DOCSLIB.ORG

Sets, Logic, Computation an Open Logic Text

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sets, Logic, Computation an Open Logic Text Sets, Logic, Computation An Open Logic Text Winter 2017 Sets, Logic, Computation The Open Logic Project Instigator Richard Zach, University of Calgary Editorial Board Aldo Antonelli,y University of California, Davis Andrew Arana, Université Paris I Panthénon–Sorbonne Jeremy Avigad, Carnegie Mellon University Walter Dean, University of Warwick Gillian Russell, University of North Carolina Nicole Wyatt, University of Calgary Audrey Yap, University of Victoria Contributors Samara Burns, University of Calgary Dana Hägg, University of Calgary Sets, Logic, Computation An Open Logic Text Remixed by Richard Zach Winter 2017 The Open Logic Project would like to acknowledge the generous support of the Faculty of Arts and the Taylor Institute of Teaching and Learning of the University of Calgary. This resource was funded by the Alberta Open Educational Re- sources (ABOER) Initiative, which is made possible through an investment from the Alberta government. Illustrations by Matthew Leadbeater, used under a Creative Com- mons Attribution-NonCommercial 4.0 International License. Typeset in Baskervald X and Universalis ADF Standard by LATEX. Sets, Logic, Computation by Richard Zach is licensed under a Creative Commons Attribution 4.0 Interna- tional License. It is based on The Open Logic Text by the Open Logic Project, used under a Creative Commons At- tribution 4.0 International License. Contents Preface xi I Sets, Relations, Functions1 1 Sets2 1.1 Basics......................... 2 1.2 Some Important Sets................ 4 1.3 Subsets........................ 5 1.4 Unions and Intersections.............. 6 1.5 Proofs about Sets .................. 8 1.6 Pairs, Tuples, Cartesian Products ......... 10 1.7 Russell’s Paradox .................. 11 Summary.......................... 12 Problems .......................... 12 2 Relations 14 2.1 Relations as Sets................... 14 2.2 Special Properties of Relations........... 16 2.3 Orders ........................ 18 2.4 Graphs........................ 21 2.5 Operations on Relations .............. 22 Summary.......................... 23 Problems .......................... 24 v CONTENTS vi 3 Functions 26 3.1 Basics......................... 26 3.2 Kinds of Functions ................. 28 3.3 Inverses of Functions................ 29 3.4 Composition of Functions ............. 30 3.5 Isomorphism..................... 31 3.6 Partial Functions................... 31 3.7 Functions and Relations .............. 32 Summary.......................... 33 Problems .......................... 34 4 The Size of Sets 35 4.1 Introduction..................... 35 4.2 Countable Sets.................... 35 4.3 Uncountable Sets .................. 41 4.4 Reduction ...................... 45 4.5 Equinumerous Sets ................. 46 4.6 Comparing Sizes of Sets .............. 48 Summary.......................... 50 Problems .......................... 51 II First-order Logic 55 5 Syntax and Semantics 56 5.1 Introduction..................... 56 5.2 First-Order Languages ............... 58 5.3 Terms and Formulas ................ 60 5.4 Unique Readability ................. 63 5.5 Main operator of a Formula ............ 67 5.6 Subformulas..................... 68 5.7 Free Variables and Sentences............ 70 5.8 Substitution ..................... 71 5.9 Structures for First-order Languages . 73 5.10 Covered Structures for First-order Languages . 75 5.11 Satisfaction of a Formula in a Structure . 76 CONTENTS vii 5.12 Extensionality.................... 82 5.13 Semantic Notions.................. 83 Summary.......................... 85 Problems .......................... 87 6 Theories and Their Models 89 6.1 Introduction..................... 89 6.2 Expressing Properties of Structures . 92 6.3 Examples of First-Order Theories......... 93 6.4 Expressing Relations in a Structure . 96 6.5 The Theory of Sets................. 98 6.6 Expressing the Size of Structures . 101 Summary..........................103 Problems ..........................103 7 Natural Deduction 105 7.1 Introduction.....................105 7.2 Rules and Derivations . 107 7.3 Examples of Derivations . 111 7.4 Proof-Theoretic Notions . 120 7.5 Properties of Derivability . 121 7.6 Soundness......................125 7.7 Derivations with Identity predicate . 128 7.8 Soundness of Identity predicate Rules . 130 Summary..........................130 Problems ..........................131 8 The Completeness Theorem 133 8.1 Introduction.....................133 8.2 Outline of the Proof . 134 8.3 Maximally Consistent Sets of Sentences . 136 8.4 Henkin Expansion..................138 8.5 Lindenbaum’s Lemma . 140 8.6 Construction of a Model . 141 8.7 Identity........................143 8.8 The Completeness Theorem . 147 CONTENTS viii 8.9 The Compactness Theorem . 147 8.10 The Löwenheim-Skolem Theorem . 150 Summary..........................152 Problems ..........................153 9 Beyond First-order Logic 154 9.1 Overview.......................154 9.2 Many-Sorted Logic . 155 9.3 Second-Order logic . 157 9.4 Higher-Order logic . 162 9.5 Intuitionistic Logic . 165 9.6 Modal Logics ....................171 9.7 Other Logics.....................173 III Turing Machines 177 10 Turing Machine Computations 178 10.1 Introduction.....................178 10.2 Representing Turing Machines . 181 10.3 Turing Machines...................186 10.4 Configurations and Computations . 187 10.5 Unary Representation of Numbers . 189 10.6 Halting States....................190 10.7 Combining Turing Machines . 191 10.8 Variants of Turing Machines . 