# Regular Languages and Finite Automata

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 model for studying the behavior of the nervous systems. Following up on the ideas of McCulloch and Pitts, Kleene [3] wrote the first paper on finite automata and regular expressions. A finite automaton can be considered as the simplest machine model in that the machine has a finite memory; that is, the memory size is independent of the input length. In a 1959 paper [5], Michael Rabin and Dana Scott presented a comprehensive study of the theory of finite automata, for which they received the Turing Award, the highest award in computer science. In this project, we study from [3] Kleene s concept of finite automata and regular expressions. In particular, we learn Kleene s own proof of the theorem (which is now called Kleene s theorem) that shows finite automata and regular expressions are equivalent in their expressiveness for denoting languages. 1.1 S. C. Kleene Stephen Cole Kleene (born on January 5, 1909 in Hartford, Connecticut; died on January 25, 1994 in Madison, Wisconsin) was an American mathematician who helped lay the foundations for theoretical computer science. He was awarded a Ph.D. in mathematics, under the supervision of Alonzo Church, from Princeton University in Kleene s research was on the theory of algorithms and recursive function theory. He developed the field of recursion theory with Church, Gödel, Turing and others. He contributed to mathematical Intuitionism, which had been founded by Brouwer. By providing methods of determining which problems are soluble, Kleene s work led to the study of which functions can be computed. Kleene developed a diverse array of topics in computability: the arithmetical hierarchy, degrees of computability, computable ordinals and hyperarithmetic 1

2 theory, finite automata and regular sets, computability on higher types, recursive realizability for intuitionistic arithmetic with consequences for philosphy and for program correctness in computer science. 2 Kleene s Finite Automata Model In [3], Kleene investigates McCulloch-Pitts nerve nets. As explained in the excerpt below, his study of nerve nets serves as an illustration of the general theory of finite automata. Finally, we repeat that we are investigating McCulloch-Pitts nerve nets only partly for their own sake as providing a simplified model of nervous activity, but also as an illustration of the general theory of automata, including robots, computing machines and the like. What a finite automaton can and cannot do is thought to be of some mathematical interest intrinsically, and may also contribute to better understanding of problems which arise on the practical level. Below is Kleene s description of the finite automata model: 8.1 CELLS. Time shall consist of a succession of discrete moments numbered by the positive integers, except in Section 10 where all the integers will be used. We shall consider automata constructed of a finite number of parts called cells, each being at each moment in one of a finite number of 2 of states. We shall distinguish two kinds of cells, input cells and inner cells. An input cell admits two states, 0 and 1 (or quiet and firing ), which one is assumed at a given moment being determined by the environment. The restriction to 2 states for input cells makes the notion of input to the automaton coincide with the notion of an input to a nerve net as formulated in Sections 4 and 6.5. But the present theory would work equally well with more than 2 states. Nothing would be gained, however, as p cells each admitting 2 states could be used to replace one cell admitting any number q (2 q 2 p ) of states 0, 1,..., q 1, where if q < 2 p we could either consider only inputs in which states q,..., 2 p 1 do not occur or identify those states with the state q 1 in all the operations of the automaton. 2

3 The number of states of an inner cell is not restricted to 2, and different inner cells may have different numbers of states. The state of each inner cell at any time t > 1 is determined by the states of all the cells at time t 1. Of course it may happen that we do not need to know the states of all the cells at time t 1 to infer the state of a given inner cell at time t. Our formulation merely leaves it unspecified what kind of a law of determination we use, except to say that nothing else than the states of the cells at t 1 can matter. For time beginning with 1, the state of each of the inner cells at that time is to be specified. 8.2 STATE. With k input cells N 1,..., N k (k 0), and m inner cells M 1,..., M m (m 1) with respective number of states s 1,..., s m, there are exactly 2 k s 1... s m possible (complete) states of the automaton. We can consider each state as a combination of an external state, of which there are 2 k possible, and an internal state, of which there are s 1... s m possible. The law by which the states of the inner cells at time t > 1 are determined by the states of all the cells at time t 1 can be given by specifying to each of the complete states at time t 1 which one of the internal states at time t shall succeed it. We notice that Kleene s model is a lot more liberal than most of the versions adopted in the textbooks ([1], [2], [6]). In fact, Kleene was well aware of the possibility for defining the finite automata model in the simpler way as in the modern textbooks. He gave the following arguments to defend his preference: We could indeed consider the entire aggregate of m internal cells as replaced by a single one admitting s 1... s m states. We shall not take advantage of this possibility, because we have in view applications of the theory of finite automata in which the cells have certain simple properties and are connected in certain simple ways. We could also (but shall not) get along with a single input cell, by scheduling the inputs on the k original input cells to come in successively in some order on the new one, which would alter the time scale so that k moments of the new scale correspond to 1 of the original. Events referring to the new time scale could then be interpreted in terms of the original. 3

