CS 482 Problem Set 2 Solutions Dan FitzGerald, dpf7
|
|
- Gerald Cross
- 5 years ago
- Views:
Transcription
1 CS 482 Problem Set 2 Solutions Dan FitzGerald, dpf7 Question : Describe the set { n 2n n } intersect { n 2n n }. Are there any strings with an odd number of blocks of zeros? Solution: Since we are dealing with set intersection, an element is in the set iff it is in both sets being intersected. Let the set we are seeking to descibe be S, the set on the LHS of the intersection be named A, and the set on the RHS be B. We start off by noticing that B begins with. Therefore every element in the set S must being with. Now we look at A. Since we know we must start with, this tells us that n=. This also requires that we add to to form to satisfy A's definition. Now we go back to B. We see that if we repeat the piece in the braces times, the outside the brace twice and then finish with, then we will satisfy both A and B and hence have an element in S(since it satisfies the definitions on each side of the intersection). Thus, we have in S. But, also notice that if we have, then for B this could mean taking the outside the braces, then letting n=2 to add on to make. Then we'd go back to A and see that we have one instance of the expression with n= and now half of the expression cocatenated with n=4. Thus, we'd have to add on 8 's and a to satisfy A. We could go back to B and either use the and outside the braced expression to the right to terminate this string, or we could use those last 8 's appended to set n=8. Continuing in this pattern of going back and forth interleaving the elements of A and B, we see that this set can be described as an even number blocks of zeros (at least 2), each separated by a and ending in, where, starting at, each successive block of zeros has twice as many zeros as the block of s preceeding it. By this description, no string can have an odd number of blocks of zeros Question 2 (2.2.6(a)) Give a DFA for the set of all strings starting with that are multiples of 5 in binary. Solution: The DFA is on the following page. To build this DFA, you must realize what happens upon reading a new character in terms of multiplication. You should recognize that if you see a, then you are multiplying your current total by 2 (this is a left shift operation). If you see a, you are multiplying your current total and adding. To ensure divisibility by 5, we should operate mod 5 and accept s when s mod 5= when it is interpreted as a binary number (and begins with ). Also, one must take care to construct a separate states for start and mod 5. The reason being, if they are the same states, then you would go to a trap state upon reading a. But what if you already read part of a string that started with and was equivalent to mod 5 (i.e. ) and then read a? Your DFA would trap when it shouldn't. Thus, two states should be made (labeled S and ) in the picture that prevent the aforementioned error from occurring.
2 3 S 2 4 trap, Question 3 This DFA accepts all strings with an odd number of ones. It tells us if we have seen an even number of ones (then we'll be in A) or an odd number of ones (we'll be in state B). Proof by induction on the length of the input string s that deltahat(a,s) iff s has an even # of 's Base case: s is of length. Clearly, if s is the empty string, then it has an even number of ones (). Thus, since we started in state A, delta(a,w)=a. Thus our base case is proven. Inductive step: Assume the DFA behaves as describes for all strings strictly shorter than s. Prove DFA behaves for strings of length s. Then our string is of the form s=xa, where x is or. Let a=. Thus, s has the same number of 's as x. If they both have an even number, by the IH, deltahat(a,x)=a. By definition of the states from the table, delta(a,)=a. If they both have an odd number of 's, then by the IH, deltahat(a,x)=b and the transition table states that delta(b,)=b. Therefore, delta(a,s)=a if and only if we have an even number of 's in the string. Let a=. Thus, s has more than x. Assume that x had an even number of ones. Then, by the IH, deltahat(a,x)=a. By the transition table, delta(a,)=b. Now assume x had an odd number of ones. By the IH, deltahat(a,x)=b. By the transition table, delta(b,)=a. Therefore, delta(a,s)=a if and only if there are an even number of 's in the string. Q.e.d.
