# CMPSCI 250: Introduction to Computation. Lecture #19: Regular Expressions and Their Languages David Mix Barrington 11 April 2013

Save this PDF as:

Size: px
Start display at page:

Download "CMPSCI 250: Introduction to Computation. Lecture #19: Regular Expressions and Their Languages David Mix Barrington 11 April 2013"

## Transcription

1 CMPSCI 250: Introduction to Computation Lecture #19: Regular Expressions and Their Languages David Mix Barrington 11 April 2013

2 Regular Expressions and Their Languages Alphabets, Strings and Languages Regular Expressions The Kleene Star Operation Finite Languages The Language (a + ab)* Logically Describable Languages Languages From Number Theory

3 Alphabets, Strings, and Languages We re going to finish the course by doing a bit of formal language theory, the branch of mathematics that deals with finite strings and with languages, where a language is defined to be a set of strings. If Σ is a finite alphabet (Rosen usually uses {0, 1}, while my book often uses {a, b}), then a string over Σ is defined to be a sequence of zero or more letters, each letter being an element of Σ. There is a special empty string, called λ, which has no letters. Two strings are equal if they have the same letters in the same order. So there is one empty string, because any two strings with no letters are equal to each other. The length of a string is the number of letters in it. The set of all finite strings over Σ is called Σ*. A language is any subset of Σ*, that is, any set of strings. Languages may be finite or infinite. We can use the operators of set theory on languages, and set notation to describe them.

4 Regular Expressions A regular expression is a string that denotes a particular language. They are defined recursively, much as boolean expressions or arithmetic expressions are defined. A regular language is a set of strings that is denoted by a regular expression. We ll see soon that some languages are regular and some are not. The regular expressions you may have seen in programming languages and operating systems are related to this concept, but the notation will be different. For example, if in Unix you want to delete all files whose names begin with q, you would say rm q*. The set of all strings beginning with q is denoted by the regular expression qσ*. Our main result about regular expressions will be Kleene s Theorem, which says that a language can be denoted by a regular expression if and only if it can be decided by a finite-state machine. We ll define finite-state machines next lecture, and prove Kleene s Theorem by the end of the course.

5 Formal Language Theory Why would we care about something like this? The study of formal languages began as an outgrowth of the study of natural languages, those such as English or Chinese that are used by humans. Natural languages obey grammatical rules, and human brains are able to determine whether different utterances are valid in the language, and if so what they mean. Computers can determine whether statements are valid for a programming language, and if so what they mean. We d like to understand how the humans do it, in part by designing computers to do similar things. There s a general phenomenon where the set of languages that can be denoted in a certain way coincides with the set that can be decided in a certain way. Kleene s Theorem is our first example, and you will see others in CMPSCI 401. Formal language theory gives a mathematical way to reason about the resources needed to carry out computational tasks.

6 The Formal Inductive Definition Fix an alphabet Σ. A regular expression over Σ is a string over the alphabet Σ {, +,, *, (, )} that can be built by the rules below. Each regular expression R denotes a language L(R), also determined by the rules below. is a regular expression and denotes the empty set. If a is any letter in Σ, then a is a regular expression and denotes the language {a}. If R and S are two regular expressions denoting languages L(R) and L(S), then R S (often written RS ) is a regular expression denoting the concatenation L(R)L(S), and R+S is a regular expression denoting the union L(R) L(S). If R is a regular expression denoting the language L(R), then R* is a regular expression denoting the Kleene star of L(R), which is written L(R)*. Nothing else is a regular expression.

