets CHAPTER 1 Chain Partitions in Ordered Sets Editors' note: The background for this chapter is based on a lecture in­ tended for undergraduates. Background R. P. DILWORTH Let us consider the sets which can be made up from the numbers 1, 2, and 3. We have the singleton sets {1}, {2}, and {3} made up of one number each. Then there are the two-element sets {1,2}, {l, 3}, and {2,3}. Finally, there is the three­ element set {l, 2, 3}. It is customary to include the empty set 0 which has no numbers belonging to it. There is a natural relation between these sets, namely, {1} is a subset of {l, 2} and {l, 3} is a subset of {l, 2, 3}. If A and B denote two of these sets, we write A = B if A and B consist of the same numbers. Thus {1, 2} = {2,1}. If every member of A is also a member of B we write A ~ B and if, in addition, there are members of B which do not belong to A we write A < B. Thus {1, 2} < {1, 2, 3}. It is convenient to diagram the containing relations existing between these sets as shown in Figure 1. Each set is represented by a small circle in the plane. Larger sets lie above smaller sets and the circles representing two sets are joined by a line if one contains the other and there is no set lying strictly between the two. The subsets of a set also have an algebraic structure. If A and B are two sets, A V B (A join B) will denote the set made up of the objects of A together with the objects of B. A A B (A meet B) will denote the set of objects belonging to both A and B. These operations are commutative, associative, distributive and idempotent (A V A = A A A = A). There is also a unary operation of complementation. A' consists of the objects in the set which do not belong to A. In the example above, 1 o {1,2,3} {1,2} O~O~O {2,3} {I} oXO~O {3} ~/ Figure 1 we have {1,2} V {3} = {1, 2, 3} {1, 2} A {2, 3} = {2} {1,3}' = {2}. The subsets of the set {1, 2, 3} under the operations of join, meet, and complemen­ tation provide a simple example of a Boolean algebra-named after George Boole­ who studied such systems in the 1800's in connection with his work in symbolic logic. The structure of a finite Boolean algebra is completely determined once the number of atoms (singleton sets) is known, namely each member of Boolean algebra is the unique join of the singleton sets contained in it. Now a singleton set together with the empty set 0 make up a two-element Boolean algebra whose diagram is shown in Figure 2. Thus the basic structure result for finite Boolean algebras states that a finite Boolean algebra is a direct product of n two-element Boolean algebras where n is the number of atoms. Figure 2 An important generalization of a Boolean algebra is a distributive lattice which has the join and meet operations with the same properties as in the case of Boolean algebras but without the complementation operation. For example, the sets 0, {l}, {2}, {l,2}, {2,3}, {l,2,3} form a distributive lattice having the diagram shown in Figure 3. It is no longer true that every member of the lattice is a join of atoms, but it is true that every member is a join of join irreducibles. A join irreducible is an element of the lattice which cannot be represented as a join of elements distinct 2 THE DILWORTH THEOREMS o {1,2,3} {1,2} /~. {2,3} {l}./~.' ~.( Figure 3 from itself. In the example, 0, {I}, {2}, and {2,3} are join irreducibles. Although {I, 2, 3} can be represented as the join of {I}, {2}, and {2,3} and also as the join of {I} and {2,3} the latter representation which is minimal is unique. Indeed, in an abstract finite distributive lattice, the minimal representations as joins of join irreducibles are unique and associating each element of the lattice with the set of join irreducibles less than or equal to it gives a representation of the lattice as a lattice of sets of join irreducibles. Conversely, if a finite ordered set is given, the subsets of the ordered set which are such that with each element in the subset also all elements less than or equal to it also belong to the subset form a distributive lattice. For example, consider the ordered set diagrammed in Figure 4. boo d a./1/ Figure 4 The sets 0, {a}, {c}, {c,d}, {a,b,c}, {a,c,d}, {b,c,d}, {a,b,c,d} form a dis­ tributive lattice diagrammed in Figure 5. The join irreducibles {a}, {a,b,c}, {c}, {c, d} correspond to the elements a, b, c, d in the ordered set. The correspondence between finite ordered sets and finite distributive lattices is one-to-one. Thus, as is frequently the case, questions concerning the structure of distributive lattices can be formulated in terms of questions concerning ordered sets. We also note from this correspondence that any representation of a finite distributive lattice as a lattice of subsets of some set requires that the set have as many members as there are join irreducibles in the distributive lattice. In other terms, the minimum number of two-element Boolean algebras whose direct product has a sublattice isomorphic to a given finite distributive lattice is the number of join irreducibles in the