3 Question 4(2.3.) Convert to a DFA this NFA: >p {p,q} {p} q {r} {r} r {s} empty set *s {s} {s} Solution: Use the subset construction. For the states of the DFA, create the power set of the NFA states {p} {p,q} {p} {q} {r} *{s} {p,q} {p,q,r} {p,r} {p,r} {p,q,s} {p} *{p,s} {p,q,s} {p,s} {q.r} *{q,s} *{r,s} {p,q,r} {p,q,r,s} {p,r} *{p,q,s} {p,q,r,s} {p,r,s} *{p,r,s} {p,q,s} {p,s} *{q,r,s} *{p,q,r,s} {p,q,r,s} {p,r,s} empty set A subset of the states of the NFA (i.e. a state in the DFA) will be a final state if the intersection of it with the final states in the NFA is non empty (i.e. it contains s). Mark these are your final states. Now, to compute the transition function, look at all the states p in the NFA, see what states the NFA goes to from p when it sees a or, and we take the union of all those states. (See pgs 6 62). We can rename the states if we wish to convince ourselves that in the DFA is a single state and not a set of states. Notice that there are unreachable states in the above table. They can be deleted. Here is a table with no unreachable states:
4 {p} {p,q} {p} {p,q} {p,q,r} {p,r} {p,r} {p,q,s} {p} *{p,s} {p,q,s} {p,s} {p,q,r} {p,q,r,s} {p,r} *{p,q,s} {p,q,r,s} {p,r,s} *{p,r,s} {p,q,s} {p,s} *{p,q,r,s} {p,q,r,s} {p,r,s} Question 5(2.3.4) a) Set of strings over {...} s.t. the final digit has been seen before Solution: See the figure. Nondeterministically guess what the final digit will be and transition accordingly. Not much else can be said in terms of words studying the picture should give you the intuition. b)set of strings over {...} s.t. the final digit has not been seen before. Solution: See the figure. This problem parallels part (a), but with different transitions out of the start stage. Once again, study the picture if you do not understand. c)set of strings of 's and 's such that there are two 's separated by a number of positions that is a multiple of 4. Solution: We need to keep track of how many positions have followed since we have seen a. So, it makes sense that we should go around in a loop mod 4 and then transition from that state to a final state upon seeing another. This is precisely what the NFA is doing. It will accept a string with right away, or if two consecutive 's do not occur, it will keep track of how many s separate the 's (mod 4). Upon seeing the 2 nd zero, if it is separated by a number of 's divisible by 4, the NFA will transition to the accept state and remain there.
5 NFA's for Question 5: a)......,2.. S b)......,2..,
6 c),, S mod4,, mod4 3mod4,, 2mod4
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 informationCSE 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 informationNondeterministic Finite Automata. Nondeterminism Subset Construction
Nondeterministic Finite Automata Nondeterminism Subset Construction 1 Nondeterminism A nondeterministic finite automaton has the ability to be in several states at once. Transitions from a state on an
More informationNondeterminism and Epsilon Transitions
Nondeterminism and Epsilon Transitions Mridul Aanjaneya Stanford University June 28, 22 Mridul Aanjaneya Automata Theory / 3 Challenge Problem Question Prove that any square with side length a power of
More informationDeterministic 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(pp ) PDAs and CFGs (Sec. 2.2)
(pp. 117-124) PDAs and CFGs (Sec. 2.2) A language is context free iff all strings in L can be generated by some context free grammar Theorem 2.20: L is Context Free iff a PDA accepts it I.e. if L is context
More informationPS2 - 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 informationLecture 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 informationDeterministic 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 informationNFA and regex. the Boolean algebra of languages. regular expressions. Informatics 1 School of Informatics, University of Edinburgh
NFA and regex cl the Boolean algebra of languages regular expressions Informatics The intersection of two regular languages is regular L = even numbers L = odd numbers L = mod L = mod Informatics The intersection
More informationNondeterministic 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 informationChapter 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(pp ) PDAs and CFGs (Sec. 2.2)
(pp. 117-124) PDAs and CFGs (Sec. 2.2) A language is context free iff all strings in L can be generated by some context free grammar Theorem 2.20: L is Context Free iff a PDA accepts it I.e. if L is context
More informationLecture 5: Minimizing DFAs
6.45 Lecture 5: Minimizing DFAs 6.45 Announcements: - Pset 2 is up (as of last night) - Dylan says: It s fire. - How was Pset? 2 DFAs NFAs DEFINITION Regular Languages Regular Expressions 3 4 Some Languages
More informationNondeterministic finite automata
Lecture 3 Nondeterministic finite automata This lecture is focused on the nondeterministic finite automata (NFA) model and its relationship to the DFA model. Nondeterminism is an important concept in the
More informationAuthor: Vivek Kulkarni ( )
Author: Vivek Kulkarni ( vivek_kulkarni@yahoo.com ) Chapter-2: Finite State Machines Solutions for Review Questions @ Oxford University Press 2013. All rights reserved. 1 Q.1 Construct Mealy and Moore
More informationT (s, xa) = T (T (s, x), a). The language recognized by M, denoted L(M), is the set of strings accepted by M. That is,
Recall A deterministic finite automaton is a five-tuple where S is a finite set of states, M = (S, Σ, T, s 0, F ) Σ is an alphabet the input alphabet, T : S Σ S is the transition function, s 0 S is the
More informationDeterministic 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 informationLecture 17: Language Recognition
Lecture 17: Language Recognition Finite State Automata Deterministic and Non-Deterministic Finite Automata Regular Expressions Push-Down Automata Turing Machines Modeling Computation When attempting to
More informationDeterministic 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 informationCS243, Logic and Computation Nondeterministic finite automata
CS243, Prof. Alvarez NONDETERMINISTIC FINITE AUTOMATA (NFA) Prof. Sergio A. Alvarez http://www.cs.bc.edu/ alvarez/ Maloney Hall, room 569 alvarez@cs.bc.edu Computer Science Department voice: (67) 552-4333
More informationClosure Properties of Regular Languages. Union, Intersection, Difference, Concatenation, Kleene Closure, Reversal, Homomorphism, Inverse Homomorphism
Closure Properties of Regular Languages Union, Intersection, Difference, Concatenation, Kleene Closure, Reversal, Homomorphism, Inverse Homomorphism Closure Properties Recall a closure property is a statement
More informationCS 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 informationCISC 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 informationCPSC 421: Tutorial #1
CPSC 421: Tutorial #1 October 14, 2016 Set Theory. 1. Let A be an arbitrary set, and let B = {x A : x / x}. That is, B contains all sets in A that do not contain themselves: For all y, ( ) y B if and only
More informationUNIT-III REGULAR LANGUAGES
Syllabus R9 Regulation REGULAR EXPRESSIONS UNIT-III REGULAR LANGUAGES Regular expressions are useful for representing certain sets of strings in an algebraic fashion. In arithmetic we can use the operations
More informationUNIT-II. NONDETERMINISTIC FINITE AUTOMATA WITH ε TRANSITIONS: SIGNIFICANCE. Use of ε-transitions. s t a r t. ε r. e g u l a r
Syllabus R9 Regulation UNIT-II NONDETERMINISTIC FINITE AUTOMATA WITH ε TRANSITIONS: In the automata theory, a nondeterministic finite automaton (NFA) or nondeterministic finite state machine is a finite
More informationIntro 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 informationLet us first give some intuitive idea about a state of a system and state transitions before describing finite automata.
Finite Automata Automata (singular: automation) are a particularly simple, but useful, model of computation. They were initially proposed as a simple model for the behavior of neurons. The concept of a
More informationChapter 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 informationSri vidya college of engineering and technology
Unit I FINITE AUTOMATA 1. Define hypothesis. The formal proof can be using deductive proof and inductive proof. The deductive proof consists of sequence of statements given with logical reasoning in order
More informationNon-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 informationSubset construction. We have defined for a DFA L(A) = {x Σ ˆδ(q 0, x) F } and for A NFA. For any NFA A we can build a DFA A D such that L(A) = L(A D )
Search algorithm Clever algorithm even for a single word Example: find abac in abaababac See Knuth-Morris-Pratt and String searching algorithm on wikipedia 2 Subset construction We have defined for a DFA
More informationProving languages to be nonregular
Proving languages to be nonregular We already know that there exist languages A Σ that are nonregular, for any choice of an alphabet Σ. This is because there are uncountably many languages in total and
More informationNondeterministic 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 informationFooling Sets and. Lecture 5
Fooling Sets and Introduction to Nondeterministic Finite Automata Lecture 5 Proving that a language is not regular Given a language, we saw how to prove it is regular (union, intersection, concatenation,
More information(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 informationHomework 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 informationJava II Finite Automata I
Java II Finite Automata I Bernd Kiefer Bernd.Kiefer@dfki.de Deutsches Forschungszentrum für künstliche Intelligenz November, 23 Processing Regular Expressions We already learned about Java s regular expression
More informationCSE 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 informationCS 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 informationTheory of Computation (VI) Yijia Chen Fudan University
Theory of Computation (VI) Yijia Chen Fudan University Review Forced handles Definition A handle h of a valid string v = xhy is a forced handle if h is the unique handle in every valid string xhŷ where
More informationAutomata 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 informationNote: In any grammar here, the meaning and usage of P (productions) is equivalent to R (rules).