7 iclicker Question #1: Concatenation of Languages If X is the language {ab, aba} and Y is the language {a, ba, aba}, what is XY? (a) {a, ab, ba, aba} (b) {aab, aaba, baab, baaba, abaab, abaaba} (c) {aba, ababa, abaaba} (d) {aba, abba, ababa, abaa, abaaba}

8 iclicker Question #2: Valid Regular Expressions Which of these is not a valid regular expression for the alphabet {0, 1}, according to the rules on the previous slide? (a) (010)* + 0*1(01)*1 (b) * + ( *)* (c) 101* + 10(*01)* (d) (011)**

9 The Kleene Star Operation If A is any language, the Kleene star of A, written A*, is the set of all strings that can be written as the concatenation of zero or more strings from A. If A =, A* = {λ} because we can only have a concatenation of zero strings from A. If A = {a}, then A* = {λ, a, aa, aaa, aaaa,...}, the set of all strings of a s. If A = Σ, then A* is just Σ*, so the star notation we have been using for Σ* is just this same Kleene star operation. A string over Σ is just the concatenation of zero or more letters from Σ. In general A* is the union of the languages A 0, A 1, A 2, A 3,... where A 0 = {λ}, A 1 = A, A 2 = AA, A 3 = AAA, and so on. (Note that some of the laws of exponents still work, like A i A j = A i+j and (A i ) j = A ij.)

10 iclicker Question #3: Kleene Star What is the relationship between the languages denoted by a*b* and (ab)*? (a) they are equal (b) a*b* (ab)* (c) (ab)* a*b* (d) None of the above

11 Finite Languages The regular expression aba denotes the concatenation {a}{b}{a} = {aba}, by the definition of concatenation of languages. Thus any language consisting of a single non-empty string has a regular expression, which is (up to a type cast) itself. The language {λ}, as we just saw, can be written *. If I have any finite language, I can denote it by a regular expression, as the union of the one-string languages for each of the strings in it. For example, the finite language {λ, aba, abbb, b} is denoted by the regular expression * + aba + abbb + b. (Note that + is not the Java concatenation operator!) A regular expression that never uses the star operator must denote a finite set of non-empty strings. (We can prove this fact using induction!) If we use the star operator on any language that contains a non-empty string, the result is an infinite language, such as (aa)* = {λ, aa, aaaa, aaaaaa,...}.

12 The Language (a + ab)* Here is a more interesting regular language, denoted by the regular expression (a + ab)*. (Note that the parentheses are important -- a + ab* and a + (ab)* denote quite different languages.) The strings in (a + ab)* are exactly those strings that can be made by concatenating zero or more strings, each of which is equal to either a or ab. We can systematically list (a + ab)* by listing (a + ab) i in turn for each natural i. We get (a + ab) 0 = {λ}, (a + ab) 1 = {a, ab}, (a + ab) 2 = {aa, aab, aba, abab}, (a + ab) 3 = (aaa, aaab, aaba, aabab, abaa, abaab, ababa, ababab}, and so forth. How can we tell whether a given string of a s and b s is in (a + ab)*? If it ends in a, we know that the last string used in the concatenation was a, and if it ends in b, the last string used was ab. So we can delete a s and ab s from the right as long as we can, and if we produce λ then the string was in the language. It turns out that (a + ab)* is the set of strings that don t begin with b and never have two b s in a row. (How would you prove this assertion?)

13 Logically Describable Languages We can say the first letter is not b and there are never two b s in a row in the predicate calculus. One way to do it is to have variables that range over positions in the string. Our atomic predicates are Ca(x) ( position x contains an a, Cb(x) ( position x contains a b ), x = y ( x and y are the same position ), and x < y ( x is to the left of y ). So we can say that the first letter is not b, x: Cb(x) y: x y, and that there are never two b s in a row, x: y: Cb(x) Cb(y) (x < y) z: (z x) (z y). One way to say both things at once is x: Cb(x) y: Pred(x, y) Ca(y), where Pred(x, y) abbreviates (x < y) z: (z x) (z y). In last year s honors section of CMPSCI 401, we proved that a language is logically describable in this way if and only if it has a certain kind of regular expression, and that (aa)* is not logically describable. Prof. Immerman was the prime developer of an extensive theory about more complicated logical descriptions.

