CS 133 : Automata Theory and Computability

Size: px
Start display at page:

Download "CS 133 : Automata Theory and Computability"

Transcription

1 CS 133 : Automata Theory and Computability Lecture Slides 1 Regular Languages and Finite Automata Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 3 1 Reference: Intro to the Theory of Computation (2nd Ed), Sipser

2 Regular Languages and Finite Automata Minimization of DFA

3 Regular Languages and Finite Automata Minimization of DFA

4 Equivalence of States We say that states p and q are equivalent if for all input strings ω, ˆδ(p, ω) is an accepting state if and only if ˆδ(q, ω) is an accepting state.

5 Equivalence of States We say that states p and q are equivalent if for all input strings ω, ˆδ(p, ω) is an accepting state if and only if ˆδ(q, ω) is an accepting state. If two states are not equivalent, then we say they are distinguishable.

6 Equivalence of States We say that states p and q are equivalent if for all input strings ω, ˆδ(p, ω) is an accepting state if and only if ˆδ(q, ω) is an accepting state. If two states are not equivalent, then we say they are distinguishable. That is, state p is distinguishable from state q if there is at least one string ω such that one of ˆδ(p, ω) and ˆδ(q, ω) is accepting, and the other is not accepting.

7 Equivalence of States We say that states p and q are equivalent if for all input strings ω, ˆδ(p, ω) is an accepting state if and only if ˆδ(q, ω) is an accepting state. If two states are not equivalent, then we say they are distinguishable. That is, state p is distinguishable from state q if there is at least one string ω such that one of ˆδ(p, ω) and ˆδ(q, ω) is accepting, and the other is not accepting.

8 Table of state inequivalences

9 Minimum-state DFA

10 3. for all q, q, r, r Q M and a alphabet(m ), if (q, r) X, (q, a, q) T M and (r, a, r) T M, then (q, r ) X. Another Example APTER Minimize 3. REGULAR the ff DFA: LANGUAGES 17 Start A 0 B 1 E 0, 1 1 F C 0, 1 D n M = M. Next, we let X be the least subset of Q M Q M 1. A M (Q M A M ) X; 2. (Q M A M ) A M X; such that:

11 Since {B, D} Z and [δ M (B, 1)] = [E] = {E, F}, we have Another Example ( B, D, 1, E, F ) T N. APTER Minimize 3. REGULAR the ff DFA: LANGUAGES 17 Since {E, F} Z and [δ M (E, 0)] = [F] = {E, F}, we have ( E, F, 0, E, F ) T 0 1 0, 1 Start A N. B E F 0 1 Since {E, F} Z and [δ M (E, 1)] = [F] = {E, F}, we have ( E, F, 1, E, F ) T N. hus our DFA N Cis: 0, 1 D n M = M. Next, we let X be the least subset of Q M Q M such that: Start A 1. A M (Q M A M ) X; 2. (Q M A M ) A M X; 1 C 0 1 B, D 0, 1 0 E, F 0, 1 3. for all q, q, r, r Q M and a alphabet(m ), if (q, r) X, (q, a, q) T M and (r, a, r) T M, then (q, r ) X.

12 Regular Languages and Finite Automata Minimization of DFA

13 A language is called a regular language if some finite automaton recognizes it.

14 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages.

15 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}.

16 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}.

17 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}.

18 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3})

19 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3}) A B =

20 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3}) A B = {a, b, 1, 2, 3}

21 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3}) A B = {a, b, 1, 2, 3} A B =

22 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3}) A B = {a, b, 1, 2, 3} A B = {a1, a2, a3, b1, b2, b3}

23 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3}) A B = {a, b, 1, 2, 3} A B = {a1, a2, a3, b1, b2, b3} A =

24 A language is called a regular language if some finite automaton recognizes it. Regular Operations Let A and B be languages. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B}. Star: A = {x 1 x 2 x k k 0 and each x i A}. Example (A = {a, b} and B = {1, 2, 3}) A B = {a, b, 1, 2, 3} A B = {a1, a2, a3, b1, b2, b3} A = {ε, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, bab, bba, bbb, }

25 A collection of objects is closed under some operation if applying that operation to members of the collection returns an object still in the collection.

26 A collection of objects is closed under some operation if applying that operation to members of the collection returns an object still in the collection. Theorem The class of regular languages is closed under the union operation. Proof.

27 A collection of objects is closed under some operation if applying that operation to members of the collection returns an object still in the collection. Theorem The class of regular languages is closed under the union operation. Proof. Theorem The class of regular languages is closed under the concatenation operation. Theorem The class of regular languages is closed under the star operation.

28 Questions? See you next meeting!

CSE 105 Homework 1 Due: Monday October 9, Instructions. should be on each page of the submission.