Note: In any grammar here, the meaning and usage of P (productions) is equivalent to R (rules). 1a) G = ({R, S, T}, {0,1}, P, S) where P is: S R0R R R0R1R R1R0R T T 0T ε (S generates the first 0. R generates
More informationCS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 3 January 9, 2017 January 9, 2017 CS21 Lecture 3 1 Outline NFA, FA equivalence Regular Expressions FA and Regular Expressions January 9, 2017 CS21 Lecture 3 2
More informationTheory of Computation p.1/?? Theory of Computation p.2/?? Unknown: Implicitly a Boolean variable: true if a word is
Abstraction of Problems Data: abstracted as a word in a given alphabet. Σ: alphabet, a finite, non-empty set of symbols. Σ : all the words of finite length built up using Σ: Conditions: abstracted as a
More informationLecture Notes On THEORY OF COMPUTATION MODULE -1 UNIT - 2
BIJU PATNAIK UNIVERSITY OF TECHNOLOGY, ODISHA Lecture Notes On THEORY OF COMPUTATION MODULE -1 UNIT - 2 Prepared by, Dr. Subhendu Kumar Rath, BPUT, Odisha. UNIT 2 Structure NON-DETERMINISTIC FINITE AUTOMATA
More informationCS 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 informationQUESTION ONE (15 points each for 75 points total) For each of the following sets, say if the set is:
HW 6, CMSC 452. Morally DUE Mar 27 This HW is 200 points and counts twice as much as other HW This HW is THREE PAGES LONG (To make the midterm SHORTER, and to give you a break, there was NOT a question
More informationTheory of Computation
Theory of Computation Dr. Sarmad Abbasi Dr. Sarmad Abbasi () Theory of Computation / Lecture 3: Overview Decidability of Logical Theories Presburger arithmetic Decidability of Presburger Arithmetic Dr.
More informationSeptember 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 informationTheory 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 informationMost General computer?
Turing Machines Most General computer? DFAs are simple model of computation. Accept only the regular languages. Is there a kind of computer that can accept any language, or compute any function? Recall
More informationCSE 311: Foundations of Computing I Autumn 2014 Practice Final: Section X. Closed book, closed notes, no cell phones, no calculators.
