DOCSLIB.ORG

A Model-Theoretic Analysis of Asher and Vieu's Mereotopology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Model-Theoretic Analysis of Asher and Vieu's Mereotopology To appear, KR-08 1 Model-Theoretic Characterization of Asher and Vieu’s Ontology of Mereotopology Torsten Hahmann Michael Gruninger Department of Computer Science Department of Mechanical and Industrial Engineering University of Toronto Department of Computer Science [email protected] University of Toronto [email protected] Abstract want to understand what kind of models the axiomatic sys- tem RT0 describes and what properties these models share. We characterize the models of Asher and Vieu’s first-order The goal is to characterize the models of RT in terms of mereotopology that evolved from Clarke’s Calculus of Indi- 0 viduals in terms of mathematical structures with well-defined classes of structures defined in topology, lattice theory, and properties: topological spaces, lattices, and graphs. Although graph theory and compare the classes to representations of for the theory RT soundness and completeness with respect to other mereotopological theories. Although the completeness a topological translation of the axioms has been proved, this and soundness of RT0 has been proven with respect to the in- provides only sparse insights into the structural properties of tended models defined by RTT over a topology T, this is a the mereotopological models. We prove that the models of the mere rephrasing of the axioms. The proofs show that the subtheory RT − are isomorphic to p-ortholattices (pseudocom- axiomatic system describes exactly the intended models, but plemented, orthocomplemented). Combining the advantages the formulation of the intended models does not reveal struc- of lattices and graphs, we give a representation theorem for tural properties that can be used to learn about practical ap- RT − the finite models of EC and show how to construct finite plicability, implicit restrictions, and hidden assumptions of models of the full mereotopology. We compare our results to representations of the Calculus of Individuals and the Region the theory. [Special emphasis in our work is put on the finite Connection Calculus (RCC). models, since these are dominant in real-world applications.] Primary motivation of this work is to give better in- sight into the axiomatic theory and to uncover problems 1 Introduction and assumptions that users of the ontology should be aware Mereotopological systems have long been considered in of. A characterization of the models of the axiomatic the- philosophic and logic communities and recently received ory allows us to reuse knowledge about the mathemati- attention from a knowledge representation perspective. cal structures in the mereotopological theory. Afterwards, Mereotopology is composed of topological (from point-set we can compare our results to the characterization of the topology) notions of connectedness and mereological no- Region Connection Calculus (RCC), as conducted in (?; tions of parthood. Point-set topology (or General Topol- ?) which uses the notion of Connection Algebras to describe ogy) relies on the definition of open (and dually closed) sets the RCC. Besides the characterization and analysis of RT, and extends standard set-theoretic notions of union, inter- the main contribution of this work is a comparison of the section, and containment with concepts such as interior, clo- suitability of different mathematical structures, in particular sure, limit points, neighborhoods, and connectedness. topological spaces, graph representations, and lattices, for a Uncertainty about differences in mereotopological sys- model-theoretic analysis and comparison of mereotopologi- tems, in particular about their implicit [inherent] assump- cal frameworks. In the long-term an exhaustive comparison tions, seem to be a major source of confusion that hinders of different mereotopological approaches within a strictly forthright application of even well-developed mereotopo- defined mathematical context is desired. It turns out that logical theories. This problem arises in the various theo- lattices and lattices as graphs are best suited because lattices ries for different reasons: some lack any formal represen- provide an intuitive way to model parthood relations. tation, leaving the user unsure about intended interpreta- Notice that we are only interested in a rigid mathemati- tions; others are formalized in first-order logic but lack a cal study to provide the community with a model-theoretic characterization of the models up to isomorphism. This pa- view on mereotopology for the example of RT0; we do per focuses on an instance of the latter problem – we an- not argue for or against underlying assumptions of different alyze the models of Asher and Vieu’s mereotopology RT0 mereotopologies. (?) in the style of a representation theorem using struc- tures from well-understood mathematical disciplines. We Serving the growing interesting in formal ontologies and upper ontologies, this kind of analysis can guide the selec- Copyright c 2008, Association for the Advancement of Artificial tion of a generic axiomatization of mereotopology for