DOCSLIB.ORG

Topology of Privacy: Lattice Structures and Information Bubbles for Inference and Obfuscation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topology of Privacy: Lattice Structures and Information Bubbles for Inference and Obfuscation Topology of Privacy: Lattice Structures and Information Bubbles for Inference and Obfuscation Michael Erdmann∗ Carnegie Mellon University December 12, 2017 c 2017 Michael Erdmann Abstract Information has intrinsic geometric and topological structure, arising from relative relationships beyond absolute values or types. For instance, the fact that two people did or did not share a meal describes a relationship independent of the meal's ingredients. Multiple such relationships give rise to relations and their lattices. Lattices have topology. That topology informs the ways in which information may be observed, hidden, inferred, and dissembled. Privacy preservation may be understood as finding isotropic topologies, in which relations appear homogeneous. Moreover, the underlying lattice structure of those topologies has a temporal aspect, which reveals how isotropy may contract over time, thereby puncturing privacy. Dowker's Theorem establishes a homotopy equivalence between two simplicial complexes derived from a relation. From a privacy perspective, one complex describes individuals with common attributes, the other describes attributes shared by individuals. The homotopy equivalence is an alignment of certain common cores of those complexes, effectively interpreting sets of individuals as sets of attributes, and vice-versa. That common core has a lattice structure. An element in the lattice consists of two components, one being a set of individuals, the other being an equivalent set of attributes. The lattice operations join and meet each amount to set intersection in one component and set union followed by a potentially privacy-puncturing inference in the other component. One objective of this research has been to understand the topology of the Dowker complexes, from a privacy perspective. First, privacy loss appears as simplicial collapse of free faces. Such collapse is local, but the property of fully preserving both attribute and association privacy requires a global condition: a particular kind of spherical hole. Second, by looking at the link of an identifiable individual in its encompassing Dowker complex, arXiv:1712.04130v1 [math.CO] 12 Dec 2017 one can characterize that individual's attribute privacy via another sphere condition. This characterization generalizes to certain groups' attribute privacy. Third, even when long- term attribute privacy is impossible, homology provides lower bounds on how an individual may defer identification, when that individual has control over how to reveal attributes. Intuitively, the idea is to first reveal information that could otherwise be inferred. This last result highlights privacy as a dynamic process. Privacy loss may be cast as gradient flow. Harmonic flow for privacy preservation may be fertile ground for future research. ∗This report is based upon work supported in part by the Air Force Office of Scientific Research under award number FA9550-14-1-0012 and in part by the National Science Foundation under award number IIS-1409003. Any opinions, findings and conclusions or recommendations expressed in this report are those of the author and do not necessarily reflect the views of the Government, the U.S. Department of Defense, or the National Science Foundation. ii Topology of Privacy: Lattice Structures and Information Bubbles Contents 1 Introduction 1 2 Outline 3 List of Primary Symbols 6 3 Privacy: Relations and Partially Ordered Sets 7 3.1 A Toy Example: Health Data and Attribute Privacy . 7 Assumption of Relational Completeness . 7 Assumption of Observational Monotonicity . 8 Assumption of Observational Accuracy . 8 3.2 A Dual Perspective: Payroll Data and Association Privacy . 10 3.3 Privacy Preservation and Loss: A Poset Model . 11 4 The Galois Connection for Modeling Privacy 13 4.1 Dowker Complexes . 13 4.2 Inference from Closure Operators . 14 4.3 Attribute and Association Privacy . 17 4.4 Disinformation Example Re-Revisited . 17 5 The Face Shape of Privacy 19 5.1 Free Faces . 19 5.2 Privacy versus Identifiability . 20 5.3 Spheres and Privacy . 21 5.4 A Spherical Non-Boundary Relation that Preserves Attribute Privacy . 21 6 Conditional Relations as Simplicial Links 23 7 Privacy Characterization via Boundary Complexes 25 8 The Meaning of Holes in Relations 27 9 Change-of-Attribute Transformations 30 10 Leveraging Lattices for Privacy Preservation 34 10.1 Attribute Release Order . 34 10.2 Inferences on a Lattice . 35 10.3 Preserving Attribute Privacy for Sets of Individuals . 36 10.4 Informative Attribute Release Sequences . 38 10.5 Isotropy, Minimal Identification, and Spheres . 39 10.6 Poset Lengths and Information Release . 41 10.7 Hidden Holes . 42 10.8 Bubbles are Lower Bounds for Privacy . 44 Contents iii 11 Experiments 47 11.1 Compare and Contrast . 48 11.2 Homology Computations . 49 11.3 Homology and Release Sequences in the Olympic Medals Dataset . 49 11.4 Homology and Release Sequences in the Jazz Dataset . 52 12 Inference in Sequence Lattices 55 12.1 Sequence Lattices for Dynamic Attribute Observations . 55 12.2 Lattices of Stochastic Observations . 57 12.3 General Inference Lattices . 58 13 Lattices for Strategy Obfuscation 62 13.1 Strategies for Nondeterministic Graphs . 62 13.2 Connecting the Topologies of Strategy Complexes and Privacy . 64 13.3 Example: Multi-State Goals and Multi-Strategy Singleton Goals . 66 13.4 Randomization . 68 14 Relations as a Category 70 14.1 Relationship-Preserving Morphisms . 70 14.2 Privacy-Establishing Morphisms . 72 14.3 Summary of Morphism Properties . 73 14.4 G-Morphisms . 75 14.5 Surjectivity Revisited . 77 15 Future Thoughts 79 15.1 Relaxing Assumptions . 79 15.2 Sensing Attributes Stochastically . 80 Acknowledgments 82 References 82 A Preliminaries 84 A.1 Simplicial Complexes . 84 A.2 Partially Ordered Sets (Posets) . 86 A.3 Semi-Lattices and Lattices . 87 A.4 Relations . 88 B Basic Tools 91 C Links, Deletions, and Inference 94 C.1 Links, Deletions, and Induced Maps . 94 C.2 Privacy Preservation in Links and Deletions . 96 C.3 Unique Identifiability, Free Faces, and Privacy Preservation . 98 D Inference Hardness 102 iv Topology of Privacy: Lattice Structures and Information Bubbles E Privacy Spheres 104 E.1 Individual Attribute Privacy . 104 E.2 Group Attribute Privacy . 106 E.3 Preserving Attribute and Association Privacy . 108 E.4 Square Relations Preserve Privacy Symmetrically . 113 F Poset Chains 118 F.1 Maximal Chains and Informative Attribute Release Sequences . 118 F.2 Chains and Links . 119 F.3 Isotropy . 121 G Many Long Chains 123 H Obfuscating Strategies 128 H.1 Source Complex . 128 H.2 Delaying Strategy Identification . 130 H.3 Delaying Goal Recognition . 132 H.4 Hamiltonian Flexibility for Strategy Obfuscation . 133 H.5 Example: A Rapidly Inferable Strategy . 134 H.6 Pure Nondeterministic Graphs and Pure Stochastic Graphs . 136 H.7 Strategy Obfuscation Summary . 137 I Morphisms and Lattice Generators 138 I.1 Morphisms . 138 I.2 G-Morphisms . 141 I.3 Lattice Generators . 142 J A Few More Examples 145 J.1 Local Spheres versus Global Contractibility . 145 J.2 Disinformation . 146 J.3 Insufficient Representation . 148 J.4 A Structural Inference Example: Passengers on Ferries . 149 Introduction 1 1 Introduction Privacy is the ability of an individual or entity to control how much that individual or entity reveals about itself to others. Fundamental research into privacy seeks to understand the limits of that ability. A brief history of privacy should include the following: • The right to privacy as a legal principle, appearing in an 1890 Harvard Law Review article [24]. The article was a reaction to the then modern technology of photography and the dissemination of gossip via print media. • A demonstration linking supposedly anonymous public information with other more specific public data, thereby revealing sensitive attributes [21]. The demonstration employed zip code, gender, and birth date to link anonymous public insurance summaries with voter registration data. Doing so produced the health record of the governor of Massachusetts. This privacy failure suggested a first form of homogenization, called k- anonymity. Roughly, the idea was to structure databases in such a way that a database could respond to any query with an answer consisting of no fewer than k individuals matching the query parameters. • The discovery that it is impossible to preserve the privacy of an individual for even a single attribute in the face of repeated statistical queries over a population [2], unless answers to those queries are purposefully perturbed with noise of magnitude on the order p of at least n. Here n is the size of the population. The significance of this discovery is to underscore how difficult it is to preserve privacy while retaining information utility. • Netflix Prize. In 2006, Netflix offered a $1M prize for an algorithm that would predict viewer preferences better than Netflix’s internal algorithm. Netflix made available some of its historical user preferences, in anonymized form, as a basis for the competition. Once again, it turned out that one could link this anonymized data with other publicly available databases, resulting in the potential (and in some cases actual) identification of Netflix viewers, thereby de-anonymizing their viewing history [17]. Whereas in the earlier health example, a few specific observables made linking possible (global coordinates, one might say, namely zip code, gender, birth date), in the Netflix example, the intrinsic geometric structure of the database facilitated linking via a wide variety of observables
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • Partially Metric Association Schemes with a Multiplicity Three
    PARTIALLY METRIC ASSOCIATION SCHEMES WITH A MULTIPLICITY THREE EDWIN R. VAN DAM, JACK H. KOOLEN, AND JONGYOOK PARK Abstract. An association scheme is called partially metric if it has a con- nected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete 4-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known 2-arc-transitive covers of the cube: the M¨obius-Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric associ- ation schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive 2- walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three. 1. Introduction Bannai and Bannai [3] showed that the association scheme of the complete graph on four vertices is the only primitive symmetric association scheme with a multiplic- ity equal to three. They also posed the problem of determining all such imprimitive symmetric association schemes. As many product constructions can give rise to such schemes with a multiplicity three, we suggest and solve a more restricted problem. Indeed, we will determine the symmetric partially metric association schemes with a multiplicity three, where an association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme.
    [Show full text]
  • Concepts of Mathematics for Students of Physics and Engineering: a Dictionary
    NASA/TP—2003-212088 Concepts of Mathematics for Students of Physics and Engineering: A Dictionary Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio August 2003 The NASA STI Program Office . in Profile Since its founding, NASA has been dedicated to • CONFERENCE PUBLICATION. Collected the advancement of aeronautics and space papers from scientific and technical science. The NASA Scientific and Technical conferences, symposia, seminars, or other Information (STI) Program Office plays a key part meetings sponsored or cosponsored by in helping NASA maintain this important role. NASA. The NASA STI Program Office is operated by • SPECIAL PUBLICATION. Scientific, Langley Research Center, the Lead Center for technical, or historical information from NASA’s scientific and technical information. The NASA programs, projects, and missions, NASA STI Program Office provides access to the often concerned with subjects having NASA STI Database, the largest collection of substantial public interest. aeronautical and space science STI in the world. The Program Office is also NASA’s institutional • TECHNICAL TRANSLATION. English- mechanism for disseminating the results of its language translations of foreign scientific research and development activities. These results and technical material pertinent to NASA’s are published by NASA in the NASA STI Report mission. Series, which includes the following report types: Specialized services that complement the STI • TECHNICAL PUBLICATION. Reports of Program Office’s diverse offerings include completed research or a major significant creating custom thesauri, building customized phase of research that present the results of databases, organizing and publishing research NASA programs and include extensive data results . even providing videos. or theoretical analysis. Includes compilations of significant scientific and technical data and For more information about the NASA STI information deemed to be of continuing Program Office, see the following: reference value.
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Propositional Quantification and Comparison in Modal Logic Permalink https://escholarship.org/uc/item/78w732g8 Author DING, YIFENG Publication Date 2021 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Propositional Quantification and Comparison in Modal Logic by Yifeng Ding A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Logic and the Methodology of Science in the Graduate Division of the University of California, Berkeley Committee in charge: Associate Professor Wesley H. Holliday, Chair Professor Shamik Dasgupta Professor Paolo Mancosu Professor Dana S. Scott Professor Seth Yalcin Spring 2021 Propositional Quantification and Comparison in Modal Logic Copyright 2021 by Yifeng Ding 1 Abstract Propositional Quantification and Comparison in Modal Logic by Yifeng Ding Doctor of Philosophy in Logic and the Methodology of Science University of California, Berkeley Associate Professor Wesley H. Holliday, Chair We make the following contributions to modal logics with propositional quantifiers and modal logics with comparative operators in this dissertation: • We define a general notion of normal modal logics with propositional quantifiers. We call them normal Π-logics. Then, as was done by Scrogg's theorem on extensions of the modal logic S5, we study in general the normal Π-logics extending S5. We show that they are all complete with respect to their algebraic semantics based on complete simple monadic algebras. We also show that the lattice formed by these logics is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N with the natural order topology.
    [Show full text]