lattice. When I completed my graduate work at Caltech and went to Yale University Chain Partitions in Ordered Sets 3 Figure 5 on a Sterling Research Fellowship, one of the most interesting problems in lattice theory at that time was the conjecture that a lattice in which every element had a unique complement was a Boolean algebra. The conjecture had been verified in several instances where one of a number of additional restrictions was imposed, namely modularity, orthocomplementation, and atomicity. The general conjecture was still open. Since the fellowship provided me with a full year in which to do research, I decided to concentrate on the conjecture. However, since it is difficult to work full time on one problem, I decided to fill the breaks by working on a problem concerning the representation of a distributive lattice as a sublattice of a direct product of totally ordered sets, i.e., chains. Since it was easy to determine the least number of two-element chains whose direct product could contain the given distributive lattice as a sublattice, it seemed natural to ask for the least number of chains of arbitrary length whose direct product could contain the given distributive lattice as a sublattice. Making use of the correspondence between finite distributive lattices and finite ordered sets it was quite straightforward to verify that imbedding a finite distributive lattice in a direct product of chains was equivalent to representing a finite ordered set as a set join of chains. One useful observation which turned up early in the investigation was the fact that the number of elements covering a given element in the distributive lattice was always a lower bound for the number of chains needed for a representation, since each covering element had to belong to a different chain. Again, making use of the correspondence between distributive lattices and ordered sets and examining many examples it became clear that the maximal number of elements covering an element in the distributive lattice was equal to the maximum number of non-comparable elements in the corresponding ordered set, i.e., to the maximal number of elements in an antichain of the ordered set. It was also clear that the maximal number of elements in an antichain was a lower bound to the minimum number of chains needed to represent the ordered set as a set join of chains. Thus the problem would be solved if it could be shown that these two numbers were 4 THE Oll...WORTH THEOREMS indeed equal. If n is the maximal number of elements in an anti chain of a finite ordered set, the most natural attack on the problem would be to take a maximal chain out of the ordered set and hope that the maximum size of an antichain in what is left is n - 1 or less. However, this doesn't work. In Figure 4, the maximum size of an anti chain is 2. The elements band c make up a maximal chain. The elements a and d which are left are an antichain of size 2. Obviously, a good place to begin would be the case n = 2, since for n = 1, the ordered set is already a chain. By starting at the bottom of the ordered set and making an inductive argument it was easy to see that the ordered set is a set join of two chains when n = 2. This special technique fails for n ~ 3 so a new approach was required. Concentrating on n = 3, I tried removing maximal chains having various special properties with the hope that the remaining elements would form an ordered set with n = 2. Although this approach worked for ordered sets of relatively small orders, the required properties became more and more complicated as the size of the ordered set increased. After several months I abandoned this approach and turned to trying inductive arguments.
Load more

Recommended publications
  • On Scattered Convex Geometries
    ON SCATTERED CONVEX GEOMETRIES KIRA ADARICHEVA AND MAURICE POUZET Abstract. A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scat- tered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice Ω(η), that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scat- teredness is established in convex geometries of relatively convex sets. 1. Introduction We call a pair X; φ of a non-empty set X and a closure operator φ 2X 2X on X a convex geometry[6], if it is a zero-closed space (i.e. ) and φ satisfies the anti-exchange axiom: ( ) ∶ → x A y and x∅ =A∅imply that y A x for all x y in X and all closed A X: ∈ ∪ { } ∉ ∉ ∪ { } The study of convex geometries in finite≠ case was inspired by their⊆ frequent appearance in mod- eling various discrete structures, as well as by their juxtaposition to matroids, see [20, 21]. More recently, there was a number of publications, see, for example, [4, 43, 44, 45, 48, 7] brought up by studies in infinite convex geometries. A convex geometry is called algebraic, if the closure operator φ is finitary. Most of interesting infinite convex geometries are algebraic, such as convex geometries of relatively convex sets, sub- semilattices of a semilattice, suborders of a partial order or convex subsets of a partially ordered set.
