Regular Languages and Finite State Machines


 Buck Whitehead
 1 years ago
 Views:
Transcription
1 Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries  some review One application formal definition of finite automata Examples 1
2 Sets A set is an unordered collection of objects without duplicates. Objects in the set are called elements or members. denotes the empty set, the set with no elements. Note: { }. Definition: For any sets S and T, S is a subset of T, denoted S T, if x[x S x T]. Example: N Z. Note: One way of proving S = T: show S T and T S. Definition: For any sets S, T: S T = {w w S or w T } S T = {w w S and w T } S T = {w w S and w T } 2
3 Review of Sets Recall: To prove a universally quantified implication with a direct proof, e.g.: x y[p(x, y) Q(x, y)] First: Let x and y be arbitrary elements of the domain. Next: Assume P(x, y) is true. Show: Q(x, y) is true. Prove: For any sets A and B, A (B C) A B c. Proof: in class Exercise: Prove: For all sets A, B and C, A (B C) = (A B) (A C). 3
4 Strings and Languages Definition: An alphabet is a finite set of symbols. Example: Σ = {0, 1}, Σ = {a, b, c,..., z}. Definition: A string over alphabet Σ is a finite sequence of symbols from Σ. Note: The string of zero symbols is ǫ. Definition: Σ is the set of all strings over Σ. Definition: A language is a set of strings over some alphabet Σ, i.e., a subset of Σ. Example: Let alphabet A = {a, b, c}. Here are some languages over A: L 1 = {a, aab} L 2 = L 3 = {xy x, y {a, b, c}, x = y and the first character in x is a} L 4 = {ǫ, aa, aaaa, aaaaaa,...} 4
5 L 5 = {w {a, b, c} # a (w) = [# b (w)] 2 } Some strings in L 5 : ǫ, c, cc, cccc, ab, aaaabb, baabaa, abc,... Describe this language in English: L = {0 i 1 j 0 i j} 5
6 Automatic door controller Consider an automatic door placed at the entrance or exit of a grocery store. The door swings open when its sensors detect an approaching person, and then stays open until the person has passed through and beyond the door. It has two sensor pads  one in front of the door and one to the rear. The controller is always in one of two states: open or closed. The controller moves from state to state, depending on the input it receives. Four possible inputs to the door: front  a person is on the front pad (in front of the door) rear  a person is on the rear pad both  people are standing on both pads neither  no one is standing on either pad 6
7 Finite Automaton  Automatic Door Controller Transition table for controller: neither front rear both closed closed open closed closed open closed open open open Notes: The door will not open if people are standing on both pads  it might knock over someone standing behind the door. If the door is open, the only input that closes it is neither  that is, if no one is on either pad. 7
8 Finite Automaton  Door Controller neither both rear front front rear both closed open neither Figure 1: state diagram for door controller 8
9 One Model of Computation: finite automata Finite automata (or finite state machines) are simple computing machines with severely restricted memories. But a computer with just a little memory can do a lot. Example: A state diagram for finite automaton M q 0 q 1 1 M has two states, q 0 and q 1. q 0 is the start state. Arrows represent transitions from one state to another on the specified input symbol. Final (or accepting) states are indicated with a double circle. 9
10 Processing Strings with a FA Question: Does M accept input string 110? 1. Processing begins in start state q M reads symbols in 110 from left to right  after reading each symbol, M moves to another state. 3. If M is in a final state after processing the last symbol, then M accepts the string. Answer: Yes, 110 is accepted by M. 10
11 Finite Automaton/Finite State Machine A finite automaton (FA) consists of: set of states start state final states input alphabet (the set of symbols on the transition arrows) rules for going from one state to another, depending on input symbol  defined by a transition function Transition function example: inclass 11
12 Formal Definition of a Deterministic Finite Automaton (DFA) Definition: A deterministic finite automaton (DFA) is a 5tuple (Q, Σ, δ, q 0, F) where 1. Q is a finite set of states 2. Σ is a finite input alphabet 3. δ : Q Σ Q is the transition function 4. q 0 Q is the start state 5. F Q is the set of final states Formal definition of M: M = ({q 0, q 1 }, {0, 1}, δ, q 0, {q 1 }), where δ is given by: 0 1 q 0 q 1 q 0 q 1 q 1 q 0 12
13 The language of a FA Definition: If S is the set of all strings the FA M accepts, then S is the language of M, denoted S = L(M). Terminology: We say that M recognizes S. Question: What is L(M)? Example: FA M 2 a b a b b a s q r Exercise: Write down the formal definition of M 2. Does M 2 accept the following strings: aaba? aaab? Question: What is L(M 2 )? 13
14 Finite Automata Example: DFA M 3 : b a a s q r b a b Exercise: Write down the formal definition of M 3. What is L(M 3 )? 14
15 Another FA Example: M 4 1 q 1 r t 1 s Question: Does M 4 accept 011? 1011? ? Question: What is L(M 4 )? 15