14 Languages From Number Theory We can easily make a regular expression for the set of even-length strings of a s, (aa)*, or the odd-length strings of a s, (aa)*a, or the set of strings of a s whose length is congruent to 3 modulo 7, a 3 (a 7 )*, or the set of strings whose length is congruent to 1, 2, or 5 modulo 6, (a + a 2 + a 5 )(a 6 )*. What about the set of strings over {a,b} that have an even number of a s? A good first guess is that such a string is a concatenation of zero or more strings, each of which has exactly two a s. This would be the language (b*ab*ab*)*. But this isn t exactly right, because bb, for example, has 0 a s and 0 is even. A correct regular expression for this language is (b + ab*a)* -- we can divide any such string into pieces which either have exactly two a s (with some number of b s between) or are just b s themselves. It s harder to get a number of a s congruent to 3 mod 7, or the strings with an even number of a s and an even number of b s, but both are possible.

15 iclicker Question #4: Counting Languages Which regular expression denotes the same language as (aa)* + (aaa)*? Note that we can also write this language as {a i : i is even or i is divisible by 3}. (a) (aaaaa)* (b) ( * + aa + aaa + aaaa)(aaaaaa)* (c) (aa)*(aaa)* (d) (aaaaaa)*

### CS103B Handout 17 Winter 2007 February 26, 2007 Languages and Regular Expressions

CS103B Handout 17 Winter 2007 February 26, 2007 Languages and Regular Expressions Theory of Formal Languages In the English language, we distinguish between three different identities: letter, word, sentence.

### CS F-02 Formal Languages 1

CS411 2015F-02 Formal Languages 1 02-0: Alphabets & Strings An alphabetσis a finite set of symbols Σ 1 = {a, b,..., z} Σ 2 = {0, 1} A string is a finite sequence of symbols from an alphabet fire, truck

### Fundamentele Informatica II

Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### C H A P T E R Regular Expressions regular expression

7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun

### 2110711 THEORY of COMPUTATION

2110711 THEORY of COMPUTATION ATHASIT SURARERKS ELITE Athasit Surarerks ELITE Engineering Laboratory in Theoretical Enumerable System Computer Engineering, Faculty of Engineering Chulalongkorn University

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch

6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 4 Nancy Lynch Today Two more models of computation: Nondeterministic Finite Automata (NFAs)

### Regular Languages and Finite State Machines

Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries - some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection

### (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.

3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### CS 341 Homework 9 Languages That Are and Are Not Regular

CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w

### Automata Theory and Languages

Automata Theory and Languages SITE : http://www.info.univ-tours.fr/ mirian/ Automata Theory, Languages and Computation - Mírian Halfeld-Ferrari p. 1/1 Introduction to Automata Theory Automata theory :

### the lemma. Keep in mind the following facts about regular languages:

CPS 2: Discrete Mathematics Instructor: Bruce Maggs Assignment Due: Wednesday September 2, 27 A Tool for Proving Irregularity (25 points) When proving that a language isn t regular, a tool that is often

### Chapter 2. Finite Automata. 2.1 The Basic Model

Chapter 2 Finite Automata 2.1 The Basic Model Finite automata model very simple computational devices or processes. These devices have a constant amount of memory and process their input in an online manner.

### Regular Expressions. Languages. Recall. A language is a set of strings made up of symbols from a given alphabet. Computer Science Theory 2

Regular Expressions Languages Recall. A language is a set of strings made up of symbols from a given alphabet. Computer Science Theory 2 1 String Recognition Machine Given a string and a definition of

### Regular Languages and Finite Automata

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

### Factorizations: Searching for Factor Strings

" 1 Factorizations: Searching for Factor Strings Some numbers can be written as the product of several different pairs of factors. For example, can be written as 1, 0,, 0, and. It is also possible to write

### Pushdown Automata. place the input head on the leftmost input symbol. while symbol read = b and pile contains discs advance head remove disc from pile

Pushdown Automata In the last section we found that restricting the computational power of computing devices produced solvable decision problems for the class of sets accepted by finite automata. But along

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Automata and Languages

Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful

### Definition: String concatenation. Definition: String. Definition: Language (cont.) Definition: Language

CMSC 330: Organization of Programming Languages Regular Expressions and Finite Automata Introduction That s it for the basics of Ruby If you need other material for your project, come to office hours or

### Sets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.

Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### If f is a 1-1 correspondence between A and B then it has an inverse, and f 1 isa 1-1 correspondence between B and A.