4 With k input cells in the finite automaton, to describe a past history of the input for t time units, we need a table of t rows by k columns. Thus, an input is described by a table. Next, Kleene defines event as follows: By an event we shall mean any property of the input. In other words, any subclass of the class of all the possible tables describing the input over all past time constitutes an event, which occurs when the table describing the actual input belongs to this subclass. In the terminology of modern textbooks ([1], [2], [6]), an event is called a language. That is, an event (or, language) is a subset of the set of all possible inputs. In what way does a finite automaton denote an event? When Kleene defines the event that a finite automaton denotes, for the sake of convenience, he makes use of what he calls aggregate states. This should be not confused with his claim that he shall not take advantage of this possibility [of replacing every aggregate state by a single state] as a finite automaton is still allowed to have multiple internal cells. Now let us call the states a 1,..., a r where r = 2 k s 1... s m and the internal states b 1,..., b q where q = s 1... s m. Let the notation be arranged so that the internal state at time 1 is b 1. With the internal state at time 1 fixed, the state at time p is a function of the input over the time 1,..., p (including the value of p, or when k = 0 only this). So each of the states a 1,..., a r represents an event, which occurs ending with time p, if and only if the input over the time 1,..., p is one which results in that one of a 1,..., a r being the state at time p. Thus the automaton can know about its past experience (inclusive of the present) only that it falls into one of r mutually exclusive classes (possibly some of them empty). Similarly an internal state at time p + 1, or a property of the complete state at time p, or a property of the internal state at time p + 1, or a property of the 4

5 internal state at time p + s for an s > 1 represents an event. Thus to say that the state at time p has a certain property is to say that the state then is one of a certain subclass of the r possible states, so that the past experience falls into the set sum (or disjunction) of the respective classes of past experiences which are separately represented by the states of the subclass. TASK: 1. Compare Kleene s finite automata model with the finite automata model that you find in your textbook. Point out the similarity and differences between the two. Argue that Kleene s model is no more powerful in denoting languages than the textbook s model. For Tasks 2 and 3, we consider the following problem. Suppose we have to maintain the value of a 16-bit binary number. Initially, the binary number has the value 0. At every time unit, we are given a new 16-bit binary number as input, which we add to the 16-bit binary number that we are maintaining. In adding two 16-bit binary numbers, overflow may occur. The higher ordered bits that cannot be represented within 16 bits are lost. For example, l Following is a summary of the logic: initialize s = ; for each time t, let t (a 16-bit binary number) be the new input s = s + t; // binary addition TASKS: 2. Give the design of a finite automaton, according to Kleene s model, with 16 input cells and 16 inner cells, for the event that the current value of the 16-bit binary number consists of all 0 s. 3. Discuss the design of a finite automaton, according to your textbook s model, that captures the same event as defined in Task 2. 5

6 For Tasks 4 and 5, we consider the chess game. Initially, all the chess pieces are positioned in their rightful places. Each input corresponds to a move, which can be described as an ordered pair of positions ((x, i), (y, j)) where x, y {a, b, c, d, e, f, g, h} and i, j {1, 2, 3, 4, 5, 6, 7, 8}. All the game rules have to be observed. Once the game is over, no more moves are allowed. TASKS: 4. Sketch the design of a finite automaton, according to Kleene s model, with 65 inner cells (one for each of the 64 squares on the chessboard and one cell for remembering whose turn, black or white, it is for the next move), for the event that the sequence of moves are valid Discuss the design of a finite automaton, according to your textbook s model, that captures the same event as defined in Task 4. 3 Regular Events The concept of regular events was introduced by Kleene via the definition of regular expressions. In the terminology of modern textbooks ([1], [2], [6]), regular events are called regular languages. Following is Kleene s definition of regular events: First we define three operations on sets of tables. If E and F are sets of tables, E F (their sum or disjunction) shall be the set of tables to which a table belongs exactly if it belongs to E or belongs to F. If E and F are sets of tables, EF (their product) shall be the set of tables to which a table belongs exactly if it is the result of writing any table of F next below any table of E. [This statement is slightly adapted from the original source. Also, some lines are omitted.] Obviously E F and EF are associative operations. We may write E 0 F for F, E 1 for E, E 2 for EE, E 3 for EEE, etc. If E and F are sets of tables, E F (the iterate of E on F, or briefly E iterate F ) shall be the infinite sum of the sets F, EF, EEF,..., or in self-explanatory symbolism F EF EEF... or n=0 En F. The regular sets (of tables) shall be the least class of sets of tables which includes the unit sets (i.e., the sets containing one table each) and the 1 It is not required to give the full design details. 6