16 The Language of a DFA Definition: Let M = (Q, Σ, δ, q 0, F) be a DFA, and let w = w 1 w 2...w n be a string over Σ. Then M accepts w if there exists a sequence of states r 0, r 1,..., r n Q such that 1. r 0 = q 0 2. δ(r i, w i+1 ) = r i+1 for i = 0, 1,..., n 1 3. r n F We say M recognizes language L = {w M accepts w}. Definition: A language is regular if some DFA recognizes it. 16
17 Designing DFAs Problem: Given a language L, define a DFA M that recognizes L. Note: What do you need to remember about a string w as you read it to determine if it s in L? This gives you the states. For a regular language, there will only be a finite amount of information you need to remember. Example: L = {w {0, 1} w ends in 1} As you read a string, one symbol at a time, left to right, what do you need to remember in order to determine if the string is in L? Remember: Last symbol seen was a 1 Last symbol seen not 1 State diagram: (in class) 17
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
More informationFormal 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
More informationCSC4510 AUTOMATA 2.1 Finite Automata: Examples and D efinitions Definitions
CSC45 AUTOMATA 2. Finite Automata: Examples and Definitions Finite Automata: Examples and Definitions A finite automaton is a simple type of computer. Itsoutputislimitedto yes to or no. It has very primitive
More informationAutomata 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
More informationDeterministic Finite Automata
1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function
More informationReading 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
More informationIntroduction to Finite Automata
Introduction to Finite Automata Our First Machine Model Captain Pedro Ortiz Department of Computer Science United States Naval Academy SI340 Theory of Computing Fall 2012 Captain Pedro Ortiz (US Naval
More informationNondeterministic Finite Automata
Chapter, Part 2 Nondeterministic Finite Automata CSC527, Chapter, Part 2 c 202 Mitsunori Ogihara Fundamental set operations Let A and B be two languages. In addition to union and intersection, we consider
More informationDefinition: 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
More information6.045: 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)
More informationSets. 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
More informationChapter 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.
More informationthe 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
More informationCMPSCI 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
More informationAutomata and Formal Languages
Automata and Formal Languages Winter 20092010 Yacov HelOr 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,
More informationFundamentele 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
More informationLecture I FINITE AUTOMATA
1. Regular Sets and DFA Lecture I Page 1 Lecture I FINITE AUTOMATA Lecture 1: Honors Theory, Spring 02, Yap We introduce finite automata (deterministic and nondeterministic) and regular languages. Some
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTATION
FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15453 SPRING 2011 2 / 20 REDUCIBILITY
More informationA set is an unordered collection of objects.
Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain
More informationCS103B 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.
More informationTuring 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 ɛtransitions,
More informationCS154. Turing Machines. Turing Machine. Turing Machines versus DFAs FINITE STATE CONTROL AI N P U T INFINITE TAPE. read write move.
CS54 Turing Machines Turing Machine q 0 AI N P U T IN TAPE read write move read write move Language = {0} q This Turing machine recognizes the language {0} Turing Machines versus DFAs TM can both write
More information2110711 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
More informationAutomata and Computability. Solutions to Exercises
Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata
More informationRegular 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
More informationSome Definitions about Sets
Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish
More informationC H A P T E R Regular Expressions regular expression
7 CHAPTER Regular Expressions Most programmers and other powerusers of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun
More informationCS F02 Formal Languages 1
CS411 2015F02 Formal Languages 1 020: 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
More informationRegular 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
More informationTheory 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
More informationCS 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
More informationThe Halting Problem is Undecidable
185 Corollary G = { M, w w L(M) } is not Turingrecognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine
More information2.1 Sets, power sets. Cartesian Products.
Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects.  used to group objects together,  often the objects with similar properties This description of a set (without
More informationHonors Class (Foundations of) Informatics. Tom Verhoeff. Department of Mathematics & Computer Science Software Engineering & Technology
Honors Class (Foundations of) Informatics Tom Verhoeff Department of Mathematics & Computer Science Software Engineering & Technology www.win.tue.nl/~wstomv/edu/hci c 2011, T. Verhoeff @ TUE.NL 1/20 Information
More information3515ICT 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) 90 Overview Turing machines: a general model of computation
More informationTheory 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, ContextFree Grammar s
More informationIntroduction 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
More informationPushdown 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
More informationFinite Automata. Reading: Chapter 2
Finite Automata Reading: Chapter 2 1 Finite Automata Informally, a state machine that comprehensively captures all possible states and transitions that a machine can take while responding to a stream (or
More informationScanner. 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
More informationApplications of Methods of Proof
CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The settheoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are
More informationMotivation. 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
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationFinite Automata. Reading: Chapter 2
Finite Automata Reading: Chapter 2 1 Finite Automaton (FA) Informally, a state diagram that comprehensively captures all possible states and transitions that a machine can take while responding to a stream
More informationAutomata Theory and Languages
Automata Theory and Languages SITE : http://www.info.univtours.fr/ mirian/ Automata Theory, Languages and Computation  Mírian HalfeldFerrari p. 1/1 Introduction to Automata Theory Automata theory :
More informationPushdown automata. Informatics 2A: Lecture 9. Alex Simpson. 3 October, 2014. School of Informatics University of Edinburgh als@inf.ed.ac.
Pushdown automata Informatics 2A: Lecture 9 Alex Simpson School of Informatics University of Edinburgh als@inf.ed.ac.uk 3 October, 2014 1 / 17 Recap of lecture 8 Contextfree languages are defined by contextfree
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More information(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
More informationSets and set operations
CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used
More information3. 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
More informationGrade 4 Mathematics Patterns, Relations, and Functions: Lesson 1
Grade 4 Mathematics Patterns, Relations, and Functions: Lesson 1 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes
More informationDeterministic PushDown Store Automata
Deterministic PushDown tore Automata 1 ContextFree Languages A B C D... A w 0 w 1 D...E The languagea n b n : a b ab 2 Finitetate Automata 3 Pushdown Automata Pushdown automata (pda s) is an fsa with
More informationCAs 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*.
More informationComputing Functions with Turing Machines
CS 30  Lecture 20 Combining Turing Machines and Turing s Thesis Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata
More information3(vi) B. Answer: False. 3(vii) B. Answer: True
Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)
More informationAutomata Theory CS F15 Undecidiability
Automata Theory CS4112015F15 Undecidiability David Galles Department of Computer Science University of San Francisco 150: Universal TM Turing Machines are Hard Wired Addition machine only adds 0 n 1
More informationAnnouncements. 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. 2.32.6 Homework 2 due Tuesday Recitation 3 on Friday
More informationSets and functions. {x R : x > 0}.
Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.
More informationCS5236 Advanced Automata Theory
CS5236 Advanced Automata Theory Frank Stephan Semester I, Academic Year 20122013 Advanced Automata Theory is a lecture which will first review the basics of formal languages and automata theory and then
More informationTuring Machines, Part I
Turing Machines, Part I Languages The $64,000 Question What is a language? What is a class of languages? Computer Science Theory 2 1 Now our picture looks like Context Free Languages Deterministic Context
More informationMACM 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.
More informationInf1A: Deterministic Finite State Machines
Lecture 2 InfA: Deterministic Finite State Machines 2. Introduction In this lecture we will focus a little more on deterministic Finite State Machines. Also, we will be mostly concerned with Finite State
More informationNotes. 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
More informationIntroduction to Turing Machines
Automata Theory, Languages and Computation  Mírian HalfeldFerrari p. 1/2 Introduction to Turing Machines SITE : http://www.sir.blois.univtours.fr/ mirian/ Automata Theory, Languages and Computation
More informationRegular 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
More informationComputability 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 110.
More informationContextFree Grammars (CFG)
ContextFree Grammars (CFG) SITE : http://www.sir.blois.univtours.fr/ mirian/ Automata Theory, Languages and Computation  Mírian HalfeldFerrari p. 1/2 An informal example Language of palindromes: L
More informationINSTITUTE OF AERONAUTICAL ENGINEERING
INSTITUTE OF AERONAUTICAL ENGINEERING DUNDIGAL 500 043, HYDERABAD COMPUTER SCIENCE AND ENGINEERING TUTORIAL QUESTION BANK Name : FORMAL LANGUAGES AND AUTOMATA THEORY Code : A40509 Class : II B. Tech II
More informationAutomata 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 20092010 winter semester Last modified:
More informationFinite Automata and Regular Languages
CHAPTER 3 Finite Automata and Regular Languages 3. Introduction 3.. States and Automata A finitestate machine or finite automaton (the noun comes from the Greek; the singular is automaton, the Greekderived
More informationSets and Logic. Chapter Sets
Chapter 2 Sets and Logic This chapter introduces sets. In it we study the structure on subsets of a set, operations on subsets, the relations of inclusion and equality on sets, and the close connection
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationGenetic programming with regular expressions
Genetic programming with regular expressions Børge Svingen Chief Technology Officer, Open AdExchange bsvingen@openadex.com 20090323 Pattern discovery Pattern discovery: Recognizing patterns that characterize
More informationASSIGNMENT ONE SOLUTIONS MATH 4805 / COMP 4805 / MATH 5605
ASSIGNMENT ONE SOLUTIONS MATH 4805 / COMP 4805 / MATH 5605 (1) (a) (0 + 1) 010 (finite automata below). (b) First observe that the following regular expression generates the binary strings with an even
More informationFoundations of Computing Discrete Mathematics Solutions to exercises for week 2
Foundations of Computing Discrete Mathematics Solutions to exercises for week 2 Agata Murawska (agmu@itu.dk) September 16, 2013 Note. The solutions presented here are usually one of many possiblities.
More informationT79.186 Reactive Systems: Introduction and Finite State Automata
T79.186 Reactive Systems: Introduction and Finite State Automata Timo Latvala 14.1.2004 Reactive Systems: Introduction and Finite State Automata 11 Reactive Systems Reactive systems are a class of software
More informationHomework Four Solution CSE 355
Homework Four Solution CSE 355 Due: 29 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 5.1.23
More informationWe 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
More informationFinite 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 Pushdown Automata Overview of the Course Undecidable and Intractable Problems
More informationComputational Models Lecture 8, Spring 2009
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance
More informationMath 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions
Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a27b. Section 8.3 on page 275 problems 1b, 8, 10a10b, 14. Section 8.4 on page 279 problems
More informationLecture 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
More informationUnit SF. Sets and Functions
Unit SF Sets and Functions Section : Sets The basic concepts of sets and functions are topics covered in high school math courses and are thus familiar to most university students. We take the intuitive
More informationDiscrete Mathematics Set Operations
Discrete Mathematics 13. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
More informationSets and set operations: cont. Functions.
CS 441 Discrete Mathematics for CS Lecture 8 Sets and set operations: cont. Functions. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Set Definition: set is a (unordered) collection of objects.
More informationLecture 18 Regular Expressions
Lecture 18 Regular Expressions Many of today s web applications require matching patterns in a text document to look for specific information. A good example is parsing a html file to extract tags
More informationωautomata Automata that accept (or reject) words of infinite length. Languages of infinite words appear:
ωautomata ωautomata Automata that accept (or reject) words of infinite length. Languages of infinite words appear: in verification, as encodings of nonterminating executions of a program. in arithmetic,
More informationAutomata and Formal Languages. Push Down Automata. Sipser pages 109114. Lecture 13. Tim Sheard 1
Automata and Formal Languages Push Down Automata Sipser pages 109114 Lecture 13 Tim Sheard 1 Push Down Automata Push Down Automata (PDAs) are εnfas with stack memory. Transitions are labeled by an input
More informationCS 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 ChurchTuring thesis Let s recap how it all started. In 1990, Hilbert stated a
More informationFormal Deænition of Finite Automaton. 1. Finite set of states, typically Q. 2. Alphabet of input symbols, typically æ.
Formal Denition of Finite Automaton 1. Finite set of states, typically Q. 2. Alphabet of input symbols, typically. 3. One state is the startèinitial state, typically q 0. 4. Zero or more nalèaccepting
More informationA 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
More informationCompiler Construction
Compiler Construction Regular expressions Scanning Görel Hedin Reviderad 2013 01 23.a 2013 Compiler Construction 2013 F021 Compiler overview source code lexical analysis tokens intermediate code generation
More informationLESSON SUMMARY. Set Operations and Venn Diagrams
LESSON SUMMARY CXC CSEC MATHEMATICS UNIT Three: Set Theory Lesson 4 Set Operations and Venn Diagrams Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volumes 1 and 2. (Some helpful exercises
More informationMATH 55: HOMEWORK #2 SOLUTIONS
MATH 55: HOMEWORK # SOLUTIONS ERIC PETERSON * 1. SECTION 1.5: NESTED QUANTIFIERS 1.1. Problem 1.5.8. Determine the truth value of each of these statements if the domain of each variable consists of all
More informationMarkov Algorithm. CHEN Yuanmi December 18, 2007
Markov Algorithm CHEN Yuanmi December 18, 2007 1 Abstract Markov Algorithm can be understood as a priority string rewriting system. In this short paper we give the definition of Markov algorithm and also
More informationCourse 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
More informationThe Language of Mathematics
CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,
More informationContextFree Grammars
ContextFree 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
More informationSections 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
More information4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.
Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,
More information