Chapter 5 Cardinality of sets 51 1-1 Correspondences A 1-1 correspondence between sets A and B is another name for a function f : A B that is 1-1 and onto If f is a 1-1 correspondence between A and B,

### 1 Die hard, once and for all

ENGG 2440A: Discrete Mathematics for Engineers Lecture 4 The Chinese University of Hong Kong, Fall 2014 6 and 7 October 2014 Number theory is the branch of mathematics that studies properties of the integers.

### CAs and Turing Machines. The Basis for Universal Computation

CAs and Turing Machines The Basis for Universal Computation What We Mean By Universal When we claim universal computation we mean that the CA is capable of calculating anything that could possibly be calculated*.

### Announcements. CompSci 230 Discrete Math for Computer Science. Test 1

CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 4-7 Prof. Rodger is out Sep 30-Oct 4 There is Recitation: Sept 27-30.

### Lecture 2: Regular Languages [Fa 14]

Caveat lector: This is the first edition of this lecture note. Please send bug reports and suggestions to jeffe@illinois.edu. But the Lord came down to see the city and the tower the people were building.

### Let s just do some examples to get the feel of congruence arithmetic.

Basic Congruence Arithmetic Let s just do some examples to get the feel of congruence arithmetic. Arithmetic Mod 7 Just write the multiplication table. 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0

### Please note that there is more than one way to answer most of these questions. The following only represents a sample solution.

HOMEWORK THREE SOLUTION CSE 355 DUE: 10 MARCH 2011 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. (1) 2.4 and 2.5 (b),(c),(e),(f):

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### Scanner. tokens scanner parser IR. source code. errors

Scanner source code tokens scanner parser IR errors maps characters into tokens the basic unit of syntax x = x + y; becomes = + ; character string value for a token is a lexeme

### Magic Squares and Modular Arithmetic

Magic Squares and Modular Arithmetic Jim Carlson November 7, 2001 1 Introduction Recall that a magic square is a square array of consecutive distinct numbers such that all row and column sums and are the

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### Context-Free Grammars

Context-Free Grammars Describing Languages We've seen two models for the regular languages: Automata accept precisely the strings in the language. Regular expressions describe precisely the strings in

### Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games V. Halava 1,2 T. Harju 1 R. Niskanen 1 I. Potapov 2 1 Department of Mathematics and Statistics University of Turku, Finland

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Computability Theory

CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.

### Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction

### Computing exponents modulo a number: Repeated squaring

Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Direct Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction

Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Finite Automata and Regular Languages

CHAPTER 3 Finite Automata and Regular Languages 3. Introduction 3.. States and Automata A finite-state machine or finite automaton (the noun comes from the Greek; the singular is automaton, the Greek-derived

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### A set is a Many that allows itself to be thought of as a One. (Georg Cantor)

Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### Homework Three Solution CSE 355

Homework Three Solution CSE 355 Due: 06 March 2012 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. Problem 1: Linz 4.1.13

### Mathematical Logic and its Application to. Computer Science. Lecture Notes

Mathematical Logic and its Application to Computer Science Lecture Notes Eitan Farchi farchi@il.ibm.com Yochai Ben-Chaim yochaib@il.ibm.com March 3, 2010 1 Contents 1 Introduction 9 1.1 The Knaves and

### MACM 101 Discrete Mathematics I

MACM 101 Discrete Mathematics I Exercises on Combinatorics, Probability, Languages and Integers. Due: Tuesday, November 2th (at the beginning of the class) Reminder: the work you submit must be your own.