CSE 105 Homework 1 Due: Monday October 9, Instructions. should be on each page of the submission. CSE 5 Homework Due: Monday October 9, 7 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in this

More information

COSE212: Programming Languages. Lecture 1 Inductive Definitions (1)

COSE212: Programming Languages. Lecture 1 Inductive Definitions (1) COSE212: Programming Languages Lecture 1 Inductive Definitions (1) Hakjoo Oh 2018 Fall Hakjoo Oh COSE212 2018 Fall, Lecture 1 September 5, 2018 1 / 10 Inductive Definitions Inductive definition (induction)

More information

Theory of Computation

Theory of Computation Fall 2002 (YEN) Theory of Computation Midterm Exam. Name:... I.D.#:... 1. (30 pts) True or false (mark O for true ; X for false ). (Score=Max{0, Right- 1 2 Wrong}.) (1) X... If L 1 is regular and L 2 L

More information

COSE212: Programming Languages. Lecture 1 Inductive Definitions (1)

COSE212: Programming Languages. Lecture 1 Inductive Definitions (1) COSE212: Programming Languages Lecture 1 Inductive Definitions (1) Hakjoo Oh 2017 Fall Hakjoo Oh COSE212 2017 Fall, Lecture 1 September 4, 2017 1 / 9 Inductive Definitions Inductive definition (induction)

More information

Theory of Computation (I) Yijia Chen Fudan University

Theory of Computation (I) Yijia Chen Fudan University Theory of Computation (I) Yijia Chen Fudan University Instructor Yijia Chen Homepage: http://basics.sjtu.edu.cn/~chen Email: yijiachen@fudan.edu.cn Textbook Introduction to the Theory of Computation Michael

More information

Harvard CS121 and CSCI E-121 Lecture 2: Mathematical Preliminaries

Harvard CS121 and CSCI E-121 Lecture 2: Mathematical Preliminaries Harvard CS121 and CSCI E-121 Lecture 2: Mathematical Preliminaries Harry Lewis September 5, 2013 Reading: Sipser, Chapter 0 Sets Sets are defined by their members A = B means that for every x, x A iff

More information

COM364 Automata Theory Lecture Note 2 - Nondeterminism

COM364 Automata Theory Lecture Note 2 - Nondeterminism COM364 Automata Theory Lecture Note 2 - Nondeterminism Kurtuluş Küllü March 2018 The FA we saw until now were deterministic FA (DFA) in the sense that for each state and input symbol there was exactly

More information

Theory of Computation

Theory of Computation Theory of Computation Lecture #2 Sarmad Abbasi Virtual University Sarmad Abbasi (Virtual University) Theory of Computation 1 / 1 Lecture 2: Overview Recall some basic definitions from Automata Theory.

More information

Automata: a short introduction

Automata: a short introduction ILIAS, University of Luxembourg Discrete Mathematics II May 2012 What is a computer? Real computers are complicated; We abstract up to an essential model of computation; We begin with the simplest possible

More information

2017/08/31 Chapter 1.2 & 1.3 in Sipser Ø Announcement:

2017/08/31 Chapter 1.2 & 1.3 in Sipser Ø Announcement: Regular Expressions Human-aware and Robo.cs Operations 2017/08/31 Chapter 1.2 & 1.3 in Sipser Ø Announcement: q q q q Many thanks to students who have responded so far! There is still time to respond to

More information

FABER Formal Languages, Automata. Lecture 2. Mälardalen University

FABER Formal Languages, Automata. Lecture 2. Mälardalen University CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 2 Mälardalen University 2010 1 Content Languages, g Alphabets and Strings Strings & String Operations Languages & Language Operations

More information

Theory of Computer Science

Theory of Computer Science Theory of Computer Science C1. Formal Languages and Grammars Malte Helmert University of Basel March 14, 2016 Introduction Example: Propositional Formulas from the logic part: Definition (Syntax of Propositional

More information

Closure Properties of Regular Languages

Closure Properties of Regular Languages Closure Properties of Regular Languages Lecture 13 Section 4.1 Robb T. Koether Hampden-Sydney College Wed, Sep 21, 2016 Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages

More information

Nondeterministic Finite Automata and Regular Expressions

Nondeterministic Finite Automata and Regular Expressions Nondeterministic Finite Automata and Regular Expressions CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Recap: Deterministic Finite Automaton A DFA is a 5-tuple M = (Q, Σ, δ, q 0, F) Q is a finite

More information

Closure under the Regular Operations

Closure under the Regular Operations September 7, 2013 Application of NFA Now we use the NFA to show that collection of regular languages is closed under regular operations union, concatenation, and star Earlier we have shown this closure

More information

CS 154. Finite Automata vs Regular Expressions, Non-Regular Languages

CS 154. Finite Automata vs Regular Expressions, Non-Regular Languages CS 154 Finite Automata vs Regular Expressions, Non-Regular Languages Deterministic Finite Automata Computation with finite memory Non-Deterministic Finite Automata Computation with finite memory and guessing

More information

CS 154, Lecture 2: Finite Automata, Closure Properties Nondeterminism,

CS 154, Lecture 2: Finite Automata, Closure Properties Nondeterminism, CS 54, Lecture 2: Finite Automata, Closure Properties Nondeterminism, Why so Many Models? Streaming Algorithms 0 42 Deterministic Finite Automata Anatomy of Deterministic Finite Automata transition: for

More information

CS Automata, Computability and Formal Languages

CS Automata, Computability and Formal Languages Automata, Computability and Formal Languages Luc Longpré faculty.utep.edu/longpre 1 - Pg 1 Slides : version 3.1 version 1 A. Tapp version 2 P. McKenzie, L. Longpré version 2.1 D. Gehl version 2.2 M. Csűrös,

More information

CS 121, Section 2. Week of September 16, 2013

CS 121, Section 2. Week of September 16, 2013 CS 121, Section 2 Week of September 16, 2013 1 Concept Review 1.1 Overview In the past weeks, we have examined the finite automaton, a simple computational model with limited memory. We proved that DFAs,

More information

Formal Languages, Automata and Models of Computation

Formal Languages, Automata and Models of Computation CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 5 School of Innovation, Design and Engineering Mälardalen University 2011 1 Content - More Properties of Regular Languages (RL)

More information

Finite Automata and Regular Languages

Finite Automata and Regular Languages Finite Automata and Regular Languages Topics to be covered in Chapters 1-4 include: deterministic vs. nondeterministic FA, regular expressions, one-way vs. two-way FA, minimization, pumping lemma for regular

More information

CMSC 330: Organization of Programming Languages. Regular Expressions and Finite Automata

CMSC 330: Organization of Programming Languages. Regular Expressions and Finite Automata CMSC 330: Organization of Programming Languages Regular Expressions and Finite Automata CMSC330 Spring 2018 1 How do regular expressions work? What we ve learned What regular expressions are What they

More information

CMSC 330: Organization of Programming Languages

CMSC 330: Organization of Programming Languages CMSC 330: Organization of Programming Languages Regular Expressions and Finite Automata CMSC 330 Spring 2017 1 How do regular expressions work? What we ve learned What regular expressions are What they

More information

C1.1 Introduction. Theory of Computer Science. Theory of Computer Science. C1.1 Introduction. C1.2 Alphabets and Formal Languages. C1.

C1.1 Introduction. Theory of Computer Science. Theory of Computer Science. C1.1 Introduction. C1.2 Alphabets and Formal Languages. C1. Theory of Computer Science March 20, 2017 C1. Formal Languages and Grammars Theory of Computer Science C1. Formal Languages and Grammars Malte Helmert University of Basel March 20, 2017 C1.1 Introduction

More information

CS:4330 Theory of Computation Spring Regular Languages. Finite Automata and Regular Expressions. Haniel Barbosa

CS:4330 Theory of Computation Spring Regular Languages. Finite Automata and Regular Expressions. Haniel Barbosa CS:4330 Theory of Computation Spring 2018 Regular Languages Finite Automata and Regular Expressions Haniel Barbosa Readings for this lecture Chapter 1 of [Sipser 1996], 3rd edition. Sections 1.1 and 1.3.

More information

Exam 1 CSU 390 Theory of Computation Fall 2007

Exam 1 CSU 390 Theory of Computation Fall 2007 Exam 1 CSU 390 Theory of Computation Fall 2007 Solutions Problem 1 [10 points] Construct a state transition diagram for a DFA that recognizes the following language over the alphabet Σ = {a, b}: L 1 =

More information

Nondeterministic Finite Automata

Nondeterministic Finite Automata Nondeterministic Finite Automata Not A DFA Does not have exactly one transition from every state on every symbol: Two transitions from q 0 on a No transition from q 1 (on either a or b) Though not a DFA,

More information

Intro to Theory of Computation

Intro to Theory of Computation Intro to Theory of Computation 1/19/2016 LECTURE 3 Last time: DFAs and NFAs Operations on languages Today: Nondeterminism Equivalence of NFAs and DFAs Closure properties of regular languages Sofya Raskhodnikova

More information

Harvard CS 121 and CSCI E-207 Lecture 10: CFLs: PDAs, Closure Properties, and Non-CFLs

Harvard CS 121 and CSCI E-207 Lecture 10: CFLs: PDAs, Closure Properties, and Non-CFLs Harvard CS 121 and CSCI E-207 Lecture 10: CFLs: PDAs, Closure Properties, and Non-CFLs Harry Lewis October 8, 2013 Reading: Sipser, pp. 119-128. Pushdown Automata (review) Pushdown Automata = Finite automaton

More information

Recap from Last Time

Recap from Last Time Regular Expressions Recap from Last Time Regular Languages A language L is a regular language if there is a DFA D such that L( D) = L. Theorem: The following are equivalent: L is a regular language. There

More information

1. (a) Explain the procedure to convert Context Free Grammar to Push Down Automata.

1. (a) Explain the procedure to convert Context Free Grammar to Push Down Automata. Code No: R09220504 R09 Set No. 2 II B.Tech II Semester Examinations,December-January, 2011-2012 FORMAL LANGUAGES AND AUTOMATA THEORY Computer Science And Engineering Time: 3 hours Max Marks: 75 Answer

More information

Author: Vivek Kulkarni ( )

Author: Vivek Kulkarni ( ) Author: Vivek Kulkarni ( vivek_kulkarni@yahoo.com ) Chapter-3: Regular Expressions Solutions for Review Questions @ Oxford University Press 2013. All rights reserved. 1 Q.1 Define the following and give

More information

TDDD65 Introduction to the Theory of Computation

TDDD65 Introduction to the Theory of Computation TDDD65 Introduction to the Theory of Computation Lecture 2 Gustav Nordh, IDA gustav.nordh@liu.se 2012-08-31 Outline - Lecture 2 Closure properties of regular languages Regular expressions Equivalence of

More information

Two-Way Automata and Descriptional Complexity

Two-Way Automata and Descriptional Complexity Two-Way Automata and Descriptional Complexity Giovanni Pighizzini Dipartimento di Informatica Università degli Studi di Milano TCS 2014 Rome, Italy September 1-3, 2014 IFIP TC1, Working Group 1.2, Descriptional

More information

Equivalence of DFAs and NFAs

Equivalence of DFAs and NFAs CS 172: Computability and Complexity Equivalence of DFAs and NFAs It s a tie! DFA NFA Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: L.von Ahn, L. Blum, M. Blum What we ll do today Prove that DFAs

More information

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY REVIEW for MIDTERM 1 THURSDAY Feb 6 Midterm 1 will cover everything we have seen so far The PROBLEMS will be from Sipser, Chapters 1, 2, 3 It will be

More information

Le rning Regular. A Languages via Alt rn ting. A E Autom ta

Le rning Regular. A Languages via Alt rn ting. A E Autom ta 1 Le rning Regular A Languages via Alt rn ting A E Autom ta A YALE MIT PENN IJCAI 15, Buenos Aires 2 4 The Problem Learn an unknown regular language L using MQ and EQ data mining neural networks geometry

More information

CS 154. Finite Automata, Nondeterminism, Regular Expressions

CS 154. Finite Automata, Nondeterminism, Regular Expressions CS 54 Finite Automata, Nondeterminism, Regular Expressions Read string left to right The DFA accepts a string if the process ends in a double circle A DFA is a 5-tuple M = (Q, Σ, δ, q, F) Q is the set

More information

Lecture 3: Nondeterministic Finite Automata

Lecture 3: Nondeterministic Finite Automata Lecture 3: Nondeterministic Finite Automata September 5, 206 CS 00 Theory of Computation As a recap of last lecture, recall that a deterministic finite automaton (DFA) consists of (Q, Σ, δ, q 0, F ) where

More information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 14 Ana Bove May 14th 2018 Recap: Context-free Grammars Simplification of grammars: Elimination of ǫ-productions; Elimination of

More information

Automata Theory Final Exam Solution 08:10-10:00 am Friday, June 13, 2008

Automata Theory Final Exam Solution 08:10-10:00 am Friday, June 13, 2008 Automata Theory Final Exam Solution 08:10-10:00 am Friday, June 13, 2008 Name: ID #: This is a Close Book examination. Only an A4 cheating sheet belonging to you is acceptable. You can write your answers

More information

EXAMPLE CFG. L = {a 2n : n 1 } L = {a 2n : n 0 } S asa aa. L = {a n b : n 0 } L = {a n b : n 1 } S asb ab S 1S00 S 1S00 100

EXAMPLE CFG. L = {a 2n : n 1 } L = {a 2n : n 0 } S asa aa. L = {a n b : n 0 } L = {a n b : n 1 } S asb ab S 1S00 S 1S00 100 EXAMPLE CFG L = {a 2n : n 1 } L = {a 2n : n 0 } S asa aa S asa L = {a n b : n 0 } L = {a n b : n 1 } S as b S as ab L { a b : n 0} L { a b : n 1} S asb S asb ab n 2n n 2n L {1 0 : n 0} L {1 0 : n 1} S

More information

CSCI 2670 Introduction to Theory of Computing

CSCI 2670 Introduction to Theory of Computing CSCI 267 Introduction to Theory of Computing Agenda Last class Reviewed syllabus Reviewed material in Chapter of Sipser Assigned pages Chapter of Sipser Questions? This class Begin Chapter Goal for the

More information

Section 1.3 Ordered Structures

Section 1.3 Ordered Structures Section 1.3 Ordered Structures Tuples Have order and can have repetitions. (6,7,6) is a 3-tuple () is the empty tuple A 2-tuple is called a pair and a 3-tuple is called a triple. We write (x 1,, x n )

More information

Topics in Timed Automata

Topics in Timed Automata 1/32 Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group 2/32 Timed Automata A theory of timed automata R. Alur and D. Dill, TCS 94 2/32 Timed Automata Language

More information

CSCI 340: Computational Models. Regular Expressions. Department of Computer Science

CSCI 340: Computational Models. Regular Expressions. Department of Computer Science CSCI 340: Computational Models Regular Expressions Chapter 4 Department of Computer Science Yet Another New Method for Defining Languages Given the Language: L 1 = {x n for n = 1 2 3...} We could easily

More information

1. Draw a parse tree for the following derivation: S C A C C A b b b b A b b b b B b b b b a A a a b b b b a b a a b b 2. Show on your parse tree u,

1. Draw a parse tree for the following derivation: S C A C C A b b b b A b b b b B b b b b a A a a b b b b a b a a b b 2. Show on your parse tree u, 1. Draw a parse tree for the following derivation: S C A C C A b b b b A b b b b B b b b b a A a a b b b b a b a a b b 2. Show on your parse tree u, v, x, y, z as per the pumping theorem. 3. Prove that

More information

CS 154, Lecture 3: DFA NFA, Regular Expressions

CS 154, Lecture 3: DFA NFA, Regular Expressions CS 154, Lecture 3: DFA NFA, Regular Expressions Homework 1 is coming out Deterministic Finite Automata Computation with finite memory Non-Deterministic Finite Automata Computation with finite memory and

More information

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 5-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY NON-DETERMINISM and REGULAR OPERATIONS THURSDAY JAN 6 UNION THEOREM The union of two regular languages is also a regular language Regular Languages Are

More information

Chapter 4. Regular Expressions. 4.1 Some Definitions

Chapter 4. Regular Expressions. 4.1 Some Definitions Chapter 4 Regular Expressions 4.1 Some Definitions Definition: If S and T are sets of strings of letters (whether they are finite or infinite sets), we define the product set of strings of letters to be

More information

1 Alphabets and Languages

1 Alphabets and Languages 1 Alphabets and Languages Look at handout 1 (inference rules for sets) and use the rules on some examples like {a} {{a}} {a} {a, b}, {a} {{a}}, {a} {{a}}, {a} {a, b}, a {{a}}, a {a, b}, a {{a}}, a {a,

More information

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY THE PUMPING LEMMA FOR REGULAR LANGUAGES and REGULAR EXPRESSIONS TUESDAY Jan 21 WHICH OF THESE ARE REGULAR? B = {0 n 1 n n 0} C = { w w has equal number

More information

In English, there are at least three different types of entities: letters, words, sentences.

In English, there are at least three different types of entities: letters, words, sentences. Chapter 2 Languages 2.1 Introduction In English, there are at least three different types of entities: letters, words, sentences. letters are from a finite alphabet { a, b, c,..., z } words are made up

More information

CFG Simplification. (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions

CFG Simplification. (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions CFG Simplification (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions 1 Eliminating useless symbols 1. A symbol X is generating if there exists: X * w, for

More information

Clarifications from last time. This Lecture. Last Lecture. CMSC 330: Organization of Programming Languages. Finite Automata.

Clarifications from last time. This Lecture. Last Lecture. CMSC 330: Organization of Programming Languages. Finite Automata. CMSC 330: Organization of Programming Languages Last Lecture Languages Sets of strings Operations on languages Finite Automata Regular expressions Constants Operators Precedence CMSC 330 2 Clarifications

More information

CS 455/555: Finite automata

CS 455/555: Finite automata CS 455/555: Finite automata Stefan D. Bruda Winter 2019 AUTOMATA (FINITE OR NOT) Generally any automaton Has a finite-state control Scans the input one symbol at a time Takes an action based on the currently

More information

Harvard CS 121 and CSCI E-207 Lecture 4: NFAs vs. DFAs, Closure Properties

Harvard CS 121 and CSCI E-207 Lecture 4: NFAs vs. DFAs, Closure Properties Harvard CS 121 and CSCI E-207 Lecture 4: NFAs vs. DFAs, Closure Properties Salil Vadhan September 13, 2012 Reading: Sipser, 1.2. How to simulate NFAs? NFA accepts w if there is at least one accepting computational