CSE 311: Foundations of Computing I Autumn 014 Practice Final: Section X YY ZZ Name: UW ID: Instructions: Closed book, closed notes, no cell phones, no calculators. You have 110 minutes to complete the
More informationCSE 105 Theory of Computation Professor Jeanne Ferrante
CSE 105 Theory of Computation http://www.jflap.org/jflaptmp/ Professor Jeanne Ferrante 1 Today s agenda NFA Review and Design NFA s Equivalence to DFA s Another Closure Property proof for Regular Languages
More informationcse303 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 informationOctober 6, Equivalence of Pushdown Automata with Context-Free Gramm
Equivalence of Pushdown Automata with Context-Free Grammar October 6, 2013 Motivation Motivation CFG and PDA are equivalent in power: a CFG generates a context-free language and a PDA recognizes a context-free
More informationClosure 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 informationHKN CS/ECE 374 Midterm 1 Review. Nathan Bleier and Mahir Morshed
HKN CS/ECE 374 Midterm 1 Review Nathan Bleier and Mahir Morshed For the most part, all about strings! String induction (to some extent) Regular languages Regular expressions (regexps) Deterministic finite
More informationFinite Universes. L is a fixed-length language if it has length n for some
Finite Universes Finite Universes When the universe is finite (e.g., the interval 0, 2 1 ), all objects can be encoded by words of the same length. A language L has length n 0 if L =, or every word of
More informationCS5371 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 informationLecture 2: Connecting the Three Models
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 2: Connecting the Three Models David Mix Barrington and Alexis Maciel July 18, 2000
More informationCS 301. Lecture 18 Decidable languages. Stephen Checkoway. April 2, 2018
CS 301 Lecture 18 Decidable languages Stephen Checkoway April 2, 2018 1 / 26 Decidable language Recall, a language A is decidable if there is some TM M that 1 recognizes A (i.e., L(M) = A), and 2 halts
More informationLecture 4 Nondeterministic Finite Accepters
Lecture 4 Nondeterministic Finite Accepters COT 4420 Theory of Computation Section 2.2, 2.3 Nondeterminism A nondeterministic finite automaton can go to several states at once. Transitions from one state
More informationFinite Automata. BİL405 - Automata Theory and Formal Languages 1
Finite Automata BİL405 - Automata Theory and Formal Languages 1 Deterministic Finite Automata (DFA) A Deterministic Finite Automata (DFA) is a quintuple A = (Q,,, q 0, F) 1. Q is a finite set of states
More informationLanguages, 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 informationAutomata and Computability. Solutions to Exercises
Automata and Computability Solutions to Exercises Fall 28 Alexis Maciel Department of Computer Science Clarkson University Copyright c 28 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTATION
FORMAL LANGUAGES, AUTOMATA AND COMPUTATION DECIDABILITY ( LECTURE 15) SLIDES FOR 15-453 SPRING 2011 1 / 34 TURING MACHINES-SYNOPSIS The most general model of computation Computations of a TM are described
More informationTheory of computation: initial remarks (Chapter 11)
Theory of computation: initial remarks (Chapter 11) For many purposes, computation is elegantly modeled with simple mathematical objects: Turing machines, finite automata, pushdown automata, and such.
More informationClosure 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 informationChap. 1.2 NonDeterministic Finite Automata (NFA)
Chap. 1.2 NonDeterministic Finite Automata (NFA) DFAs: exactly 1 new state for any state & next char NFA: machine may not work same each time More than 1 transition rule for same state & input Any one
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
5-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 5-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY YOU NEED TO PICK UP THE SYLLABUS, THE COURSE SCHEDULE, THE PROJECT INFO SHEET, TODAY S CLASS NOTES
More informationClarifications 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 informationCS 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 informationCS5371 Theory of Computation. Lecture 12: Computability III (Decidable Languages relating to DFA, NFA, and CFG)
CS5371 Theory of Computation Lecture 12: Computability III (Decidable Languages relating to DFA, NFA, and CFG) Objectives Recall that decidable languages are languages that can be decided by TM (that means,
More informationInduction and Recursion
. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Induction and Recursion
More informationCSE 135: Introduction to Theory of Computation Equivalence of DFA and NFA
CSE 135: Introduction to Theory of Computation Equivalence of DFA and NFA Sungjin Im University of California, Merced 02-03-2015 Expressive Power of NFAs and DFAs Is there a language that is recognized
More informationInf2A: Converting from NFAs to DFAs and Closure Properties
1/43 Inf2A: Converting from NFAs to DFAs and Stuart Anderson School of Informatics University of Edinburgh October 13, 2009 Starter Questions 2/43 1 Can you devise a way of testing for any FSM M whether
More informationBefore we show how languages can be proven not regular, first, how would we show a language is regular?
CS35 Proving Languages not to be Regular Before we show how languages can be proven not regular, first, how would we show a language is regular? Although regular languages and automata are quite powerful
More informationECS 120: Theory of Computation UC Davis Phillip Rogaway February 16, Midterm Exam
ECS 120: Theory of Computation Handout MT UC Davis Phillip Rogaway February 16, 2012 Midterm Exam Instructions: The exam has six pages, including this cover page, printed out two-sided (no more wasted
More informationComputational Theory
Computational Theory Finite Automata and Regular Languages Curtis Larsen Dixie State University Computing and Design Fall 2018 Adapted from notes by Russ Ross Adapted from notes by Harry Lewis Curtis Larsen
More informationComputational 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 informationCS 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 information1 More finite deterministic automata
CS 125 Section #6 Finite automata October 18, 2016 1 More finite deterministic automata Exercise. Consider the following game with two players: Repeatedly flip a coin. On heads, player 1 gets a point.
