Constructive Galois Connections
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Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
GALOIS THEORY for ARBITRARY FIELD EXTENSIONS Contents 1
GALOIS THEORY FOR ARBITRARY FIELD EXTENSIONS PETE L. CLARK Contents 1. Introduction 1 1.1. Kaplansky's Galois Connection and Correspondence 1 1.2. Three flavors of Galois extensions 2 1.3. Galois theory for algebraic extensions 3 1.4. Transcendental Extensions 3 2. Galois Connections 4 2.1. The basic formalism 4 2.2. Lattice Properties 5 2.3. Examples 6 2.4. Galois Connections Decorticated (Relations) 8 2.5. Indexed Galois Connections 9 3. Galois Theory of Group Actions 11 3.1. Basic Setup 11 3.2. Normality and Stability 11 3.3. The J -topology and the K-topology 12 4. Return to the Galois Correspondence for Field Extensions 15 4.1. The Artinian Perspective 15 4.2. The Index Calculus 17 4.3. Normality and Stability:::and Normality 18 4.4. Finite Galois Extensions 18 4.5. Algebraic Galois Extensions 19 4.6. The J -topology 22 4.7. The K-topology 22 4.8. When K is algebraically closed 22 5. Three Flavors Revisited 24 5.1. Galois Extensions 24 5.2. Dedekind Extensions 26 5.3. Perfectly Galois Extensions 27 6. Notes 28 References 29 Abstract. 1. Introduction 1.1. Kaplansky's Galois Connection and Correspondence. For an arbitrary field extension K=F , define L = L(K=F ) to be the lattice of 1 2 PETE L. CLARK subextensions L of K=F and H = H(K=F ) to be the lattice of all subgroups H of G = Aut(K=F ). Then we have Φ: L!H;L 7! Aut(K=L) and Ψ: H!F;H 7! KH : For L 2 L, we write c(L) := Ψ(Φ(L)) = KAut(K=L): One immediately verifies: L ⊂ L0 =) c(L) ⊂ c(L0);L ⊂ c(L); c(c(L)) = c(L); these properties assert that L 7! c(L) is a closure operator on the lattice L in the sense of order theory. -
Formal Contexts, Formal Concept Analysis, and Galois Connections
Formal Contexts, Formal Concept Analysis, and Galois Connections Jeffrey T. Denniston Austin Melton Department of Mathematical Sciences Departments of Computer Science and Mathematical Sciences Kent State University Kent State University Kent, Ohio, USA 44242 Kent, Ohio, USA 44242 [email protected] [email protected] Stephen E. Rodabaugh College of Science, Technology, Engineering, Mathematics (STEM) Youngstown State University Youngstown, OH, USA 44555-3347 [email protected] Formal concept analysis (FCA) is built on a special type of Galois connections called polarities. We present new results in formal concept analysis and in Galois connections by presenting new Galois connection results and then applying these to formal concept analysis. We also approach FCA from the perspective of collections of formal contexts. Usually, when doing FCA, a formal context is fixed. We are interested in comparing formal contexts and asking what criteria should be used when determining when one formal context is better than another formal context. Interestingly, we address this issue by studying sets of polarities. 1 Formal Concept Analysis and Order-Reversing Galois Connections We study formal concept analysis (FCA) from a “larger” perspective than is commonly done. We em- phasize formal contexts. For example, we are interested in questions such as if we are working with a given formal context K , that is, we are working with a set of objects G, a set of properties M, and a relation R ⊂ G×M, what do we do if we want to replace K with a better formal context. Of course, this raises the question: what makes one formal context better than another formal context. -
Logics from Galois Connections
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Journal of Approximate Reasoning 49 (2008) 595–606 Contents lists available at ScienceDirect International Journal of Approximate Reasoning journal homepage: www.elsevier.com/locate/ijar Logics from Galois connections Jouni Järvinen a,*, Michiro Kondo b, Jari Kortelainen c a Turku Centre for Computer Science (TUCS), University of Turku, FI-20014 Turku, Finland b School of Information Environment, Tokyo Denki University, Inzai 270-1382, Japan c Mikkeli University of Applied Sciences, P.O. Box 181, FI-50101 Mikkeli, Finland article info abstract Article history: In this paper, Information Logic of Galois Connections (ILGC) suited for approximate rea- Received 13 June 2007 soning about knowledge is introduced. In addition to the three classical propositional logic Received in revised form 29 May 2008 axioms and the inference rule of modus ponens, ILGC contains only two auxiliary rules of Accepted 16 June 2008 inference mimicking the performance of Galois connections of lattice theory, and this Available online 27 June 2008 makes ILGC comfortable to use due to the flip-flop property of the modal connectives. Kripke-style semantics based on information relations is defined for ILGC. It is also shown that ILGC is equivalent to the minimal tense logic K , and decidability and completeness of Keywords: t ILGC follow from this observation. Additionally, relationship of ILGC to the so-called clas- Rough sets Fuzzy sets sical modal logics is studied. Namely, a certain composition of Galois connection mappings Approximate reasoning forms a lattice-theoretical interior operator, and this motivates us to axiomatize a logic of Knowledge representation these compositions. -