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Mathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition)

Mathematical Induction Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathmatical Induction Mathmatical induction can be used to prove statements that assert that P(n) is true for all positive

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range {0,1,...,N

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### MATH20302 Propositional Logic. Mike Prest School of Mathematics Alan Turing Building Room

MATH20302 Propositional Logic Mike Prest School of Mathematics Alan Turing Building Room 1.120 mprest@manchester.ac.uk April 10, 2015 Contents I Propositional Logic 3 1 Propositional languages 4 1.1 Propositional

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20

CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of

### Integers and division

CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

### Induction and Recursion

Induction and Recursion Winter 2010 Mathematical Induction Mathematical Induction Principle (of Mathematical Induction) Suppose you want to prove that a statement about an integer n is true for every positive

### MATHS 315 Mathematical Logic

MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................

### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

### Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy

Regular Expressions with Nested Levels of Back Referencing Form a Hierarchy Kim S. Larsen Odense University Abstract For many years, regular expressions with back referencing have been used in a variety

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Exam #5 Sequences and Series

College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 Don-Hickethier@Eureka.redwoods.cc.ca.us Last Revision Date:

### Automata on Infinite Words and Trees

Automata on Infinite Words and Trees Course notes for the course Automata on Infinite Words and Trees given by Dr. Meghyn Bienvenu at Universität Bremen in the 2009-2010 winter semester Last modified:

### Motivation. Automata = abstract computing devices. Turing studied Turing Machines (= computers) before there were any real computers

Motivation Automata = abstract computing devices Turing studied Turing Machines (= computers) before there were any real computers We will also look at simpler devices than Turing machines (Finite State

### List of Standard Counting Problems

List of Standard Counting Problems Advanced Problem Solving This list is stolen from Daniel Marcus Combinatorics a Problem Oriented Approach (MAA, 998). This is not a mathematical standard list, just a

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Introduction to mathematical arguments

Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math

### Turing Machines: An Introduction

CIT 596 Theory of Computation 1 We have seen several abstract models of computing devices: Deterministic Finite Automata, Nondeterministic Finite Automata, Nondeterministic Finite Automata with ɛ-transitions,

### Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB

Problem Set 5 1. (a) Four-digit number S = aabb is a square. Find it; (hint: 11 is a factor of S) (b) If n is a sum of two square, so is 2n. (Frank) Solution: (a) Since (A + B) (A + B) = 0, and 11 0, 11

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li Kishan Mehrotra 3 L A TEX at July 18, 007 1 No part of this book can be reproduced without permission from the authors Illinois State University, Normal,

### Lecture 1. Basic Concepts of Set Theory, Functions and Relations

September 7, 2005 p. 1 Lecture 1. Basic Concepts of Set Theory, Functions and Relations 0. Preliminaries...1 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### Properties of Context-Free Languages. Decision Properties Closure Properties

Properties of Context-Free Languages Decision Properties Closure Properties 1 Summary of Decision Properties As usual, when we talk about a CFL we really mean a representation for the CFL, e.g., a CFG

### Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

### Notes. Sets. Notes. Introduction II. Notes. Definition. Definition. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry.

Sets Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### In mathematics there are endless ways that two entities can be related

CHAPTER 11 Relations In mathematics there are endless ways that two entities can be related to each other. Consider the following mathematical statements. 5 < 10 5 5 6 = 30 5 5 80 7 > 4 x y 8 3 a b ( mod

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Chapter 3. if 2 a i then location: = i. Page 40

Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)

### Automata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi

Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the

### Course notes on Number Theory

Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that

### Basic Properties of Rings

Basic Properties of Rings A ring is an algebraic structure with an addition operation and a multiplication operation. These operations are required to satisfy many (but not all!) familiar properties. Some

### SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections