CpSc 421 Homework 1 Solutions

Size: px
Start display at page:

Download "CpSc 421 Homework 1 Solutions"

Transcription

1 CpSc 421 Homework 1 Solutions 1. (15 points) Let Σ = {a, b, c}. Figure 7 depicts two finite state machines that read Let L a and L b denote the languages recognized by DFA (a) and DFA (b) respectively. (a) (6 points) For each of L a and L b list three strings in Σ that are in the language and three strings in Σ that are not in the language. Solution: {abc, aabbcc, aaaabccc} L a {ɛ, acabc, c} L a = {ɛ, aab, aaacaab} L b {aaab, aaacc, a 7 } L b = (b) (3 points) Write a short, English description of language, L a. Solution: Language L a is the set of all strings starting with one or more a followed by one or more b followed by one or more c. (c) (6 points) Write a short, English description of language, L b. Solution: Language L b is the set of all strings s such that (1) the #a(s) 2#b(s) 3#c(s) = 0; (2) For all prefixes x of s, 0 #a(x) 2#b(x) 3#c(x) (15 points) (inspired by Sipser exercise 1.6 (p. 84): Give state diagrams of DFAs recognizing the following languages. For each language, the alphabet is {0, 1}. (a) {w w begins and ends with the same symbol}. (b) {w w contains three consecutive 1s}. (c) {w w contains neither the substring 010 nor the substring 101}. Solution: See below. 3. (20 points) (inspired by Sipser exercise 1.7 (p. 84): Give state diagrams of NFAs with the specified number of states recognizing each of the following languages. For each language, the alphabet is Σ with Σ = {0, 1}. Solution: See below. (a) The set of strings w that end with the substring 010: {w x Σ. w = x010} Use four states. (b) The set of strings w that contain the substring 010: {w x, y Σ. w = x010y} Use four states. (c) The set of strings w that can be written as x 1 x 2 x k for some k 0, with each x i is an element of {01, 10, 001, 0011} Use eleven states. (d) The set of strings whose length is a multiple of three plus a multiple of five: {w m, n N. w = 3m + 5n} Use eight states.

2 Figure 1: Solution for question 2a Figure 2: Solution for question 2b

3 Figure 3: Solution for question 2c Figure 4: Solution for question 3a Figure 5: Solution for question 3b Figure 6: Solution for question 3c

4 Figure 7: Solution for question 3d 4. (20 points): Let Σ = {0, 1}. Let L k be the set of all strings whose k th from end symbol is a 1: L k = {w Σ x Σ, y Σ k 1. w = x1y} In the September 12 notes, we claimed that any DFA that recognizes L k must have at least 2 k states. Prove this claim. Solution: Suppose the claim does not hold, i.e. there exists a DFA A k = (Q k, Σ, δ, q 0, F ) recognizing L k with Q k < 2 k. Since there are 2 k strings of length k over Σ = {0, 1}, there exists two strings s 1, s 2 Σ k such that s 1 s 2 and A k is in the same state after processing both s 1 and s 2. Without loss of generarilty, assume that s 1 and s 2 differ in symbol i, i.e. s i 1 = 0 and s i 2 = 1 with 0 i k 1. Then A k is in the same state after processing strings s 1 0 i and s 2 0 i, but s 1 0 i / L k and s 2 0 i L k, a contradiction. Therefore, Q k 2 k. 5. (20 points, extra credit): In this exercise, you will prove the equivalence of NFA acceptances as defined in Sipser and the formulation that I gave in class. This problem statement is long because I first summarize both Sipser s and my definitions of acceptance. If you are comfortable with both, you can skip to the end of the problem statement where I ask you to prove the equivalence of the two formulations. Let N mrg = (Q, Σ,, q 0, F ) be an NFA as defined in the September 12 notes. Q is a finite set of states; Σ is a finite alphabet; Q Σ ɛ Q is the transition relation; q 0 Q is the initial state; and F Q is the set of accepting states. where Σ ɛ = Σ {ɛ}. Here are the formulas that we used to define NFA acceptance (see the slides for explanations). close ɛ (q) is the subset of Q where p close ɛ (q) iff p = q q close ɛ (q). (q, ɛ, p) We extended close ɛ to sets as close ɛ (G) = close ɛ (q) q G

5 Then we defined: step(q, c) = close ɛ ({q (q, c, q ) }), c Σ step(g, c) = step(q, c), G Q, c Σ q G (G, ɛ) = close ɛ (G), G Q (G, x c) = step( (G, x), c), G Q, x Σ, c Σ Finally, we said that N mrg accepts s iff ({q 0 }, s) F Now, for Sipser s version. Given N mrg = (Q, Σ,, q 0, F ) as described above, let N ms = (Q, Σ, δ, q 0, F ) with δ : Q Σ ɛ 2 Q where 2 Q is the power set of Q and δ(q, c) = {q (q, c, q ) } Sipser says that NFA N ms accepts string s iff we can find y 1, y 2,..., y m Σ ɛ and r 0, r 2,... r m Q such that s = y 1 y 2 y m ; r 0 = q 0 ; for i in 0... m 1, r i+1 δ(r i, y i + 1); r m F. Prove that N ms accepts a string (i.e. acceptance by Sipser s condition), iff N mrg accepts the string (i.e. acceptance by the condition given in class).

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

CpSc 421 Homework 9 Solution

CpSc 421 Homework 9 Solution CpSc 421 Homework 9 Solution Attempt any three of the six problems below. The homework is graded on a scale of 100 points, even though you can attempt fewer or more points than that. Your recorded grade

More information

CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata (cont )

CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata (cont ) CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata (cont ) Sungjin Im University of California, Merced 2-3-214 Example II A ɛ B ɛ D F C E Example II A ɛ B ɛ D F C E NFA accepting

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

Equivalence of NFAs and DFAs

Equivalence of NFAs and DFAs Equivalence of NFAs and DFAs Mark Greenstreet, CpSc 421, Term 1, 2008/09 15 September 2008 p.1/15 Lecture Outline Equivalence of NFAs and DFAs Implementing NFAs in software using exhaustive enumeration

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

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

CS 154 Formal Languages and Computability Assignment #2 Solutions

CS 154 Formal Languages and Computability Assignment #2 Solutions CS 154 Formal Languages and Computability Assignment #2 Solutions Department of Computer Science San Jose State University Spring 2016 Instructor: Ron Mak www.cs.sjsu.edu/~mak Assignment #2: Question 1

More information

September 7, Formal Definition of a Nondeterministic Finite Automaton

September 7, Formal Definition of a Nondeterministic Finite Automaton Formal Definition of a Nondeterministic Finite Automaton September 7, 2014 A comment first The formal definition of an NFA is similar to that of a DFA. Both have states, an alphabet, transition function,

More information

Recap DFA,NFA, DTM. Slides by Prof. Debasis Mitra, FIT.

Recap DFA,NFA, DTM. Slides by Prof. Debasis Mitra, FIT. Recap DFA,NFA, DTM Slides by Prof. Debasis Mitra, FIT. 1 Formal Language Finite set of alphabets Σ: e.g., {0, 1}, {a, b, c}, { {, } } Language L is a subset of strings on Σ, e.g., {00, 110, 01} a finite

More information

Finite Automata and Regular languages

Finite Automata and Regular languages Finite Automata and Regular languages Huan Long Shanghai Jiao Tong University Acknowledgements Part of the slides comes from a similar course in Fudan University given by Prof. Yijia Chen. http://basics.sjtu.edu.cn/

More information

Deterministic Finite Automata (DFAs)

Deterministic Finite Automata (DFAs) Algorithms & Models of Computation CS/ECE 374, Fall 27 Deterministic Finite Automata (DFAs) Lecture 3 Tuesday, September 5, 27 Sariel Har-Peled (UIUC) CS374 Fall 27 / 36 Part I DFA Introduction Sariel