inclu- Intelligence (www.aaai.org). All rights reserved. sion in upper ontologies such as SUMO, DOLCE, and BFO. To appear, KR-08 2 2 The Mereotopology RT of Asher and Vieu have been identified with it, we focus on Asher and Vieu’s Mereology investigates parthood structures and relative version of the theory where the completeness and soundness complementation, dating back to Whitehead (?) and proof ease the model-theoretic analysis. Notice that Clarke’s Lesniewski´ (?). The first formal specification of extensional and Asher and Vieu’s theories are more sophisticated that mereology was (?). For an overview of extensional mere- the RCC which does not distinguish individuals from their ology, we invite the reader to consult (?). The primitive re- interiors and closures, claiming such distinction superfluous lation in mereology is parthood (an entity being part of an- for spatial reasoning. But contradictory, tangential and other) expressed as irreflexive proper parthood, < or PP, or non-tangential parts as well as regular overlap and external as reflexive parthood, ≤ or P. The latter is usually a standard connection which all rely on open and closed properties are partial order that is reflexive, anti-symmetric, and transitive, distinguished in RCC. coined Ground Mereology M in (?). Moreover, most mere- ologies define concepts of overlap, union, and intersection 2.1 Axiomatization RT0 of entities. General sums (fusion), i.e. the union of arbitrar- The first-order theory RT0 of (?) uses the connection rela- ily many individuals, are also widespread. In all mereolog- tion C as only primitive. The theory is based on Clarke’s ical theories a whole (universe) can be defined as the entity Calculus of Individuals (?; ?), with changes to make the that everything else is part of. If differences are defined, a theory first-order definable: (1) the explicit fusion operator complement exists for every individual relative to the mere- is eliminated, it is claimed unnecessary; and (2) the con- ological whole. More controversial is whether mereology cept of weak contact, WCont, is added. To eliminate trivial should allow atoms, i.e. individuals without proper parts that models, RT0 requires at least one external connection and are the smallest entities of interest. Some theories are atom- one weak contact (A11, A12). Some ontological and cogni- less while others explicitly force the existence of atoms (?); tive issues are also addressed, see (?). RT0 follows the strat- mereotopology inherits this controversy: it can be defined egy called “Topology as Basis for Mereology” for defining atomless, atomic, or make no assumption about atomism at mereotopology and does not contain an explicit mereology. all. Consequently, the parthood relation P is defined solely in Neither topology nor mereology are by themselves pow- terms of the primitive C which limits the whole theory to erful enough to express part-whole relations: Connection the expressiveness of C. For consequences of this kind of does not imply parthood between two individuals and dis- axiomatization, see (?; ?). connection does not prevent parthood as well as mereo- To construct models of the theory RT0 the following def- logical wholes do not imply topological (self-connected) initions are necessary. Except for WCont, these were al- wholes. To be able to reason about integral, self-connected ready defined in (?) and are comparable to those of other individuals, ways to combine mereology with topology are mereotopological systems. necessary. The different options to merge the two inde- (D1) P(x,y) ≡ ∀z[C(z,x) → C(z,y)] (Parthood as reflexive pendent theories are used in (?) to classify mereotopolo- partial order satisfying the axioms of M) gies. Bridging the gap between mereology and topology (D3) O(x,y) ≡ ∃z[P(z,x) ∧ P(z,y)] (Two individuals over- can be achieved by extending mereology with a topolog- lap iff they have a common part) ical primitive as applied in (?; ?; ?). More widespread (D4) EC(x,y) ≡ C(x,y)∧¬O(x,y) (Two individuals are ex- is the reverse: assuming topology to be more fundamental ternally connected iff they are connected but share no and defining mereology on top of it using only topological common part) primitives (“Topology as Basis for Mereology”). (?; ?; ?; ?) and the RCC (?; ?) use this method with a connection (or (D6) NTP(x,y) ≡ P(x,y)∧¬∃z[EC(z,x) ∧ EC(z,y)] (Non- contact) relation as only primitive and parthood expressed tangential parts do not touch the border of the larger in terms of connection. A third, less common way to merge individuals) topology and mereology was presented in (?) extending the (D8) OP(x) ≡ x = i(x) (Open individuals) mereological framework of (?) by quasi-mereological no- (D9) CL(x) ≡ x = c(x) (Closed individuals) tions (combining mereology with some topological idea) to (D11) WCont(x,y) ≡ ¬C(c(x),c(y)) ∧ ∀z[(OP(z) ∧ define topological wholeness. P(x,z)) → C(c(z),y)] (Weak contact requires the clo- As mentioned before, our focus are first-order sures of x and y to be disconnected, but any neighbor- mereotopologies. However, most of these theories either hood containing cl(x) to be connected to y) entirely lack soundness and completeness proofs, e.g.