More information

How do regular expressions work? CMSC 330: Organization of Programming Languages

How do regular expressions work? CMSC 330: Organization of Programming Languages How do regular expressions work? CMSC 330: Organization of Programming Languages Regular Expressions and Finite Automata What we ve learned What regular expressions are What they can express, and cannot

More information

The Probability of Winning a Series. Gregory Quenell

The Probability of Winning a Series. Gregory Quenell The Probability of Winning a Series Gregory Quenell Exercise: Team A and Team B play a series of n + games. The first team to win n + games wins the series. All games are independent, and Team A wins any

More information

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION FORMAL LANGUAGES, AUTOMATA AND COMPUTATION IDENTIFYING NONREGULAR LANGUAGES PUMPING LEMMA Carnegie Mellon University in Qatar (CARNEGIE MELLON UNIVERSITY IN QATAR) SLIDES FOR 15-453 LECTURE 5 SPRING 2011

More information

COMP4141 Theory of Computation

COMP4141 Theory of Computation COMP4141 Theory of Computation Lecture 4 Regular Languages cont. Ron van der Meyden CSE, UNSW Revision: 2013/03/14 (Credits: David Dill, Thomas Wilke, Kai Engelhardt, Peter Höfner, Rob van Glabbeek) Regular

More information

CSEP 590 Data Compression Autumn Arithmetic Coding

CSEP 590 Data Compression Autumn Arithmetic Coding CSEP 590 Data Compression Autumn 2007 Arithmetic Coding Reals in Binary Any real number x in the interval [0,1) can be represented in binary as.b 1 b 2... where b i is a bit. x 0 0 1 0 1... binary representation

More information

Lecture 7 Properties of regular languages

Lecture 7 Properties of regular languages Lecture 7 Properties of regular languages COT 4420 Theory of Computation Section 4.1 Closure properties of regular languages If L 1 and L 2 are regular languages, then we prove that: Union: L 1 L 2 Concatenation:

More information

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz Compiler Design Spring 2011 Lexical Analysis Sample Exercises and Solutions Prof. Pedro C. Diniz USC / Information Sciences Institute 4676 Admiralty Way, Suite 1001 Marina del Rey, California 90292 pedro@isi.edu

More information

Regular Expressions [1] Regular Expressions. Regular expressions can be seen as a system of notations for denoting ɛ-nfa

Regular Expressions [1] Regular Expressions. Regular expressions can be seen as a system of notations for denoting ɛ-nfa Regular Expressions [1] Regular Expressions Regular expressions can be seen as a system of notations for denoting ɛ-nfa They form an algebraic representation of ɛ-nfa algebraic : expressions with equations

More information

CS 530: Theory of Computation Based on Sipser (second edition): Notes on regular languages(version 1.1)

CS 530: Theory of Computation Based on Sipser (second edition): Notes on regular languages(version 1.1) CS 530: Theory of Computation Based on Sipser (second edition): Notes on regular languages(version 1.1) Definition 1 (Alphabet) A alphabet is a finite set of objects called symbols. Definition 2 (String)

More information

UNIT-I. Strings, Alphabets, Language and Operations

UNIT-I. Strings, Alphabets, Language and Operations UNIT-I Strings, Alphabets, Language and Operations Strings of characters are fundamental building blocks in computer science. Alphabet is defined as a non empty finite set or nonempty set of symbols. The

More information

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 11 CHAPTER 3 CONTEXT-FREE LANGUAGES 1. Context Free Grammars 2. Pushdown Automata 3. Pushdown automata and context -free

More information

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 5 CHAPTER 2 FINITE AUTOMATA 1. Deterministic Finite Automata DFA 2. Nondeterministic Finite Automata NDFA 3. Finite Automata

More information

Büchi Automata and their closure properties. - Ajith S and Ankit Kumar

Büchi Automata and their closure properties. - Ajith S and Ankit Kumar Büchi Automata and their closure properties - Ajith S and Ankit Kumar Motivation Conventional programs accept input, compute, output result, then terminate Reactive program : not expected to terminate

More information

Chapter Five: Nondeterministic Finite Automata

Chapter Five: Nondeterministic Finite Automata Chapter Five: Nondeterministic Finite Automata From DFA to NFA A DFA has exactly one transition from every state on every symbol in the alphabet. By relaxing this requirement we get a related but more

More information

Unit 6. Non Regular Languages The Pumping Lemma. Reading: Sipser, chapter 1

Unit 6. Non Regular Languages The Pumping Lemma. Reading: Sipser, chapter 1 Unit 6 Non Regular Languages The Pumping Lemma Reading: Sipser, chapter 1 1 Are all languages regular? No! Most of the languages are not regular! Why? A finite automaton has limited memory. How can we

More information

Pushdown automata. Twan van Laarhoven. Institute for Computing and Information Sciences Intelligent Systems Radboud University Nijmegen