A Galois Connection Calculus for Abstract Interpretation∗
A Galois Connection Calculus for Abstract Interpretation⇤ Patrick Cousot Radhia Cousot CIMS⇤⇤, NYU, USA [email protected] CNRS Emeritus, ENS, France s [email protected] rfru Abstract We introduce a Galois connection calculus for language independent 3. Basic GC semantics Basic GCs are primitive abstractions of specification of abstract interpretations used in programming language semantics, properties. Classical examples are the identity abstraction 1[ , formal verification, and static analysis. This Galois connection calculus and its type λ Q . Q S hC system are typed by abstract interpretation. ] , , , , the top abstraction [ , vi hC vi −− − λ − −P − − −. − −P !−!− − hC vi S > hC Categories and Subject Descriptors D.2.4 [Software/Program Verification] λ Q . J General Terms Algorithms, Languages, Reliability, Security, Theory, Verification. , ] , , > , , the join abstraction [C] vi K> hC vi −−−−−λ P . hC vi S J[ Keywords Abstract Interpretation, Galois connection, Static Analysis, Verification. −−−−−!γ} > }(}(C)), }(C), with ↵}(P ) P , γ}(Q) In Abstract , } , , 1. Galois connections in Abstract Interpretation h K ✓i −−! −−↵ h ✓i J K interpretation [3, 4, 6, 7] concrete properties (for example (e.g.) }(Q), the complement abstraction [C] , }(SC), ¬ of computations) are related to abstract properties (e.g. types). The S ¬ h ✓i −−! −¬ }(C), , the finite/infinite sequence abstraction [C] , abstract properties are always sound approximations of the con- h ◆i γ S 1 1 J K crete properties (abstract proofs/static analyzes are always correct }(C1), }(C), with ↵1(P ) , σi σ P i h ✓i ↵ −1 −− h ✓i { | 2 ^ 2 in the concrete) and are sometimes complete (proofs/analyzes of −− − −! J K dom(σ) and γ1(Q) , σ C1 i dom(σ):σi Q , the abstract properties can all be done in the abstract only). -
The Computational Complexity Column by V
The Computational Complexity Column by V. Arvind Institute of Mathematical Sciences, CIT Campus, Taramani Chennai 600113, India [email protected] http://www.imsc.res.in/~arvind The mid-1960’s to the early 1980’s marked the early epoch in the field of com- putational complexity. Several ideas and research directions which shaped the field at that time originated in recursive function theory. Some of these made a lasting impact and still have interesting open questions. The notion of lowness for complexity classes is a prominent example. Uwe Schöning was the first to study lowness for complexity classes and proved some striking results. The present article by Johannes Köbler and Jacobo Torán, written on the occasion of Uwe Schöning’s soon-to-come 60th birthday, is a nice introductory survey on this intriguing topic. It explains how lowness gives a unifying perspective on many complexity results, especially about problems that are of intermediate complexity. The survey also points to the several questions that still remain open. Lowness results: the next generation Johannes Köbler Humboldt Universität zu Berlin [email protected] Jacobo Torán Universität Ulm [email protected] Abstract Our colleague and friend Uwe Schöning, who has helped to shape the area of Complexity Theory in many decisive ways is turning 60 this year. As a little birthday present we survey in this column some of the newer results related with the concept of lowness, an idea that Uwe translated from the area of Recursion Theory in the early eighties. Originally this concept was applied to the complexity classes in the polynomial time hierarchy. -
The Galois Correspondence Between a Ring and Its Spectrum
THE GALOIS CORRESPONDENCE BETWEEN A RING AND ITS SPECTRUM Let A and X be two sets equipped with a relation R ⊆ A × X. The notation is intended to remind you of the case where A is a commutative ring, X is its spectrum Spec A, and R is given by (f; x) 2 R () f(x) = 0 (i.e., f 2 px): But mathematics is full of other examples, including Galois theory (which is where the phrase \Galois correspondence" in the title comes from). For any subset S ⊆ A, define S0 ⊆ X by S0 := fx 2 X j (f; x) 2 R for all f 2 Sg : Similarly, for any subset Y ⊆ X, define Y 0 ⊆ A by Y 0 := ff 2 A j (f; x) 2 R for all x 2 Y g : In our canonical example, S0 = V (S) (the \zero set" of S), and Y 0 = I(Y ) := T 0 y2Y py. In more intuitive language, Y is the ideal of functions that vanish on Y . These \prime" operations are order reversing: S ⊆ T =) T 0 ⊆ S0; and similarly for subsets of X. Any subset of X of the form S0 will be called closed. (This gives the Zariski topology in our canonical example.) Similarly, any subset of A of the form Y 0 will be called closed. (So the closed subsets of A are the radical ideals in our canonical example.) Note that we always have S ⊆ S00 and Y ⊆ Y 00 for S ⊆ A and Y ⊆ X. I claim that equality holds for closed sets. -
Physics 212B, Ken Intriligator Lecture 6, Jan 29, 2018 • Parity P Acts As P
Physics 212b, Ken Intriligator lecture 6, Jan 29, 2018 Parity P acts as P = P † = P −1 with P~vP = ~v and P~aP = ~a for vectors (e.g. ~x • − and ~p) and axial pseudo-vectors (e.g. L~ and B~ ) respectively. Note that it is not a rotation. Parity preserves all the basic commutation relations, but it is not necessarily a symmetry of the Hamiltonian. The names scalar and pseudo scalar are often used to denote parity even vs odd: P P = for “scalar” operators (like ~x2 or L~ 2, and P ′P = ′ for pseudo O O O −O scalar operators (like S~ ~x). · Nature respects parity aside from some tiny effects at the fundamental level. E.g. • in electricity and magnetism, we can preserve Maxwell’s equations via V~ ~ V and → − V~ +V~ . axial → axial In QM, parity is represented by a unitary operator such that P ~r = ~r and • | i | − i P 2 = 1. So the eigenvectors of P have eigenvalue 1. A state is even or odd under parity ± if P ψ = ψ ; in position space, ψ( ~x) = ψ(~x). In radial coordinates, parity takes | i ±| i − ± cos θ cos θ and φ φ + π, so P α,ℓ,m =( 1)ℓ α,ℓ,m . This is clear also from the → − → | i − | i ℓ = 1 case, any getting general ℓ from tensor products. If [H,P ] = 0, then the non-degenerate energy eigenstates can be written as P eigen- states. Parity in 1d: PxP = x and P pP = p. The Hamiltonian respects parity if V ( x) = − − − V (x). -
Galois Connections
1 Galois Connections Roland Backhouse 3rd December, 2002 2 Fusion Many problems are expressed in the form evaluate generate ◦ where generate generates a (possibly infinite) candidate set of solutions, and evaluate selects a best solution. Examples: shortest path ; ◦ (x ) L: 2 ◦ Solution method is to fuse the generation and evaluation processes, eliminating the need to generate all candidate solutions. 3 Conditions for Fusion Fusion is made possible when evaluate is an adjoint in a Galois connection, • generate is expressed as a fixed point. • Solution method typically involves generalising the problem. 4 Definition Suppose = (A; ) and = (B; ) are partially ordered sets and A v B suppose F A B and G B A . Then (F;G) is a Galois connection 2 2 of and iff, for all x B and y A, A B 2 2 F:x y x G:y : v ≡ F is called the lower adjoint. G is the upper adjoint. 5 Examples | Propositional Calculus :p q p :q ≡ p ^ q r q (p r) ) ≡ ( p _ q r (p r) ^ (q r) ) ≡ ) ) p q _ r p ^ :q r ) ≡ ) ) ) ) Examples | Set Theory :S T S :T ⊆ ≡ ⊇ S T U T :S U \ ⊆ ≡ ⊆ [ S T U S U ^ T U [ ⊆ ≡ ⊆ ⊆ S T U S :T U ⊆ [ ≡ \ ⊆ 6 Examples | Number Theory -x y x -y ≤ ≡ ≥ x+y z y z-x ≤ ≡ ≤ x n x n d e ≤ ≡ ≤ x y z x z ^ y z ≤ ≡ ≤ ≤ x y z x z=y for all y > 0 ×" ≤ ≡ ≤ 7 Examples | predicates even:m b (if b then 2 else 1) n m ≡ odd:m b (if b then 1 else 2) n m ( ≡ x S b S if b then fxg else φ 2 ) ≡ ⊇ x S b S if b then U else Unfxg 2 ( ≡ ⊆ S = φ b S if b then φ else U ) ≡ ⊆ S = φ b S if b then U else φ 6 ( ≡ ⊆ ) 8 Examples | programming algebra Languages ("factors") L M N L N=M · ⊆ ≡ -
An Algebraic Theory of Complexity for Discrete Optimization Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny
An Algebraic Theory of Complexity for Discrete Optimization Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny To cite this version: Davis Cohen, Martin Cooper, Paidi Creed, Peter Jeavons, Stanislas Zivny. An Algebraic Theory of Complexity for Discrete Optimization. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2013, vol. 42 (n° 5), pp. 1915-1939. 10.1137/130906398. hal-01122748 HAL Id: hal-01122748 https://hal.archives-ouvertes.fr/hal-01122748 Submitted on 4 Mar 2015 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. Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 12741 To link to this article : DOI :10.1137/130906398 URL : http://dx.doi.org/10.1137/130906398 To cite this version : Cohen, Davis and Cooper, Martin C. and Creed, Paidi and Jeavons, Peter and Zivny, Stanislas An Algebraic Theory of Complexity for Discrete Optimization. (2013) SIAM Journal on Computing, vol. 42 (n° 5). -
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”.