193 10.9 The Church-Turing Thesis . 195 Summary..........................196 Problems ..........................197 11 Undecidability 199 11.1 Introduction.....................199 11.2 Enumerating Turing Machines . 201 11.3 The Halting Problem . 203 11.4 The Decision Problem . 205 11.5 Representing Turing Machines . 206 CONTENTS ix 11.6 Verifying the Representation . 209 11.7 The Decision Problem is Unsolvable . 214 Summary..........................215 Problems ..........................216 A Induction 219 A.1 Introduction.....................219 A.2 Induction on N . 220 A.3 Strong Induction ..................223 A.4 Inductive Definitions . 224 A.5 Structural Induction . 227 B Biographies 229 B.1 Georg Cantor ....................229 B.2 Alonzo Church ...................230 B.3 Gerhard Gentzen ..................231 B.4 Kurt Gödel......................233 B.5 Emmy Noether ...................235 B.6 Bertrand Russell...................236 B.7 Alfred Tarski.....................238 B.8 Alan Turing .....................239 B.9 Ernst Zermelo....................241 Glossary 245 Photo Credits 251 Bibliography 253 About the Open Logic Project 258 Preface This book is an introduction to meta-logic, aimed especially at students of computer science and philosophy. “Meta-logic” is so- called because it is the discipline that studies logic itself. Logic proper is concerned with canons of valid inference, and its sym- bolic or formal version presents these canons using formal lan- guages, such as those of propositional and predicate, a.k.a., first- order logic. Meta-logic investigates the properties of these lan- guage, and of the canons of correct inference that use them. It studies topics such as how to give precise meaning to the ex- pressions of these formal languages, how to justify the canons of valid inference, what the properties of various proof systems are, including their computational properties. These questions are important and interesting in their own right, because the lan- guages and proof systems investigated are applied in many dier- ent areas—in mathematics, philosophy, computer science, and linguistics, especially—but they also serve as examples of how to study formal systems in general. The logical languages we study here are not the only ones people are interested in. For instance, linguists and philosophers are interested in languages that are much more complicated than those of propositional and first-order logic, and computer scientists are interested in other kinds of languages altogether, such as programming languages. And the methods we discuss here—how to give semantics for for- mal languages, how to prove results about formal languages, how xi PREFACE xii to investigate the properties of formal languages—are applicable in those cases as well. Like any discipline, meta-logic both has a set of results or facts, and a store of methods and techniques, and this text cov- ers both. Some students won’t need to know some of the results we discuss outside of this course, but they will need and use the methods we use to establish them. The Löwenheim-Skolem the- orem, say, does not often make an appearance in computer sci- ence, but the methods we use to prove it do. On the other hand, many of the results we discuss do have relevance for certain de- bates, say, in the philosophy of science and in metaphysics. Phi- losophy students may not need to be able to prove these results outside this course, but they do need to understand what the results are—and you really only understand these results if you have thought through the definitions and proofs needed to es- tablish them. These are, in part, the reasons for why the results and the methods covered in this text are recommended study—in some cases even required—for students of computer science and philosophy. The material is divided into three parts. Part 1 concerns it- self with the theory of sets. Logic and meta-logic is historically connected very closely to what’s called the “foundations of math- ematics.” Mathematical
Load more

Recommended publications
  • Basic Structures: Sets, Functions, Sequences, and Sums 2-2
    CHAPTER Basic Structures: Sets, Functions, 2 Sequences, and Sums 2.1 Sets uch of discrete mathematics is devoted to the study of discrete structures, used to represent discrete objects. Many important discrete structures are built using sets, which 2.2 Set Operations M are collections of objects. Among the discrete structures built from sets are combinations, 2.3 Functions unordered collections of objects used extensively in counting; relations, sets of ordered pairs that represent relationships between objects; graphs, sets of vertices and edges that connect 2.4 Sequences and vertices; and finite state machines, used to model computing machines. These are some of the Summations topics we will study in later chapters. The concept of a function is extremely important in discrete mathematics. A function assigns to each element of a set exactly one element of a set. Functions play important roles throughout discrete mathematics. They are used to represent the computational complexity of algorithms, to study the size of sets, to count objects, and in a myriad of other ways. Useful structures such as sequences and strings are special types of functions. In this chapter, we will introduce the notion of a sequence, which represents ordered lists of elements. We will introduce some important types of sequences, and we will address the problem of identifying a pattern for the terms of a sequence from its first few terms. Using the notion of a sequence, we will define what it means for a set to be countable, namely, that we can list all the elements of the set in a sequence.