    [Show full text]
  • Advanced Discrete Mathematics Mm-504 &
    1 ADVANCED DISCRETE MATHEMATICS M.A./M.Sc. Mathematics (Final) MM-504 & 505 (Option-P3) Directorate of Distance Education Maharshi Dayanand University ROHTAK – 124 001 2 Copyright © 2004, Maharshi Dayanand University, ROHTAK All Rights Reserved. No part of this publication may be reproduced or stored in a retrieval system or transmitted in any form or by any means; electronic, mechanical, photocopying, recording or otherwise, without the written permission of the copyright holder. Maharshi Dayanand University ROHTAK – 124 001 Developed & Produced by EXCEL BOOKS PVT. LTD., A-45 Naraina, Phase 1, New Delhi-110 028 3 Contents UNIT 1: Logic, Semigroups & Monoids and Lattices 5 Part A: Logic Part B: Semigroups & Monoids Part C: Lattices UNIT 2: Boolean Algebra 84 UNIT 3: Graph Theory 119 UNIT 4: Computability Theory 202 UNIT 5: Languages and Grammars 231 4 M.A./M.Sc. Mathematics (Final) ADVANCED DISCRETE MATHEMATICS MM- 504 & 505 (P3) Max. Marks : 100 Time : 3 Hours Note: Question paper will consist of three sections. Section I consisting of one question with ten parts covering whole of the syllabus of 2 marks each shall be compulsory. From Section II, 10 questions to be set selecting two questions from each unit. The candidate will be required to attempt any seven questions each of five marks. Section III, five questions to be set, one from each unit. The candidate will be required to attempt any three questions each of fifteen marks. Unit I Formal Logic: Statement, Symbolic representation, totologies, quantifiers, pradicates and validity, propositional logic. Semigroups and Monoids: Definitions and examples of semigroups and monoids (including those pertaining to concentration operations).
    [Show full text]
  • Completely Representable Lattices
    Completely representable lattices Robert Egrot and Robin Hirsch Abstract It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be jCRL ∩ mCRL. We prove CRL ⊂ biCRL = mCRL ∩ jCRL ⊂ mCRL =6 jCRL ⊂ DL where the marked inclusions are proper. Let L be a DL. Then L ∈ mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L ∈ jCRL iff L has a distinguishing set of completely prime filters, and L ∈ CRL iff L has a distinguishing set of complete, completely prime filters. Each of the classes above is shown to be pseudo-elementary hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary. 1 Introduction An atomic representation h of a Boolean algebra B is a representation h: B → ℘(X) (some set X) where h(1) = {h(a): a is an atom of B}. It is known that a representation of a Boolean algebraS is a complete representation (in the sense of a complete embedding into a field of sets) if and only if it is an atomic repre- sentation and hence that the class of completely representable Boolean algebras is precisely the class of atomic Boolean algebras, and hence is elementary [6]. arXiv:1201.2331v3 [math.RA] 30 Aug 2016 This result is not obvious as the usual definition of a complete representation is thoroughly second order.
    [Show full text]
  • On Birkhoff's Common Abstraction Problem
    F. Paoli On Birkho®'s Common C. Tsinakis Abstraction Problem Abstract. In his milestone textbook Lattice Theory, Garrett Birkho® challenged his readers to develop a \common abstraction" that includes Boolean algebras and lattice- ordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of BA and LG their join BA_LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for BA_LG relative to FL. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing BA _ LG. Keywords: Residuated lattice, FL-algebra, Substructural logics, Boolean algebra, Lattice- ordered group, Birkho®'s problem, History of 20th C. algebra. 1. Introduction In his milestone textbook Lattice Theory [2, Problem 108], Garrett Birkho® challenged his readers by suggesting the following project: Develop a common abstraction that includes Boolean algebras (rings) and lattice ordered groups as special cases. Over the subsequent decades, several mathematicians tried their hands at Birkho®'s intriguing problem. Its very formulation, in fact, intrinsically seems to call for reiterated attempts: unlike most problems contained in the book, for which it is manifest what would count as a correct solution, this one is stated in su±ciently vague terms as to leave it open to debate whether any proposed answer is really adequate. It appears to us that Rama Rao puts things right when he remarks [28, p.
    [Show full text]
  • Partial Orders — Basics
    Partial Orders — Basics Edward A. Lee UC Berkeley — EECS EECS 219D — Concurrent Models of Computation Last updated: January 23, 2014 Outline Sets Join (Least Upper Bound) Relations and Functions Meet (Greatest Lower Bound) Notation Example of Join and Meet Directed Sets, Bottom Partial Order What is Order? Complete Partial Order Strict Partial Order Complete Partial Order Chains and Total Orders Alternative Definition Quiz Example Partial Orders — Basics Sets Frequently used sets: • B = {0, 1}, the set of binary digits. • T = {false, true}, the set of truth values. • N = {0, 1, 2, ···}, the set of natural numbers. • Z = {· · · , −1, 0, 1, 2, ···}, the set of integers. • R, the set of real numbers. • R+, the set of non-negative real numbers. Edward A. Lee | UC Berkeley — EECS3/32 Partial Orders — Basics Relations and Functions • A binary relation from A to B is a subset of A × B. • A partial function f from A to B is a relation where (a, b) ∈ f and (a, b0) ∈ f =⇒ b = b0. Such a partial function is written f : A*B. • A total function or just function f from A to B is a partial function where for all a ∈ A, there is a b ∈ B such that (a, b) ∈ f. Edward A. Lee | UC Berkeley — EECS4/32 Partial Orders — Basics Notation • A binary relation: R ⊆ A × B. • Infix notation: (a, b) ∈ R is written aRb. • A symbol for a relation: • ≤⊂ N × N • (a, b) ∈≤ is written a ≤ b. • A function is written f : A → B, and the A is called its domain and the B its codomain.
    [Show full text]
  • Cayley's and Holland's Theorems for Idempotent Semirings and Their
    Cayley's and Holland's Theorems for Idempotent Semirings and Their Applications to Residuated Lattices Nikolaos Galatos Department of Mathematics University of Denver [email protected] Rostislav Horˇc´ık Institute of Computer Sciences Academy of Sciences of the Czech Republic [email protected] Abstract We extend Cayley's and Holland's representation theorems to idempotent semirings and residuated lattices, and provide both functional and relational versions. Our analysis allows for extensions of the results to situations where conditions are imposed on the order relation of the representing structures. Moreover, we give a new proof of the finite embeddability property for the variety of integral residuated lattices and many of its subvarieties. 1 Introduction Cayley's theorem states that every group can be embedded in the (symmetric) group of permutations on a set. Likewise, every monoid can be embedded into the (transformation) monoid of self-maps on a set. C. Holland [10] showed that every lattice-ordered group can be embedded into the lattice-ordered group of order-preserving permutations on a totally-ordered set. Recall that a lattice-ordered group (`-group) is a structure G = hG; _; ^; ·;−1 ; 1i, where hG; ·;−1 ; 1i is group and hG; _; ^i is a lattice, such that multiplication preserves the order (equivalently, it distributes over joins and/or meets). An analogous representation was proved also for distributive lattice-ordered monoids in [2, 11]. We will prove similar theorems for resid- uated lattices and idempotent semirings in Sections 2 and 3. Section 4 focuses on the finite embeddability property (FEP) for various classes of idempotent semirings and residuated lat- tices.