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]
  • 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 Introduction to the Theory of Lattice Ý Jinfang Wang £
    An Introduction to the Theory of Lattice Ý Jinfang Wang £ Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan May 11, 2006 £ Fax: 81-43-290-3663. E-Mail addresses: [email protected] Ý Partially supported by Grand-in-Aid 15500179 of the Japanese Ministry of Education, Science, Sports and Cul- ture. 1 1 Introduction A lattice1 is a partially ordered set (or poset), in which all nonempty finite subsets have both a supremum (join) and an infimum (meet). Lattices can also be characterized as algebraic structures that satisfy certain identities. Since both views can be used interchangeably, lattice theory can draw upon applications and methods both from order theory and from universal algebra. Lattices constitute one of the most prominent representatives of a series of “lattice-like” structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, and Boolean algebras. 2 Semilattice A semilattice is a partially ordered set within which either all binary sets have a supremum (join) or all binary sets have an infimum (meet). Consequently, one speaks of either a join-semilattice or a meet-semilattice. Semilattices may be regarded as a generalization of the more prominent concept of a lattice. In the literature, join-semilattices sometimes are sometimes additionally required to have a least element (the join of the empty set). Dually, meet-semilattices may include a greatest element. 2.1 Definitions Semilattices as posets Ë µ Ü DEFINITION 2.1 (MEET-SEMILATTICE). A poset ´ is a meet-semilattice if for all elements Ë Ü Ý and Ý of , the greatest lower bound (meet) of the set exists.
    [Show full text]
  • Lattice Theory
    Chapter 1 Lattice theory 1.1 Partial orders 1.1.1 Binary Relations A binary relation R on a set X is a set of pairs of elements of X. That is, R ⊆ X2. We write xRy as a synonym for (x, y) ∈ R and say that R holds at (x, y). We may also view R as a square matrix of 0’s and 1’s, with rows and columns each indexed by elements of X. Then Rxy = 1 just when xRy. The following attributes of a binary relation R in the left column satisfy the corresponding conditions on the right for all x, y, and z. We abbreviate “xRy and yRz” to “xRyRz”. empty ¬(xRy) reflexive xRx irreflexive ¬(xRx) identity xRy ↔ x = y transitive xRyRz → xRz symmetric xRy → yRx antisymmetric xRyRx → x = y clique xRy For any given X, three of these attributes are each satisfied by exactly one binary relation on X, namely empty, identity, and clique, written respectively ∅, 1X , and KX . As sets of pairs these are respectively the empty set, the set of all pairs (x, x), and the set of all pairs (x, y), for x, y ∈ X. As square X × X matrices these are respectively the matrix of all 0’s, the matrix with 1’s on the leading diagonal and 0’s off the diagonal, and the matrix of all 1’s. Each of these attributes holds of R if and only if it holds of its converse R˘, defined by xRy ↔ yRx˘ . This extends to Boolean combinations of these attributes, those formed using “and,” “or,” and “not,” such as “reflexive and either not transitive or antisymmetric”.
    [Show full text]
  • Arxiv:2004.12171V1 [Cs.LO]
    Algebraic Approach to Directed Rough Sets A Mani∗ HBCSE, Tata Institute of Fundamental Research 9/1B, Jatin Bagchi Road Kolkata (Calcutta)-700029, India Homepage: http: // www. logicamani. in Orchid: https: // orcid. org/ 0000-0002-0880-1035 S´andor Radeleczki Institute of Mathematics, University of Miskolc 3515 Miskolc-Ergyetemv´aros Miskolc, Hungary Homepage: https: // www. uni-miskolc. hu/ ~matradi arXiv:2004.12171v1 [cs.LO] 25 Apr 2020 ∗Corresponding author Email addresses: [email protected] (A Mani), [email protected] (S´andor Radeleczki) Preprint submitted to Elsevier Algebraic Approach to Directed Rough Sets A Mani∗ HBCSE, Tata Institute of Fundamental Research 9/1B, Jatin Bagchi Road Kolkata (Calcutta)-700029, India Homepage: http: // www. logicamani. in Orchid: https: // orcid. org/ 0000-0002-0880-1035 S´andor Radeleczki Institute of Mathematics, University of Miskolc 3515 Miskolc-Ergyetemv´aros Miskolc, Hungary Homepage: https: // www. uni-miskolc. hu/ ~matradi Abstract In relational approach to general rough sets, ideas of directed relations are sup- plemented with additional conditions for multiple algebraic approaches in this research paper. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric for a better be- haved groupoidal semantics over the set of roughly equivalent objects by the first author. Another distinct algebraic semantics over the set of approxima- tions, and a new knowledge interpretation are also invented in this research by her. Because of minimal conditions imposed on the relations, neighborhood granulations are used in the construction of all approximations (granular and pointwise). Necessary and sufficient conditions for the lattice of local upper approximations to be completely distributive are proved by the second author.