Pushdown automata. Twan van Laarhoven. Institute for Computing and Information Sciences Intelligent Systems Radboud University Nijmegen Pushdown automata Twan van Laarhoven Institute for Computing and Information Sciences Intelligent Systems Version: fall 2014 T. van Laarhoven Version: fall 2014 Formal Languages, Grammars and Automata

More information

CS311 Computational Structures Regular Languages and Regular Expressions. Lecture 4. Andrew P. Black Andrew Tolmach

CS311 Computational Structures Regular Languages and Regular Expressions. Lecture 4. Andrew P. Black Andrew Tolmach CS311 Computational Structures Regular Languages and Regular Expressions Lecture 4 Andrew P. Black Andrew Tolmach 1 Expressions Weʼre used to using expressions to describe mathematical objects Example:

More information

Theory of Computation (II) Yijia Chen Fudan University

Theory of Computation (II) Yijia Chen Fudan University Theory of Computation (II) Yijia Chen Fudan University Review A language L is a subset of strings over an alphabet Σ. Our goal is to identify those languages that can be recognized by one of the simplest

More information

Context Free Languages. Automata Theory and Formal Grammars: Lecture 6. Languages That Are Not Regular. Non-Regular Languages

Context Free Languages. Automata Theory and Formal Grammars: Lecture 6. Languages That Are Not Regular. Non-Regular Languages Context Free Languages Automata Theory and Formal Grammars: Lecture 6 Context Free Languages Last Time Decision procedures for FAs Minimum-state DFAs Today The Myhill-Nerode Theorem The Pumping Lemma Context-free

More information

Introduction to Formal Languages, Automata and Computability p.1/51

Introduction to Formal Languages, Automata and Computability p.1/51 Introduction to Formal Languages, Automata and Computability Finite State Automata K. Krithivasan and R. Rama Introduction to Formal Languages, Automata and Computability p.1/51 Introduction As another

More information

download instant at Assume that (w R ) R = w for all strings w Σ of length n or less.

download instant at  Assume that (w R ) R = w for all strings w Σ of length n or less. Chapter 2 Languages 3. We prove, by induction on the length of the string, that w = (w R ) R for every string w Σ. Basis: The basis consists of the null string. In this case, (λ R ) R = (λ) R = λ as desired.

More information

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro Diniz

Compiler Design. Spring Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro Diniz Compiler Design Spring 2010 Lexical Analysis Sample Exercises and Solutions Prof. Pedro Diniz USC / Information Sciences Institute 4676 Admiralty Way, Suite 1001 Marina del Rey, California 90292 pedro@isi.edu

More information

More Properties of Regular Languages

More Properties of Regular Languages More Properties of Regular anguages 1 We have proven Regular languages are closed under: Union Concatenation Star operation Reverse 2 Namely, for regular languages 1 and 2 : Union 1 2 Concatenation Star

More information

2.1 Solution. E T F a. E E + T T + T F + T a + T a + F a + a

2.1 Solution. E T F a. E E + T T + T F + T a + T a + F a + a . Solution E T F a E E + T T + T F + T a + T a + F a + a E E + T E + T + T T + T + T F + T + T a + T + T a + F + T a + a + T a + a + F a + a + a E T F ( E) ( T ) ( F) (( E)) (( T )) (( F)) (( a)) . Solution

More information

CS375 Midterm Exam Solution Set (Fall 2017)

CS375 Midterm Exam Solution Set (Fall 2017) CS375 Midterm Exam Solution Set (Fall 2017) Closed book & closed notes October 17, 2017 Name sample 1. (10 points) (a) Put in the following blank the number of strings of length 5 over A={a, b, c} that

More information

Intro to Theory of Computation

Intro to Theory of Computation Intro to Theory of Computation 1/26/2016 LECTURE 5 Last time: Closure properties. Equivalence of NFAs, DFAs and regular expressions Today: Conversion from NFAs to regular expressions Proving that a language

More information

60-354, Theory of Computation Fall Asish Mukhopadhyay School of Computer Science University of Windsor

60-354, Theory of Computation Fall Asish Mukhopadhyay School of Computer Science University of Windsor 60-354, Theory of Computation Fall 2013 Asish Mukhopadhyay School of Computer Science University of Windsor Pushdown Automata (PDA) PDA = ε-nfa + stack Acceptance ε-nfa enters a final state or Stack is

More information

Great Theoretical Ideas in Computer Science. Lecture 4: Deterministic Finite Automaton (DFA), Part 2

Great Theoretical Ideas in Computer Science. Lecture 4: Deterministic Finite Automaton (DFA), Part 2 5-25 Great Theoretical Ideas in Computer Science Lecture 4: Deterministic Finite Automaton (DFA), Part 2 January 26th, 27 Formal definition: DFA A deterministic finite automaton (DFA) M =(Q,,,q,F) M is

More information

Critical CS Questions

Critical CS Questions Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Critical CS Questions What is a computer? And What is a Computation? real computers too complex for any

More information

Figure 1: NFA N. Figure 2: Equivalent DFA N obtained through function nfa2dfa

Figure 1: NFA N. Figure 2: Equivalent DFA N obtained through function nfa2dfa CS 3100 Models of Computation Fall 2011 FIRST MIDTERM CLOSED BOOK 100 points I ve standardized on @ for representing Epsilons in all my figures as well as my code (liked equally by dot and our Python programs).