7 empty set and which is closed under the operations of passing from E and F to E F, to EF and to E F. An event shall be regular, if there is a regular set of tables which describes it in the sense that the event occurs or not according as the input is described by one of the tables of the set or by none of them. Kleene defines a related concept of regular expressions as follows 2 : By a regular expression, we shall mean a particular way of expressing a regular set of tables starting with single-table sets and applying zero or more times the three operations (passing from E and F to E F, EF or E F ). TASK: 6. Compare Kleene s definitions of regular events and expressions to the corresponding definitions of regular sets (equivalently, regular languages) and regular expressions given in your textbook. The definitions differ slightly. In what way do the definitions differ? Are the differences significant? Why are regular expressions and regular events introduced? Kleene explains the reasons in the following excerpt: Our reason for introducing the regular events, as given by regular sets of tables described by regular expressions, is Theorem 5, which we discovered before Theorem 3. By using the notion of regular events, we thus demonstrate that a McCulloch-Pitts nerve net can represent any event which any other kind of finite digital automaton (in the sense to be developed in detail in Section 8) can represent. This of course includes a number of special results which McCulloch and Pitts obtained for alternative kinds of nerve nets, but is more general. The way is open to attempt similarly to verify the like for other kinds of cells in place of neurons, or to seek some characterization of the properties of the cells in order that aggregates of them have the capacity for representing all representable (i.e., all regular) events. 2 The excerpt is slightly adapted from the original source without introducing new terminologies that are present in the original text. 7

8 That is, regular expressions and regular events are introduced so that we can prove the equivalence with other finite digital automata by showing that the events that they can represent are exactly the regular events. In Theorem 5, Kleene shows that finite automata denote regular events, which are defined in terms of regular expressions. What sorts of events can be represented? As the concept of input is the same as in Part I, we can use the notion of regular event which was introduced in Section 7. The following theorem, together with Theorem 3 referring to a special kind of finite automaton, answer the question. Theorem 5. In any finite automaton (in particular, in a McCulloch-Pitts nerve net), started at time 1 in a given internal state b 1, the event represented by a given state existing at time p is regular. Next, we give Kleene s proof of Theorem 5. PROOF. Since the initial internal state is specified, there are 2 k possible initial states (the results of combining the given initial internal state b 1 with each of the 2 k possible external states at time 1). So if we can show that the automaton starting from a given state at time 1 will reach a given state at time p, if and only if a certain regular event occurs ending with time p, then the theorem will follow by taking the disjunction of 2 k respective regular events, which is itself a regular event. Given any state a at time t 1 (t 2), exactly 2 k states are possible at time t, since the internal part of the state at time t is determined by a, and the external part can happen in 2 k ways. Let us say each of these 2 k states is in relation R to a. At this point, Kleene decided to give a generalized treatment for the main proof technique behind the proof of Theorem 5. This is in contrast with the treatments that we find in modern textbooks ([1], [2], [6]). Besides, the general treatment is interesting in its own right. 8

9 The next part of our analysis will apply to any binary relation R defined on a given set of r 1 objects a 1,..., a r (called states ), whether or not it arises in the manner just described. Consider any two a and ā of the states, not necessarily distinct. We shall study the strings of states d p d p 1... d 1 (p 1) for which d p is a, d 1 is ā, and for each t (t = 2,..., p) d t is in relation R to d t 1 (in symbols, d t Rd t 1 ); say such strings connect a to ā. We say a set of strings is regular under the following definition (chosen analogously to the definition of regular sets of tables in 7.1). The empty set and for each i (i = 1,..., r) the unit set {a i } having as only member a i considered as a string of length 1 are regular. If A and B are regular, so is their sum, written A B. If A and B are regular, so is the set, written AB, of the strings obtainable by writing a string belonging to A just left of a string belonging to B. If A and B are regular, so is the sum, written A B, for n = 0, 1, 2,... of the sets A... AB with n A s preceding the B. Lemma 7. The strings d p... d 1 connecting a to ā constitute a regular set. PROOF OF LEMMA, by induction on r. BASIS: r = 1. Then ā is a. If ara (i.e., if R is an irreflexive relation), the set of the strings connecting a to a is the unit set {a}, which is regular. If ara, then the set is {a, aa, aaa,...}, which is regular, since it can be written A A where A = {a}. INDUCTION STEP: r > 1. CASE 1: a = ā. In this case any string connecting a to ā is of the form a a a... a a, where the number of a s is 0 and each represents independently the empty string (this being possible only if ara) or a non-empty string without any a in it. Let e 1,..., e g (g 0) be the states e such that are but e a, and f 1,..., f h (h 0) the states f such that fra but f a. Now any non-empty string represented by an must start with one of e 1,..., e g and end with one of f 1,..., f h. For each pair e i f j, by the hypothesis of the induction the set of the strings connecting e i to f j without a in it is regular. Say B 1,..., B gh are these regular sets; and let A be {a}. Now if ara, the set of the possible strings a is A A(B 1... B gh ) (which reduces to A if gh = 0 or all the B s are empty); and if ara, it is A(B 1... B gh ) (which is empty if gh = 0 or all the B s are empty). Let this set be C. Then the set of the strings leading from a to a is C A (which reduces to A if C is empty). 9