    [Show full text]
  • LATTICE THEORY of CONSENSUS (AGGREGATION) an Overview
    Workshop Judgement Aggregation and Voting September 9-11, 2011, Freudenstadt-Lauterbad 1 LATTICE THEORY of CONSENSUS (AGGREGATION) An overview Bernard Monjardet CES, Université Paris I Panthéon Sorbonne & CAMS, EHESS Workshop Judgement Aggregation and Voting September 9-11, 2011, Freudenstadt-Lauterbad 2 First a little precision In their kind invitation letter, Klaus and Clemens wrote "Like others in the judgment aggregation community, we are aware of the existence of a sizeable amount of work of you and other – mainly French – authors on generalized aggregation models". Indeed, there is a sizeable amount of work and I will only present some main directions and some main results. Now here a list of the main contributors: Workshop Judgement Aggregation and Voting September 9-11, 2011, Freudenstadt-Lauterbad 3 Bandelt H.J. Germany Barbut, M. France Barthélemy, J.P. France Crown, G.D., USA Day W.H.E. Canada Janowitz, M.F. USA Mulder H.M. Germany Powers, R.C. USA Leclerc, B. France Monjardet, B. France McMorris F.R. USA Neumann, D.A. USA Norton Jr. V.T USA Powers, R.C. USA Roberts F.S. USA Workshop Judgement Aggregation and Voting September 9-11, 2011, Freudenstadt-Lauterbad 4 LATTICE THEORY of CONSENSUS (AGGREGATION) : An overview OUTLINE ABSTRACT AGGREGATION THEORIES: WHY? HOW The LATTICE APPROACH LATTICES: SOME RECALLS The CONSTRUCTIVE METHOD The federation consensus rules The AXIOMATIC METHOD Arrowian results The OPTIMISATION METHOD Lattice metric rules and the median procedure The "good" lattice structures for medians: Distributive lattices Median semilattice Workshop Judgement Aggregation and Voting September 9-11, 2011, Freudenstadt-Lauterbad 5 ABSTRACT CONSENSUS THEORIES: WHY? "since Arrow’s 1951 theorem, there has been a flurry of activity designed to prove analogues of this theorem in other contexts, and to establish contexts in which the rather dismaying consequences of this theorem are not necessarily valid.
    [Show full text]
  • Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions
    Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions Section 10. 1, Problems: 1, 2, 3, 4, 10, 11, 29, 36, 37 (fifth edition); Section 11.1, Problems: 1, 2, 5, 6, 12, 13, 31, 40, 41 (sixth edition) The notation ""forOR is bad and misleading. Just think that in the context of boolean functions, the author uses instead of ∨.The integers modulo 2, that is ℤ2 0,1, have an addition where 1 1 0 while 1 ∨ 1 1. AsetA is partially ordered by a binary relation ≤, if this relation is reflexive, that is a ≤ a holds for every element a ∈ S,it is transitive, that is if a ≤ b and b ≤ c hold for elements a,b,c ∈ S, then one also has that a ≤ c, and ≤ is anti-symmetric, that is a ≤ b and b ≤ a can hold for elements a,b ∈ S only if a b. The subsets of any set S are partially ordered by set inclusion. that is the power set PS,⊆ is a partially ordered set. A partial ordering on S is a total ordering if for any two elements a,b of S one has that a ≤ b or b ≤ a. The natural numbers ℕ,≤ with their ordinary ordering are totally ordered. A bounded lattice L is a partially ordered set where every finite subset has a least upper bound and a greatest lower bound.The least upper bound of the empty subset is defined as 0, it is the smallest element of L.
    [Show full text]
  • ON DISCRETE IDEMPOTENT PATHS Luigi Santocanale
    ON DISCRETE IDEMPOTENT PATHS Luigi Santocanale To cite this version: Luigi Santocanale. ON DISCRETE IDEMPOTENT PATHS. Words 2019, Sep 2019, Loughborough, United Kingdom. pp.312–325, 10.1007/978-3-030-28796-2_25. hal-02153821 HAL Id: hal-02153821 https://hal.archives-ouvertes.fr/hal-02153821 Submitted on 12 Jun 2019 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. ON DISCRETE IDEMPOTENT PATHS LUIGI SANTOCANALE Laboratoire d’Informatique et des Syst`emes, UMR 7020, Aix-Marseille Universit´e, CNRS Abstract. The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w x, y such that w x = w y = n) has a canonical monoid structure inher- ∈ { }∗ | | | | ited from the bijection with the set of join-continuous maps from the chain 0, 1,..., n to { } itself. We explicitly describe this monoid structure and, relying on a general characteriza- tion of idempotent join-continuous maps from a complete lattice to itself, we characterize idempotent paths as upper zigzag paths. We argue that these paths are counted by the odd Fibonacci numbers. Our method yields a geometric/combinatorial proof of counting results, due to Howie and to Laradji and Umar, for idempotents in monoids of monotone endomaps on finite chains.
    [Show full text]
  • Hopf Monoids of Ordered Simplicial Complexes
    HOPF MONOIDS OF ORDERED SIMPLICIAL COMPLEXES FEDERICO CASTILLO, JEREMY L. MARTIN, AND JOSE´ A. SAMPER ABSTRACT. We study pure ordered simplicial complexes (i.e., simplicial complexes with a linear order on their ground sets) from the Hopf-theoretic point of view. We define a Hopf class to be a family of pure ordered simplicial complexes that give rise to a Hopf monoid under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class allows us to give a systematic study of simpli- cial complexes related to matroids, including shifted complexes, broken-circuit complexes, and unbounded matroids (which arise from unbounded generalized permutohedra with 0/1 coordinates). We compute the antipodes in two cases: facet-initial complexes (a much larger class than shifted complexes) and unbounded ordered matroids. In the latter case, we embed the Hopf monoid of ordered matroids into the Hopf monoid of ordered generalized permuto- hedra, enabling us to compute the antipode using the topological method of Aguiar and Ardila. The calculation is complicated by the appearance of certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control certain coeffi- cients of the antipode. The resulting antipode formula is multiplicity-free and cancellation- free. CONTENTS 1. Introduction2 1.1. Background: Matroids and combinatorial Hopf theory2 1.2. Ordered simplicial complexes3 1.3. Polyhedra5 1.4. Antipodes6 2. Background and notation7 2.1. Simplicial complexes8 2.2. Matroids9 2.3. Set compositions, preposets, and the braid fan9 2.4. Generalized permutohedra 11 2.5. Species and Hopf monoids 14 2.6.
    [Show full text]
  • Lattice Duality: the Origin of Probability and Entropy
    , 1 Lattice Duality: The Origin of Probability and Entropy Kevin H. Knuth NASA Ames Research Center, Mail Stop 269-3, Moffett Field CA 94035-1000, USA Abstract Bayesian probability theory is an inference calculus, which originates from a gen- eralization of inclusion on the Boolean lattice of logical assertions to a degree of inclusion represented by a real number. Dual to this lattice is the distributive lat- tice of questions constructed from the ordered set of down-sets of assertions, which forms the foundation of the calculus of inquiry-a generalization of information theory. In this paper we introduce this novel perspective on these spaces in which machine learning is performed and discuss the relationship between these results and several proposed generalizations of information theory in the literature. Key words: probability, entropy, lattice, information theory, Bayesian inference, inquiry PACS: 1 Introduction It has been known for some time that probability theory can be derived as a generalization of Boolean implication to degrees of implication repre- sented by real numbers [11,12]. Straightforward consistency requirements dic- tate the form of the sum and product rules of probability, and Bayes’ thee rem [11,12,47,46,20,34],which forms the basis of the inferential calculus, also known as inductive inference. However, in machine learning applications it is often times more useful to rely on information theory [45] in the design of an algorithm. On the surface, the connection between information theory and probability theory seems clear-information depends on entropy and entropy Email address: kevin.h. [email protected] (Kevin H.
    [Show full text]
  • An Efficient Algorithm for Fully Robust Stable Matchings Via Join
    An Efficient Algorithm for Fully Robust Stable Matchings via Join Semi-Sublattices Tung Mai∗1 and Vijay V. Vazirani2 1Adobe Research 2University of California, Irvine Abstract We are given a stable matching instance A and a set S of errors that can be introduced into A. Each error consists of applying a specific permutation to the preference list of a chosen boy or a chosen girl. Assume that A is being transmitted over a channel which introduces one error from set S; as a result, the channel outputs this new instance. We wish to find a matching that is stable for A and for each of the jSj possible new instances. If S is picked from a special class of errors, we give an O(jSjp(n)) time algorithm for this problem. We call the obtained matching a fully robust stable matching w.r.t. S. In particular, if S is polynomial sized, then our algorithm runs in polynomial time. Our algorithm is based on new, non-trivial structural properties of the lattice of stable matchings; these properties pertain to certain join semi-sublattices of the lattice. Birkhoff’s Representation Theorem for finite distributive lattices plays a special role in our algorithms. 1 Introduction The two topics, of stable matching and the design of algorithms that produce solutions that are robust to errors, have been studied extensively for decades and there are today several books on each of them, e.g., see [Knu97, GI89, Man13] and [CE06, BTEGN09]. Yet, there is a paucity of results at the intersection of these two topics.
    [Show full text]