More information

CS5371 Theory of Computation. Lecture 5: Automata Theory III (Non-regular Language, Pumping Lemma, Regular Expression)

CS5371 Theory of Computation. Lecture 5: Automata Theory III (Non-regular Language, Pumping Lemma, Regular Expression) CS5371 Theory of Computation Lecture 5: Automata Theory III (Non-regular Language, Pumping Lemma, Regular Expression) Objectives Prove the Pumping Lemma, and use it to show that there are non-regular languages

More information

Theory of Computation Lecture 1. Dr. Nahla Belal

Theory of Computation Lecture 1. Dr. Nahla Belal Theory of Computation Lecture 1 Dr. Nahla Belal Book The primary textbook is: Introduction to the Theory of Computation by Michael Sipser. Grading 10%: Weekly Homework. 30%: Two quizzes and one exam. 20%:

More information

Semigroup presentations via boundaries in Cayley graphs 1

Semigroup presentations via boundaries in Cayley graphs 1 Semigroup presentations via boundaries in Cayley graphs 1 Robert Gray University of Leeds BMC, Newcastle 2006 1 (Research conducted while I was a research student at the University of St Andrews, under

More information

Finite-State Machines (Automata) lecture 12

Finite-State Machines (Automata) lecture 12 Finite-State Machines (Automata) lecture 12 cl a simple form of computation used widely one way to find patterns 1 A current D B A B C D B C D A C next 2 Application Fields Industry real-time control,

More information

TAFL 1 (ECS-403) Unit- III. 3.1 Definition of CFG (Context Free Grammar) and problems. 3.2 Derivation. 3.3 Ambiguity in Grammar

TAFL 1 (ECS-403) Unit- III. 3.1 Definition of CFG (Context Free Grammar) and problems. 3.2 Derivation. 3.3 Ambiguity in Grammar TAFL 1 (ECS-403) Unit- III 3.1 Definition of CFG (Context Free Grammar) and problems 3.2 Derivation 3.3 Ambiguity in Grammar 3.3.1 Inherent Ambiguity 3.3.2 Ambiguous to Unambiguous CFG 3.4 Simplification

More information

Algebraic Approach to Automata Theory

Algebraic Approach to Automata Theory Algebraic Approach to Automata Theory Deepak D Souza Department of Computer Science and Automation Indian Institute of Science, Bangalore. 20 September 2016 Outline 1 Overview 2 Recognition via monoid

More information

CDM Closure Properties

CDM Closure Properties CDM Closure Properties Klaus Sutner Carnegie Mellon Universality 15-closure 2017/12/15 23:19 1 Nondeterministic Machines 2 Determinization 3 Closure Properties Where Are We? 3 We have a definition of regular

More information

CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Sipser Ch 1.4 Explain the limits of the class of regular languages Justify why the Pumping

More information

CS21 Decidability and Tractability

CS21 Decidability and Tractability CS21 Decidability and Tractability Lecture 2 January 5, 2018 January 5, 2018 CS21 Lecture 2 1 Outline Finite Automata Nondeterministic Finite Automata Closure under regular operations NFA, FA equivalence

More information

CMSC 330: Organization of Programming Languages. Theory of Regular Expressions Finite Automata

CMSC 330: Organization of Programming Languages. Theory of Regular Expressions Finite Automata : Organization of Programming Languages Theory of Regular Expressions Finite Automata Previous Course Review {s s defined} means the set of string s such that s is chosen or defined as given s A means

More information

Computability Theory

Computability Theory CS:4330 Theory of Computation Spring 2018 Computability Theory Decidable Languages Haniel Barbosa Readings for this lecture Chapter 4 of [Sipser 1996], 3rd edition. Section 4.1. Decidable Languages We

More information

Computer Sciences Department

Computer Sciences Department 1 Reference Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER 3 objectives Finite automaton Infinite automaton Formal definition State diagram Regular and Non-regular

More information

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 14 SMALL REVIEW FOR FINAL SOME Y/N QUESTIONS Q1 Given Σ =, there is L over Σ Yes: = {e} and L = {e} Σ Q2 There are uncountably

More information