More information

CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata

CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata Sungjin Im University of California, Merced 1-27-215 Nondeterminism Michael Rabin and Dana Scott (1959) Michael Rabin Dana

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

Closure under the Regular Operations

Closure under the Regular Operations Closure under the Regular Operations 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

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

CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Section 1.4 Explain the limits of the class of regular languages Justify why the

More information

CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION CSE 105 THEORY OF COMPUTATION Spring 2018 http://cseweb.ucsd.edu/classes/sp18/cse105-ab/ Today's learning goals Sipser Section 1.1 Prove closure properties of the class of regular languages Apply closure

More information

Finite Automata. Seungjin Choi

Finite Automata. Seungjin Choi Finite Automata Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr 1 / 28 Outline

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

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

CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Section 1.1 Design an automaton that recognizes a given language. Specify each of

More information

Please give details of your answer. A direct answer without explanation is not counted.

Please give details of your answer. A direct answer without explanation is not counted. Please give details of your answer. A direct answer without explanation is not counted. Your answers must be in English. Please carefully read problem statements. During the exam you are not allowed to

More information

Computational Models - Lecture 1 1

Computational Models - Lecture 1 1 Computational Models - Lecture 1 1 Handout Mode Ronitt Rubinfeld and Iftach Haitner. Tel Aviv University. February 29/ March 02, 2016 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames

More information

Deterministic Finite Automata (DFAs)

Deterministic Finite Automata (DFAs) CS/ECE 374: Algorithms & Models of Computation, Fall 28 Deterministic Finite Automata (DFAs) Lecture 3 September 4, 28 Chandra Chekuri (UIUC) CS/ECE 374 Fall 28 / 33 Part I DFA Introduction Chandra Chekuri

More information

CISC 4090: Theory of Computation Chapter 1 Regular Languages. Section 1.1: Finite Automata. What is a computer? Finite automata

CISC 4090: Theory of Computation Chapter 1 Regular Languages. Section 1.1: Finite Automata. What is a computer? Finite automata CISC 4090: Theory of Computation Chapter Regular Languages Xiaolan Zhang, adapted from slides by Prof. Werschulz Section.: Finite Automata Fordham University Department of Computer and Information Sciences

More information

(Refer Slide Time: 0:21)

(Refer Slide Time: 0:21) Theory of Computation Prof. Somenath Biswas Department of Computer Science and Engineering Indian Institute of Technology Kanpur Lecture 7 A generalisation of pumping lemma, Non-deterministic finite automata

More information

Non-deterministic Finite Automata (NFAs)

Non-deterministic Finite Automata (NFAs) Algorithms & Models of Computation CS/ECE 374, Fall 27 Non-deterministic Finite Automata (NFAs) Part I NFA Introduction Lecture 4 Thursday, September 7, 27 Sariel Har-Peled (UIUC) CS374 Fall 27 / 39 Sariel

More information

Outline. Nondetermistic Finite Automata. Transition diagrams. A finite automaton is a 5-tuple (Q, Σ,δ,q 0,F)

Outline. Nondetermistic Finite Automata. Transition diagrams. A finite automaton is a 5-tuple (Q, Σ,δ,q 0,F) Outline Nondeterminism Regular expressions Elementary reductions http://www.cs.caltech.edu/~cs20/a October 8, 2002 1 Determistic Finite Automata A finite automaton is a 5-tuple (Q, Σ,δ,q 0,F) Q is a finite

More information

CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Section 1.1 Determine if a language is regular Apply closure properties to conclude

More information

CSE 105 Homework 3 Due: Monday October 23, Instructions. should be on each page of the submission.

CSE 105 Homework 3 Due: Monday October 23, Instructions. should be on each page of the submission. CSE 5 Homework 3 Due: Monday October 23, 27 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

More information

Chapter 6: NFA Applications

Chapter 6: NFA Applications Chapter 6: NFA Applications Implementing NFAs The problem with implementing NFAs is that, being nondeterministic, they define a more complex computational procedure for testing language membership. To

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

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

Introduction to Turing Machines. Reading: Chapters 8 & 9

Introduction to Turing Machines. Reading: Chapters 8 & 9 Introduction to Turing Machines Reading: Chapters 8 & 9 1 Turing Machines (TM) Generalize the class of CFLs: Recursively Enumerable Languages Recursive Languages Context-Free Languages Regular Languages

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

Examples of Regular Expressions. Finite Automata vs. Regular Expressions. Example of Using flex. Application

Examples of Regular Expressions. Finite Automata vs. Regular Expressions. Example of Using flex. Application Examples of Regular Expressions 1. 0 10, L(0 10 ) = {w w contains exactly a single 1} 2. Σ 1Σ, L(Σ 1Σ ) = {w w contains at least one 1} 3. Σ 001Σ, L(Σ 001Σ ) = {w w contains the string 001 as a substring}

More information

Proofs, Strings, and Finite Automata. CS154 Chris Pollett Feb 5, 2007.

Proofs, Strings, and Finite Automata. CS154 Chris Pollett Feb 5, 2007. Proofs, Strings, and Finite Automata CS154 Chris Pollett Feb 5, 2007. Outline Proofs and Proof Strategies Strings Finding proofs Example: For every graph G, the sum of the degrees of all the nodes in G

More information

September 11, Second Part of Regular Expressions Equivalence with Finite Aut

September 11, Second Part of Regular Expressions Equivalence with Finite Aut Second Part of Regular Expressions Equivalence with Finite Automata September 11, 2013 Lemma 1.60 If a language is regular then it is specified by a regular expression Proof idea: For a given regular language

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

Homework 02 Solution updated

Homework 02 Solution updated CS3-Automata and Complexity Theory Homework 2 Solution updated Due On: 5hrs Wednesday, December 2, 25 Max Points: 25 Problem [5+5+5+ points] Give DFA for the following languages, over the alphabet {,}

More information

Languages, regular languages, finite automata

Languages, regular languages, finite automata Notes on Computer Theory Last updated: January, 2018 Languages, regular languages, finite automata Content largely taken from Richards [1] and Sipser [2] 1 Languages An alphabet is a finite set of characters,

More information

ECS120 Fall Discussion Notes. October 25, The midterm is on Thursday, November 2nd during class. (That is next week!)

ECS120 Fall Discussion Notes. October 25, The midterm is on Thursday, November 2nd during class. (That is next week!) ECS120 Fall 2006 Discussion Notes October 25, 2006 Announcements The midterm is on Thursday, November 2nd during class. (That is next week!) Homework 4 Quick Hints Problem 1 Prove that the following languages

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

Deterministic Finite Automata. Non deterministic finite automata. Non-Deterministic Finite Automata (NFA) Non-Deterministic Finite Automata (NFA)

Deterministic Finite Automata. Non deterministic finite automata. Non-Deterministic Finite Automata (NFA) Non-Deterministic Finite Automata (NFA) Deterministic Finite Automata Non deterministic finite automata Automata we ve been dealing with have been deterministic For every state and every alphabet symbol there is exactly one move that the machine

More information

Nondeterministic Finite Automata

Nondeterministic Finite Automata Nondeterministic Finite Automata Mahesh Viswanathan Introducing Nondeterminism Consider the machine shown in Figure. Like a DFA it has finitely many states and transitions labeled by symbols from an input

More information

Introduction to the Theory of Computation. Automata 1VO + 1PS. Lecturer: Dr. Ana Sokolova.

Introduction to the Theory of Computation. Automata 1VO + 1PS. Lecturer: Dr. Ana Sokolova. Introduction to the Theory of Computation Automata 1VO + 1PS Lecturer: Dr. Ana Sokolova http://cs.uni-salzburg.at/~anas/ Setup and Dates Lectures and Instructions 23.10. 3.11. 17.11. 24.11. 1.12. 11.12.

More information

CPSC 313 Introduction to Computability

CPSC 313 Introduction to Computability CPSC 313 Introduction to Computability Grammars in Chomsky Normal Form (Cont d) (Sipser, pages 109-111 (3 rd ed) and 107-109 (2 nd ed)) Renate Scheidler Fall 2018 Chomsky Normal Form A context-free grammar

More information

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

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

Lecture 1: Finite State Automaton

Lecture 1: Finite State Automaton Lecture 1: Finite State Automaton Instructor: Ketan Mulmuley Scriber: Yuan Li January 6, 2015 1 Deterministic Finite Automaton Informally, a deterministic finite automaton (DFA) has finite number of s-

More information

Introduction to the Theory of Computation. Automata 1VO + 1PS. Lecturer: Dr. Ana Sokolova.

Introduction to the Theory of Computation. Automata 1VO + 1PS. Lecturer: Dr. Ana Sokolova. Introduction to the Theory of Computation Automata 1VO + 1PS Lecturer: Dr. Ana Sokolova http://cs.uni-salzburg.at/~anas/ Setup and Dates Lectures Tuesday 10:45 pm - 12:15 pm Instructions Tuesday 12:30

More information

Theory of Languages and Automata

Theory of Languages and Automata Theory of Languages and Automata Chapter 1- Regular Languages & Finite State Automaton Sharif University of Technology Finite State Automaton We begin with the simplest model of Computation, called finite

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

Deterministic Finite Automata (DFAs)

Deterministic Finite Automata (DFAs) Algorithms & Models of Computation CS/ECE 374, Spring 29 Deterministic Finite Automata (DFAs) Lecture 3 Tuesday, January 22, 29 L A TEXed: December 27, 28 8:25 Chan, Har-Peled, Hassanieh (UIUC) CS374 Spring

More information

PS2 - Comments. University of Virginia - cs3102: Theory of Computation Spring 2010

PS2 - Comments. University of Virginia - cs3102: Theory of Computation Spring 2010 University of Virginia - cs3102: Theory of Computation Spring 2010 PS2 - Comments Average: 77.4 (full credit for each question is 100 points) Distribution (of 54 submissions): 90, 12; 80 89, 11; 70-79,

More information

1. Induction on Strings

1. Induction on Strings CS/ECE 374: Algorithms & Models of Computation Version: 1.0 Fall 2017 This is a core dump of potential questions for Midterm 1. This should give you a good idea of the types of questions that we will ask

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

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

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

CPS 220 Theory of Computation REGULAR LANGUAGES

CPS 220 Theory of Computation REGULAR LANGUAGES CPS 22 Theory of Computation REGULAR LANGUAGES Introduction Model (def) a miniature representation of a thing; sometimes a facsimile Iraq village mockup for the Marines Scientific modelling - the process

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

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

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

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 Definition of Computation. August 28, 2013

Formal Definition of Computation. August 28, 2013 August 28, 2013 Computation model The model of computation considered so far is the work performed by a finite automaton Finite automata were described informally, using state diagrams, and formally, as

More information

11-711: Algorithms for NLP Homework Assignment #1: Formal Language Theory Solutions

11-711: Algorithms for NLP Homework Assignment #1: Formal Language Theory Solutions -: Algorithms for NLP Homework Assignment #: Formal Language Theory Solutions Out: September 4, 204 Due: September 8, 204 Problem [5 points] Prove that, for any deterministic FSA A = (Q, Σ, δ, q 0, F,

More information

Computational Models #1

Computational Models #1 Computational Models #1 Handout Mode Nachum Dershowitz & Yishay Mansour March 13-15, 2017 Nachum Dershowitz & Yishay Mansour Computational Models #1 March 13-15, 2017 1 / 41 Lecture Outline I Motivation

More information

Chapter 6. Properties of Regular Languages

Chapter 6. Properties of Regular Languages Chapter 6 Properties of Regular Languages Regular Sets and Languages Claim(1). The family of languages accepted by FSAs consists of precisely the regular sets over a given alphabet. Every regular set is

More information

Deterministic Finite Automata (DFAs)

Deterministic Finite Automata (DFAs) Algorithms & Models of Computation CS/ECE 374, Fall 27 Deterministic Finite Automata (DFAs) Lecture 3 Tuesday, September 5, 27 Part I DFA Introduction Sariel Har-Peled (UIUC) CS374 Fall 27 / 36 Sariel

More information

Nondeterminism. September 7, Nondeterminism

Nondeterminism. September 7, Nondeterminism September 7, 204 Introduction is a useful concept that has a great impact on the theory of computation Introduction is a useful concept that has a great impact on the theory of computation So far in our

More information

Languages. Non deterministic finite automata with ε transitions. First there was the DFA. Finite Automata. Non-Deterministic Finite Automata (NFA)

Languages. Non deterministic finite automata with ε transitions. First there was the DFA. Finite Automata. Non-Deterministic Finite Automata (NFA) Languages Non deterministic finite automata with ε transitions Recall What is a language? What is a class of languages? Finite Automata Consists of A set of states (Q) A start state (q o ) A set of accepting

More information

Finite Automata. Theorems - Unit I SUSAN ELIAS. Professor Department of Computer Science & Engineering Sri Venkateswara College of Engineering

Finite Automata. Theorems - Unit I SUSAN ELIAS. Professor Department of Computer Science & Engineering Sri Venkateswara College of Engineering Finite Automata Theorems - Unit I SUSAN ELIAS Professor Department of Computer Science & Engineering Sri Venkateswara College of Engineering September 17, 2012 Unit I - Guidelines Formal Definitions Definition

More information

Reducability. Sipser, pages

Reducability. Sipser, pages Reducability Sipser, pages 187-214 Reduction Reduction encodes (transforms) one problem as a second problem. A solution to the second, can be transformed into a solution to the first. We expect both transformations

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

Introduction to Theory of Computing

Introduction to Theory of Computing CSCI 2670, Fall 2012 Introduction to Theory of Computing Department of Computer Science University of Georgia Athens, GA 30602 Instructor: Liming Cai www.cs.uga.edu/ cai 0 Lecture Note 3 Context-Free Languages

More information

CS500 Homework #2 Solutions

CS500 Homework #2 Solutions CS500 Homework #2 Solutions 1. Consider the two languages Show that L 1 is context-free but L 2 is not. L 1 = {a i b j c k d l i = j k = l} L 2 = {a i b j c k d l i = k j = l} Answer. L 1 is the concatenation

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

Finite Automata Theory and Formal Languages TMV026/TMV027/DIT321 Responsible: Ana Bove

Finite Automata Theory and Formal Languages TMV026/TMV027/DIT321 Responsible: Ana Bove Finite Automata Theory and Formal Languages TMV026/TMV027/DIT321 Responsible: Ana Bove Tuesday 28 of May 2013 Total: 60 points TMV027/DIT321 registration VT13 TMV026/DIT321 registration before VT13 Exam

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

Outline. CS21 Decidability and Tractability. Machine view of FA. Machine view of FA. Machine view of FA. Machine view of FA.

Outline. CS21 Decidability and Tractability. Machine view of FA. Machine view of FA. Machine view of FA. Machine view of FA. Outline CS21 Decidability and Tractability Lecture 5 January 16, 219 and Languages equivalence of NPDAs and CFGs non context-free languages January 16, 219 CS21 Lecture 5 1 January 16, 219 CS21 Lecture

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

CSE 311: Foundations of Computing. Lecture 23: Finite State Machine Minimization & NFAs

CSE 311: Foundations of Computing. Lecture 23: Finite State Machine Minimization & NFAs CSE : Foundations of Computing Lecture : Finite State Machine Minimization & NFAs State Minimization Many different FSMs (DFAs) for the same problem Take a given FSM and try to reduce its state set by

More information

CS 322 D: Formal languages and automata theory

CS 322 D: Formal languages and automata theory CS 322 D: Formal languages and automata theory Tutorial NFA DFA Regular Expression T. Najla Arfawi 2 nd Term - 26 Finite Automata Finite Automata. Q - States 2. S - Alphabets 3. d - Transitions 4. q -