More information6.2 Deeper Properties of Continuous Functions
6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 69 6.2 Deeper Properties of Continuous Functions 6.2. Intermediate Value Theorem and Consequences When one studies a function, one is usually interested in
More informationRegular Expressions and Language Properties
Regular Expressions and Language Properties Mridul Aanjaneya Stanford University July 3, 2012 Mridul Aanjaneya Automata Theory 1/ 47 Tentative Schedule HW #1: Out (07/03), Due (07/11) HW #2: Out (07/10),
More informationTuring Machines (TM) Deterministic Turing Machine (DTM) Nondeterministic Turing Machine (NDTM)
Turing Machines (TM) Deterministic Turing Machine (DTM) Nondeterministic Turing Machine (NDTM) 1 Deterministic Turing Machine (DTM).. B B 0 1 1 0 0 B B.. Finite Control Two-way, infinite tape, broken into
More information3515ICT: 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 informationTAFL 1 (ECS-403) Unit- II. 2.1 Regular Expression: The Operators of Regular Expressions: Building Regular Expressions
TAFL 1 (ECS-403) Unit- II 2.1 Regular Expression: 2.1.1 The Operators of Regular Expressions: 2.1.2 Building Regular Expressions 2.1.3 Precedence of Regular-Expression Operators 2.1.4 Algebraic laws for
More informationTheory of Computation p.1/?? Theory of Computation p.2/?? We develop examples of languages that are decidable
Decidable Languages We use languages to represent various computational problems because we have a terminology for dealing with languages Definition: A language is decidable if there is an algorithm (i.e.
More informationAssignment #2 COMP 3200 Spring 2012 Prof. Stucki
Assignment #2 COMP 3200 Spring 2012 Prof. Stucki 1) Construct deterministic finite automata accepting each of the following languages. In (a)-(c) the alphabet is = {0,1}. In (d)-(e) the alphabet is ASCII
More informationCMPSCI 250: Introduction to Computation. Lecture #22: From λ-nfa s to NFA s to DFA s David Mix Barrington 22 April 2013
CMPSCI 250: Introduction to Computation Lecture #22: From λ-nfa s to NFA s to DFA s David Mix Barrington 22 April 2013 λ-nfa s to NFA s to DFA s Reviewing the Three Models and Kleene s Theorem The Subset
More informationCSE 2001: Introduction to Theory of Computation Fall Suprakash Datta
CSE 2001: Introduction to Theory of Computation Fall 2012 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/2001 9/6/2012 CSE
More informationCSC173 Workshop: 13 Sept. Notes
CSC173 Workshop: 13 Sept. Notes Frank Ferraro Department of Computer Science University of Rochester September 14, 2010 1 Regular Languages and Equivalent Forms A language can be thought of a set L of
More informationTDDD65 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 informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automata Theory and Formal Languages TMV27/DIT32 LP4 28 Lecture 5 Ana Bove March 26th 28 Recap: Inductive sets, (terminating) recursive functions, structural induction To define an inductive set
More informationCSC 344 Algorithms and Complexity. Proof by Mathematical Induction
CSC 344 Algorithms and Complexity Lecture #1 Review of Mathematical Induction Proof by Mathematical Induction Many results in mathematics are claimed true for every positive integer. Any of these results
More informationAutomata & languages. A primer on the Theory of Computation. Laurent Vanbever. ETH Zürich (D-ITET) September,
Automata & languages A primer on the Theory of Computation Laurent Vanbever www.vanbever.eu ETH Zürich (D-ITET) September, 24 2015 Last week was all about Deterministic Finite Automaton We saw three main
More informationAutomata 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 informationComputability Crib Sheet
Computer Science and Engineering, UCSD Winter 10 CSE 200: Computability and Complexity Instructor: Mihir Bellare Computability Crib Sheet January 3, 2010 Computability Crib Sheet This is a quick reference
More information