    [Show full text]
  • Arxiv:2007.04099V2 [Math.AT] 22 Apr 2021 Ope Oamnmlcleto Htitrwnswt H H the with [31]
    Cellular Sheaves of Lattices and the Tarski Laplacian Robert Ghrist* Hans Riess† Abstract This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications. 1 Introduction The goal of this paper is to initiate a theory of sheaf cohomology for cellu- lar sheaves valued in a category of lattices. Lattices are algebraic structures with a rich history [41] and a wide array of applications [13, 2, 17, 42, 34, 16]. Cellular sheaves are data structures that stitch together algebraic entities according to the pattern of a cell complex [43]. Sheaf cohomology is a compression that collapses all the data over a topological space — or cell complex — to a minimal collection that intertwines with the homological features of the base space [31]. arXiv:2007.04099v2 [math.AT] 22 Apr 2021 *Department of Mathematics, Department of Electrical & Systems Engineering University of Pennsylvania [email protected] †Department of Electrical & Systems Engineering University of Pennsylvania [email protected] 1 1.1 Contributions Our approach is to set up a Hodge-style theory, developing analogues of the combinatorial Laplacian adapted to sheaves of lattices. Specific contri- butions of this work include the following. 1. In §2, we review posets, lattices, lattice connections, cellular sheaves, and Hodge theory. It is nontrivial to define cohomology for sheaves valued in the (nonabelian) category of lattices and connections.
    [Show full text]
  • Submodular Functions, Optimization, and Applications to Machine Learning — Spring Quarter, Lecture 14 —
    Submodular Functions, Optimization, and Applications to Machine Learning | Spring Quarter, Lecture 14 | http://j.ee.washington.edu/~bilmes/classes/ee596b_spring_2014/ Prof. Jeff Bilmes University of Washington, Seattle Department of Electrical Engineering http://melodi.ee.washington.edu/~bilmes May 18th, 2014 f (A) + f (B) f (A B) + f (A B) Clockwise from top left:v Lásló Lovász Jack Edmonds ∪ ∩ Satoru Fujishige George Nemhauser ≥ Laurence Wolsey = f (A ) + 2f (C) + f (B ) = f (A ) +f (C) + f (B ) = f (A B) András Frank r r r r Lloyd Shapley ∩ H. Narayanan Robert Bixby William Cunningham William Tutte Richard Rado Alexander Schrijver Garrett Birkho Hassler Whitney Richard Dedekind Prof. Jeff Bilmes EE596b/Spring 2014/Submodularity - Lecture 14 - May 18th, 2014 F1/54 (pg.1/57) Logistics Review Cumulative Outstanding Reading Good references for today: Birkhoff, \Lattice Theory", 1967. Prof. Jeff Bilmes EE596b/Spring 2014/Submodularity - Lecture 14 - May 18th, 2014 F2/54 (pg.2/57) Logistics Review Announcements, Assignments, and Reminders This is a special extra lecture on lattices that is not given during regular lecture time. Prof. Jeff Bilmes EE596b/Spring 2014/Submodularity - Lecture 14 - May 18th, 2014 F3/54 (pg.3/57) Logistics Review Class Road Map - IT-I L1 (3/31): Motivation, Applications, & L11: More properties of polymatroids, Basic Definitions SFM special cases L2: (4/2): Applications, Basic L12: polymatroid properties, extreme Definitions, Properties points polymatroids, L3: More examples and properties (e.g., L13: sat, dep, supp, exchange capacity, closure properties), and examples, examples spanning trees L14: Lattice theory: partially ordered L4: proofs of equivalent definitions, sets; lattices; distributive, modular, independence, start matroids submodular, and boolean lattices; ideals L5: matroids, basic definitions and and join irreducibles.
    [Show full text]