More information

This lecture covers Chapter 6 of HMU: Pushdown Automata

This lecture covers Chapter 6 of HMU: Pushdown Automata This lecture covers Chapter 6 of HMU: ushdown Automata ushdown Automata (DA) Language accepted by a DA Equivalence of CFs and the languages accepted by DAs Deterministic DAs Additional Reading: Chapter

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 6a REVIEW for Q2 Q2 covers Lecture 5 and Lecture 6 Chapter 2 - Deterministic Finite Automata DFA Chapter 2 - Nondeterministic

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

Computability Theory

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

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 6 CHAPTER 2 FINITE AUTOMATA 2. Nondeterministic Finite Automata NFA 3. Finite Automata and Regular Expressions 4. Languages

More information

CIT 596 Theory of Computation Spring 2005, 212 Moore, TR PM. Homework Assignment 2

CIT 596 Theory of Computation Spring 2005, 212 Moore, TR PM. Homework Assignment 2 CIT 596 Theory of Computation Spring 2005, 212 Moore, TR 12-1.30PM Homework Assignment 2 Due: Tuesday, February 1, 2005, by 12 PM (IN CLASS) Name: Student ID (8 digits): Email: Signature: PLEASE, READ

More information

What we have done so far

What we have done so far What we have done so far DFAs and regular languages NFAs and their equivalence to DFAs Regular expressions. Regular expressions capture exactly regular languages: Construct a NFA from a regular expression.

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

CS 321 Solutions to Homework 4 Due October 22, 5pm

CS 321 Solutions to Homework 4 Due October 22, 5pm CS 321 Solutions to Homework 4 Due October 22, 5pm 1. Convert the following NFAs from HW3 to (a) {ab, aba} (b) bitstrings with 0 as the third last symbol (c) bitstrings that contain 0100 How do these converted

More information

CS 133 : Automata Theory and Computability

CS 133 : Automata Theory and Computability 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

More information

Automata and Computability. Solutions to Exercises

Automata and Computability. Solutions to Exercises Automata and Computability Solutions to Exercises Spring 27 Alexis Maciel Department of Computer Science Clarkson University Copyright c 27 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata

More information

Context-Free Languages

Context-Free Languages CS:4330 Theory of Computation Spring 2018 Context-Free Languages Pushdown Automata Haniel Barbosa Readings for this lecture Chapter 2 of [Sipser 1996], 3rd edition. Section 2.2. Finite automaton 1 / 13

More information

Foreword. Grammatical inference. Examples of sequences. Sources. Example of problems expressed by sequences Switching the light

Foreword. Grammatical inference. Examples of sequences. Sources. Example of problems expressed by sequences Switching the light Foreword Vincent Claveau IRISA - CNRS Rennes, France In the course of the course supervised symbolic machine learning technique concept learning (i.e. 2 classes) INSA 4 Sources s of sequences Slides and

More information

Automata and Formal Languages - CM0081 Non-Deterministic Finite Automata

Automata and Formal Languages - CM0081 Non-Deterministic Finite Automata Automata and Formal Languages - CM81 Non-Deterministic Finite Automata Andrés Sicard-Ramírez Universidad EAFIT Semester 217-2 Non-Deterministic Finite Automata (NFA) Introduction q i a a q j a q k The

More information

3515ICT: Theory of Computation. Regular languages

3515ICT: Theory of Computation. Regular languages 3515ICT: Theory of Computation Regular languages Notation and concepts concerning alphabets, strings and languages, and identification of languages with problems (H, 1.5). Regular expressions (H, 3.1,

More information

The Power of One-State Turing Machines

The Power of One-State Turing Machines The Power of One-State Turing Machines Marzio De Biasi Jan 15, 2018 Abstract At first glance, one state Turing machines are very weak: the Halting problem for them is decidable, and, without memory, they

More information