    [Show full text]
  • I0 and Rank-Into-Rank Axioms
    I0 and rank-into-rank axioms Vincenzo Dimonte∗ July 11, 2017 Abstract Just a survey on I0. Keywords: Large cardinals; Axiom I0; rank-into-rank axioms; elemen- tary embeddings; relative constructibility; embedding lifting; singular car- dinals combinatorics. 2010 Mathematics Subject Classifications: 03E55 (03E05 03E35 03E45) 1 Informal Introduction to the Introduction Ok, that’s it. People who know me also know that it is years that I am ranting about a book about I0, how it is important to write it and publish it, etc. I realized that this particular moment is still right for me to write it, for two reasons: time is scarce, and probably there are still not enough results (or anyway not enough different lines of research). This has the potential of being a very nasty vicious circle: there are not enough results about I0 to write a book, but if nobody writes a book the diffusion of I0 is too limited, and so the number of results does not increase much. To avoid this deadlock, I decided to divulge a first draft of such a hypothetical book, hoping to start a “conversation” with that. It is literally a first draft, so it is full of errors and in a liquid state. For example, I still haven’t decided whether to call it I0 or I0, both notations are used in literature. I feel like it still lacks a vision of the future, a map on where the research could and should going about I0. Many proofs are old but never published, and therefore reconstructed by me, so maybe they are wrong.
    [Show full text]
  • Domain and Range of a Function
    4.1 Domain and Range of a Function How can you fi nd the domain and range of STATES a function? STANDARDS MA.8.A.1.1 MA.8.A.1.5 1 ACTIVITY: The Domain and Range of a Function Work with a partner. The table shows the number of adult and child tickets sold for a school concert. Input Number of Adult Tickets, x 01234 Output Number of Child Tickets, y 86420 The variables x and y are related by the linear equation 4x + 2y = 16. a. Write the equation in function form by solving for y. b. The domain of a function is the set of all input values. Find the domain of the function. Domain = Why is x = 5 not in the domain of the function? 1 Why is x = — not in the domain of the function? 2 c. The range of a function is the set of all output values. Find the range of the function. Range = d. Functions can be described in many ways. ● by an equation ● by an input-output table y 9 ● in words 8 ● by a graph 7 6 ● as a set of ordered pairs 5 4 Use the graph to write the function 3 as a set of ordered pairs. 2 1 , , ( , ) ( , ) 0 09321 45 876 x ( , ) , ( , ) , ( , ) 148 Chapter 4 Functions 2 ACTIVITY: Finding Domains and Ranges Work with a partner. ● Copy and complete each input-output table. ● Find the domain and range of the function represented by the table. 1 a. y = −3x + 4 b. y = — x − 6 2 x −2 −10 1 2 x 01234 y y c.
    [Show full text]
  • Adaptive Lower Bound for Testing Monotonicity on the Line
    Adaptive Lower Bound for Testing Monotonicity on the Line Aleksandrs Belovs∗ Abstract In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of ε-testing monotonicity of a function f :[n] → [r]. All our lower bounds are for adaptive two-sided testers. log r • We prove a nearly tight lower bound for this problem in terms of r. TheboundisΩ log log r when ε =1/2. No previous satisfactory lower bound in terms of r was known. • We completely characterise query complexity of this problem in terms of n for smaller values of ε. The complexity is Θ ε−1 log(εn) . Apart from giving the lower bound, this improves on the best known upper bound. Finally, we give an alternative proof of the Ω(ε−1d log n−ε−1 log ε−1) lower bound for testing monotonicity on the hypergrid [n]d due to Chakrabarty and Seshadhri (RANDOM’13). 1 Introduction The framework of property testing was formulated by Rubinfeld and Sudan [19] and Goldreich et al. [16]. A property testing problem is specified by a property P, which is a class of functions mapping some finite set D into some finite set R, and proximity parameter ε, which is a real number between 0 and 1. An ε-tester is a bounded-error randomised query algorithm which, given oracle access to a function f : D → R, distinguishes between the case when f belongs to P and the case when f is ε-far from P.