10 CASE 2: a ā. Now we have instead a a a... a a ā where the number of a s is 0 and each and the represents independently the empty string or a non-empty string without any a in it. If D is the set of the possible strings a, and E = {ā}, the set of the strings connecting a to ā is C DE, which is regular. With the proof of Lemma 7 done, Kleene is ready to complete the proof of Theorem 5. PROOF OF THEOREM (completed). We need to show that, for a given state a and each of 2 k states ā, the state is a at time p and ā at time 1, if and only if a certain regular event occurs over the time 1,..., p. By the lemma, the set of the strings which can connect a to ā is regular. Consider an expression for this regular set in terms of the empty set and the sets {a i } as the units (cf. 7.2). In this expression let us replace each unit {a i } by the unit set consisting of the k 1 table which (if k > 0) describes the external part of the state a i, labeled initial or non-initial according to whether that unit {a i } was earliest or not. Each empty set as unit we replace by itself (but write it Ī). There results a regular expression. The state changes from ā at time 1 to a at time p, exactly if the event described by this regular expression occurs over the time 1,..., p. TASKS: 7. Consider Case 2 of the induction proof for Lemma 7. Is it justified to claim that C DE is regular? Carefully explain your answer. 8. Compare critically Kleene s proof that the events [languages] denoted by finite automata are regular with the proof that you find in your textbook. Which proof do you think is easier to follow? Which proof do you think offer a more rigorous correctness argument? 9. Kleene shows that any finite automaton, as defined according to Kleene s definition, denotes regular event. You are asked to re-prove the result for the modern day definition of finite automata by making use of Kleene s Lemma 7. 10

11 4 Notes to the Instructor Notes on Task 2: The usual logic for addition requires the carry bit to be passed sequentially from the lower ordered digits to the higher ordered digits as the addition process is performed. Recall that, with Kleene s model, the logic for determining the i-th bit (an inner cell) at time t is said to be determined by the states of all the cells [that include the input cells and inner cells] at time t 1. However, Kleene s model leaves it wide open how the states of all the cells at time t 1 decide the state of an inner cell at time t. So, the mechanism for determining the state of an inner cell can indeed be summarized by a sequential procedure (as employed in the addition logic). Notes on Task 8: In [1] and [6], both textbooks use the state elimination method in constructing a regular expression from a given finite automaton. The construction requires the introduction of the concept of generalized nondeterministic finite automata. In [2], a dynamic programming approach, that resembles Floyd-Warshall s algorithm for computing transitive closure of a graph, is used. On the other hand, Kleene s approach seems to be more in line with the usual mathematical induction technique in that the assertion is proved for a small number of states, and an induction step is used to prove the claim as the number of states increases. So, it is quite possible that the students may find Kleene s approach more natural-looking assuming that they have had a good training in writing mathematical induction proofs. References [1] J. E. Hopcroft, R. Motwani and J. D. Ullman, Introduction to automata theory, languages, and computation, 2nd Edition, Addison- Wesley, [2] J. E. Hopcroft and J. D. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley, [3] S. C. Kleene, Representation of events in nerve nets and finite automata, in: C. Shannon and J. McCarthy, (eds.) Automata Studies, Princeton University Press, NJ, 1956,

12 [4] W. S. McCulloch and W. Pitts, A logical calculus of ideas immanent in nervous activity, Bull. Math. BioPhys., 5 (1943), [5] M. O. Rabin and D. Scott, Finite automata and their decision problems, IBM Journal of Research and Development, 3 (1959), [6] M. Sipser, Introduction to the theory of computation, 2nd Edition, Thomson Course Technology,

### 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

### 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 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### diffy boxes (iterations of the ducci four number game) 1

diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game.