    [Show full text]
  • Elementary Functions Part 1, Functions Lecture 1.6D, Function Inverses: One-To-One and Onto Functions
    Elementary Functions Part 1, Functions Lecture 1.6d, Function Inverses: One-to-one and onto functions Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 26 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be \one-to-one" and \onto". Smith (SHSU) Elementary Functions 2013 27 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output. Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C: Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions* A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25.
    [Show full text]
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show full text]
  • Module 1 Lecture Notes
    Module 1 Lecture Notes Contents 1.1 Identifying Functions.............................1 1.2 Algebraically Determining the Domain of a Function..........4 1.3 Evaluating Functions.............................6 1.4 Function Operations..............................7 1.5 The Difference Quotient...........................9 1.6 Applications of Function Operations.................... 10 1.7 Determining the Domain and Range of a Function Graphically.... 12 1.8 Reading the Graph of a Function...................... 14 1.1 Identifying Functions In previous classes, you should have studied a variety basic functions. For example, 1 p f(x) = 3x − 5; g(x) = 2x2 − 1; h(x) = ; j(x) = 5x + 2 x − 5 We will begin this course by studying functions and their properties. As the course progresses, we will study inverse, composite, exponential, logarithmic, polynomial and rational functions. Math 111 Module 1 Lecture Notes Definition 1: A relation is a correspondence between two variables. A relation can be ex- pressed through a set of ordered pairs, a graph, a table, or an equation. A set containing ordered pairs (x; y) defines y as a function of x if and only if no two ordered pairs in the set have the same x-coordinate. In other words, every input maps to exactly one output. We write y = f(x) and say \y is a function of x." For the function defined by y = f(x), • x is the independent variable (also known as the input) • y is the dependent variable (also known as the output) • f is the function name Example 1: Determine whether or not each of the following represents a function. Table 1.1 Chicken Name Egg Color Emma Turquoise Hazel Light Brown George(ia) Chocolate Brown Isabella White Yvonne Light Brown (a) The set of ordered pairs of the form (chicken name, egg color) shown in Table 1.1.
    [Show full text]
  • A Denotational Semantics Approach to Functional and Logic Programming
    A Denotational Semantics Approach to Functional and Logic Programming TR89-030 August, 1989 Frank S.K. Silbermann The University of North Carolina at Chapel Hill Department of Computer Science CB#3175, Sitterson Hall Chapel Hill, NC 27599-3175 UNC is an Equal OpportunityjAfflrmative Action Institution. A Denotational Semantics Approach to Functional and Logic Programming by FrankS. K. Silbermann A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in par­ tial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science. Chapel Hill 1989 @1989 Frank S. K. Silbermann ALL RIGHTS RESERVED 11 FRANK STEVEN KENT SILBERMANN. A Denotational Semantics Approach to Functional and Logic Programming (Under the direction of Bharat Jayaraman.) ABSTRACT This dissertation addresses the problem of incorporating into lazy higher-order functional programming the relational programming capability of Horn logic. The language design is based on set abstraction, a feature whose denotational semantics has until now not been rigorously defined. A novel approach is taken in constructing an operational semantics directly from the denotational description. The main results of this dissertation are: (i) Relative set abstraction can combine lazy higher-order functional program­ ming with not only first-order Horn logic, but also with a useful subset of higher­ order Horn logic. Sets, as well as functions, can be treated as first-class objects. (ii) Angelic powerdomains provide the semantic foundation for relative set ab­ straction. (iii) The computation rule appropriate for this language is a modified parallel­ outermost, rather than the more familiar left-most rule. (iv) Optimizations incorporating ideas from narrowing and resolution greatly improve the efficiency of the interpreter, while maintaining correctness.
    [Show full text]
  • The Domain of Solutions to Differential Equations
    connect to college success™ The Domain of Solutions To Differential Equations Larry Riddle available on apcentral.collegeboard.com connect to college success™ www.collegeboard.com The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 4,700 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three and a half million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. Equity Policy Statement The College Board and the Advanced Placement Program encourage teachers, AP Coordinators, and school administrators to make equitable access a guiding principle for their AP programs. The College Board is committed to the principle that all students deserve an opportunity to participate in rigorous and academically challenging courses and programs. All students who are willing to accept the challenge of a rigorous academic curriculum should be considered for admission to AP courses. The Board encourages the elimination of barriers that restrict access to AP courses for students from ethnic, racial, and socioeconomic groups that have been traditionally underrepresented in the AP Program.
    [Show full text]
  • Monomorphism - Wikipedia, the Free Encyclopedia
    Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G.
    [Show full text]
  • Math 475 Homework #3 March 1, 2010 Section 4.6
    Student: Yu Cheng (Jade) Math 475 Homework #3 March 1, 2010 Section 4.6 Exercise 36-a Let ͒ be a set of ͢ elements. How many different relations on ͒ are there? Answer: On set ͒ with ͢ elements, we have the following facts. ) Number of two different element pairs ƳͦƷ Number of relations on two different elements ) ʚ͕, ͖ʛ ∈ ͌, ʚ͖, ͕ʛ ∈ ͌ 2 Ɛ ƳͦƷ Number of relations including the reflexive ones ) ʚ͕, ͕ʛ ∈ ͌ 2 Ɛ ƳͦƷ ƍ ͢ ġ Number of ways to select these relations to form a relation on ͒ 2ͦƐƳvƷͮ) ͦƐ)! ġ ͮ) ʚ ʛ v 2ͦƐƳvƷͮ) Ɣ 2ʚ)ͯͦʛ!Ɛͦ Ɣ 2) )ͯͥ ͮ) Ɣ 2) . b. How many of these relations are reflexive? Answer: We still have ) number of relations on element pairs to choose from, but we have to 2 Ɛ ƳͦƷ ƍ ͢ include the reflexive one, ʚ͕, ͕ʛ ∈ ͌. There are ͢ relations of this kind. Therefore there are ͦƐ)! ġ ġ ʚ ʛ 2ƳͦƐƳvƷͮ)Ʒͯ) Ɣ 2ͦƐƳvƷ Ɣ 2ʚ)ͯͦʛ!Ɛͦ Ɣ 2) )ͯͥ . c. How many of these relations are symmetric? Answer: To select only the symmetric relations on set ͒ with ͢ elements, we have the following facts. ) Number of symmetric relation pairs between two elements ƳͦƷ Number of relations including the reflexive ones ) ʚ͕, ͕ʛ ∈ ͌ ƳͦƷ ƍ ͢ ġ Number of ways to select these relations to form a relation on ͒ 2ƳvƷͮ) )! )ʚ)ͯͥʛ )ʚ)ͮͥʛ ġ ͮ) ͮ) 2ƳvƷͮ) Ɣ 2ʚ)ͯͦʛ!Ɛͦ Ɣ 2 ͦ Ɣ 2 ͦ . d.
    [Show full text]
  • Extending Partial Isomorphisms of Finite Graphs
    S´eminaire de Master 2`emeann´ee Sous la direction de Thomas Blossier C´edricMilliet — 1er d´ecembre 2004 Extending partial isomorphisms of finite graphs Abstract : This paper aims at proving a theorem given in [4] by Hrushovski concerning finite graphs, and gives a generalization (with- out any proof) of this theorem for a finite structure in a finite relational language. Hrushovski’s therorem is the following : any finite graph G embeds in a finite supergraph H so that any local isomorphism of G extends to an automorphism of H. 1 Introduction In this paper, we call finite graph (G, R) any finite structure G with one binary symmetric reflexive relation R (that is ∀x ∈ G xRx and ∀x, y xRy =⇒ yRx). We call vertex of such a graph any point of G, and edge, any couple (x, y) such that xRy. Geometrically, a finite graph (G, R) is simply a finite set of points, some of them being linked by edges (see picture 1 ). A subgraph (F, R0) of (G, R) is any subset F of G along with the binary relation R0 induced by R on F . H H H J H J H J H J Hqqq HJ Hqqq HJ ¨ ¨ q q ¨ q q q q q Picture 1 — A graph (G, R) and a subgraph (F, R0) of (G, R). We call isomorphism between two graphs (G, R) and (G0,R0) any bijection that preserves the binary relations, that is, any bijection σ that sends an edge on an edge along with σ−1. If (G, R) = (G0,R0), then such a σ is called an automorphism of (G, R).
    [Show full text]