### Properties of Stabilizing Computations

Theory and Applications of Mathematics & Computer Science 5 (1) (2015) 71 93 Properties of Stabilizing Computations Mark Burgin a a University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA

### 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,

### 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,

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### 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 :

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 2 / 20 REDUCIBILITY

### 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

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### 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

### 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 Expressions and Automata using Haskell

Regular Expressions and Automata using Haskell Simon Thompson Computing Laboratory University of Kent at Canterbury January 2000 Contents 1 Introduction 2 2 Regular Expressions 2 3 Matching regular expressions

### 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

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### 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

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

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

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### 3515ICT Theory of Computation Turing Machines

Griffith University 3515ICT Theory of Computation Turing Machines (Based loosely on slides by Harald Søndergaard of The University of Melbourne) 9-0 Overview Turing machines: a general model of computation

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### (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

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Finite Automata and Formal Languages

Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Lecture 14 May 19th 2011 Overview of today s lecture: Turing Machines Push-down Automata Overview of the Course Undecidable and Intractable Problems

### Theory of Computation

Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

### For the purposes of this course, the natural numbers are the positive integers. We denote by N, the set of natural numbers.

Lecture 1: Induction and the Natural numbers Math 1a is a somewhat unusual course. It is a proof-based treatment of Calculus, for all of you who have already demonstrated a strong grounding in Calculus

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### Discrete mathematics is the study of techniques, ideas and modes

CHAPTER 1 Discrete Systems Discrete mathematics is the study of techniques, ideas and modes of reasoning that are indispensable in applied disciplines such as computer science or information technology.

### 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

### 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

### COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1)

COMP 50 Fall 016 9 - Mathematical induction Sept 6, 016 You will see many examples in this course and upcoming courses of algorithms for solving various problems It many cases, it will be obvious that

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### Introduction Number Systems and Conversion

UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### DISCRETE MATHEMATICS W W L CHEN

DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### 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

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### 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

### (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

### 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

### Euclid s Algorithm for the Greatest Common Divisor Desh Ranjan Department of Computer Science New Mexico State University

Euclid s Algorithm for the Greatest Common Divisor Desh Ranjan Department of Computer Science New Mexico State University 1 Numbers, Division and Euclid People have been using numbers, and operations on

### Mathematical Induction

Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### Introduction to Automata Theory. Reading: Chapter 1

Introduction to Automata Theory Reading: Chapter 1 1 What is Automata Theory? Study of abstract computing devices, or machines Automaton = an abstract computing device Note: A device need not even be a

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours)

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University April 17, 2009 4.1 Mathematical Induction 4.1 Mathematical

### CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

### Models of Sequential Computation

Models of Sequential Computation 13 C H A P T E R Programming is often considered as an art, but it raises mostly practical questions like which language should I choose, which algorithm is adequate, and

### ECS 120 Lesson 17 Church s Thesis, The Universal Turing Machine

ECS 120 Lesson 17 Church s Thesis, The Universal Turing Machine Oliver Kreylos Monday, May 7th, 2001 In the last lecture, we looked at the computation of Turing Machines, and also at some variants of Turing

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

### 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,

### A Formalization of the Turing Test Evgeny Chutchev

A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### 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.

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### 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

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### A Fixed-Point Approach towards Efficient Models Conversion

ISSN: 1940-8978 A Fixed-Point Approach towards Efficient Models Conversion Stefan Andrei, Hikyoo Koh No. 2, June 2008 Computer Science Department Technical Reports - Lamar University Computer Science Department,

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Cartesian Products and Relations

Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special

### Finite Automata and Formal Languages

Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Ana Bove Lecture 1 March 21st 2011 Course Organisation Overview of the Course Overview of today s lecture: Course Organisation Level: This course

### CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo

Propositional Logic CSE 191, Class Note 01 Propositional Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 37 Discrete Mathematics What is Discrete

### SECTION 10-2 Mathematical Induction

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

### Lecture 1: Elementary Number Theory

Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers

### 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)

### 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

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### 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.

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### 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,

### 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

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Induction and the Division Theorem 2.1 The Method of Induction

2 Induction and the Division Theorem 2.1 The Method of Induction In the Introduction we discussed a mathematical problem whose solution required the verification of an infinite family of statements. We

### 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

### Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural