Automata Theory. Lecture on Discussion Course of CS120. Runzhe SJTU ACM CLASS
|
|
- Willa Gray
- 5 years ago
- Views:
Transcription
1 Automata Theory Lecture on Discussion Course of CS2
2 This Lecture is about Mathematical Models of Computation.
3 Why Should I Care? - Ways of thinking. - Theory can drive practice. - Don t be an Instrumentalist.
4 Mathematical Models of Computation (predated computers as we know them) Automata and Languages: (94 s) finite automata, regular languages, pushdown automata, context-free languages, pumping lemmas. 2 Computability Theory: (93-4 s) Turing Machines, decidability, reducibility, the arithmetic hierarchy, the recursion theorem, the Post correspondence problem. 3 Complexity Theory and Applications: (96-7 s) time complexity, classes P and NP, NP-completeness, space complexity PSPACE, PSPACE-completeness, the polynomial hierarchy, randomized complexity, classes RP and BPP.
5
6 Let me emphasize Proofs a bit more. A good proof:. Correct 2. Easy to understand
7 Suppose with. True or False: There are always two numbers in one divides the other. such that
8 LEVEL HINT : THE PIGEONHOLE PRINCIPLE If you put 6 pigeons in 5 holes then at least one hole will have more than one pigeon
9 LEVEL HINT : THE PIGEONHOLE PRINCIPLE If you put 6 pigeons in 5 holes then at least one hole will have more than one pigeon
10 LEVEL HINT : THE PIGEONHOLE PRINCIPLE If you put 6 pigeons in 5 holes then at least one hole will have more than one pigeon HINT 2: Every integer can be written as, where is an odd number
11 LEVEL 2 Proof Idea: Given: with. Show: There is an integer and and. such that
12 LEVEL 3 Proof: Suppose with. Write every number in A as a = 2 k m, where m is an odd number. How many odd numbers in {,,2n-}? n Since A = n+, there must be two numbers in A with the same odd part. Say a and a 2 have the same odd part m. Then a = 2 i m and a 2 = 2 j m, so one must divide the other.
13 We expect your proofs to have three levels: The first level should be a one-word or one-phrase HINT of the proof (e.g. Proof by contradiction, Proof by induction, Follows from the pigeonhole principle ) The second level should be a short oneparagraph description or KEY IDEA The third level should be the FULL PROOF
14 Double standards :-) HINT + KEY IDEA (+ FULL PROOF)
15 Deterministic Finite Automata (DFA)
16 NOTATION An alphabet Σ is a finite set (e.g., Σ = {,}) A string over Σ is a finite-length sequence of elements of Σ Σ* denotes the set of finite length sequences of elements of Σ For x a string, x is the length of x The unique string of length will be denoted by ε and will be called the empty or null string A language over Σ is a set of strings over Σ, ie, a subset of Σ*
17 Finite Automata finite # of internal states Input (String) -> -> Output (Yes/No) Divice
18 A Deterministic Finite Automata, Read string left to right The machine accepts a string if the process ends in a double circle.
19 A Deterministic Finite Automata states q, q q 3 q 2 states The machine accepts a string if the process ends in a double circle.
20 A Deterministic Finite Automata states final states (F) q, q q 3 start state (q ) q 2 states The machine accepts a string if the process ends in a double circle.
21 A Deterministic Finite Automata is represented by a 5-tuple M = (Q, Σ, δ, q, F) : Q is the set of states (finite) Σ is the alphabet (finite) δ : Q Σ Q is the transition function q Q is the start state F Q is the set of accept states Let w,..., w n Σ and w = w... w n Σ* Then M accepts w if there are r, r,..., r n Q, s.t. r =q δ(r i, w i+ ) = r i+, for i =,..., n-, and r n F
22 A Deterministic Finite Automata is represented by a 5-tuple M = (Q, Σ, δ, q, F) : Q is the set of states (finite) Σ is the alphabet (finite) δ : Q Σ Q is the transition function q Q is the start state F Q is the set of accept states L(M) = the language of machine M = set of all strings machine M accepts
23 Warming-up q, L(M) ={,}*
24 Warming-up L(M) ={,}*
25 Warming-up L(M) = { w w has an even number of s}
26 Warming-up q q L(M) = { w w has an even number of s}
27 Warming-up Build an automaton that accepts all and only those strings that contain
28 Warming-up Build an automaton that accepts all and only those strings that contain, q q q q
29 Regular Language A language L is regular if it is recognized by a deterministic finite automaton (DFA), i.e. if there is a DFA M such that L = L (M). L = { w w contains } is regular L = { w w has an even number of s} is regular
30 UNION THEOREM Given two languages, L and L 2, define the union of L and L 2 as L L 2 = { w w L or w L 2 } Theorem: The union of two regular languages is also a regular language.
31 Theorem: The union of two regular languages is also a regular language. Proof: Let M = (Q, Σ, δ, q, F ) be finite automaton for L and 2 M 2 = (Q 2, Σ, δ 2, q, F 2 ) be finite automaton for L 2 We want to construct a finite automaton M = (Q, Σ, δ, q, F) that recognizes L = L L 2
32 Theorem: The union of two regular languages is also a regular language. Idea: Run both M and M 2 at the same time! Q = pairs of states, one from M and one from M 2 = { (q, q 2 ) q Q and q 2 Q 2 } = Q Q 2 2 q = (q, q ) F = { (q, q 2 ) q F or q 2 F 2 } δ( (q,q 2 ), σ) = (δ (q, σ), δ 2 (q 2, σ))
33 Theorem: The union of two regular languages is also a regular language. M = q q M 2 = p p
34 Theorem: The union of two regular languages is also a regular language. q,p q,p M = q,p q,p
35 INTERSECTION THEOREM Given two languages, L and L 2, define the intersection of L and L 2 as L L 2 = { w w L and w L 2 } Theorem: The intersection of two regular languages is also a regular language.
36 COMPLEMENT THEOREM Given language L, define the complement of L as L= { w Σ* w L } Theorem: The complement of a regular languages is also a regular language.
37 THE REGULAR OPERATIONS Union: L L 2 = { w w L or w L 2 } Intersection: L L 2 = { w w L and w L 2 } Negation: L = { w Σ* w L } Reverse: L R = { w w k w k w L } Concatenation: L L 2 = { vw v L and w L 2 } Star: L* = { w w k k and each w i L }
38 REVERSE THEOREM Given language L, define the reverse of L R as L R = { w w k w k w L } Theorem: The reverse of a regular languages is also a regular language.
39 Theorem: The reverse of a regular languages is also a regular language. Proof: Let M = (Q, Σ, δ, q, F) that recognizes L. If M accepts w then w describes a directed path in M from start to an accept state We want to construct a finite automaton M R as M with the arrows reversed.
40 M R IS NOT ALWAYS A DFA! - It may have many start states - Some states may have too many outgoing edges, or none
41 M R IS NOT ALWAYS A DFA!,
42 M R IS NOT ALWAYS A DFA!,
43 Non-Determinism, We will say that the machine accepts if there is some way to make it reach an accept state
44 Example,,,ε At each state, possibly zero, one or many out arrows for each σ Σ or with label ε
45 Example, ε Possibly many start states
46 Example L(M)={,}
47 A Non-Deterministic Finite Automata (NFA) is represented by a 5-tuple N = (Q, Σ, δ, Q, F) : Q is the set of states (finite) Σ is the alphabet (finite) δ : Q Σ {ε} 2 Q is the transition function Q Q is the set of start state F Q is the set of accept states Let w,..., w n Σ {ε} and w = w... w n Σ ε * Then N accepts w if there are r, r,..., r n Q, s.t. r =Q r i+ δ(r i, w i+ ), for i =,..., n-, and r n F
48 A Non-Deterministic Finite Automata (NFA) is represented by a 5-tuple N = (Q, Σ, δ, Q, F) : Q is the set of states (finite) Σ is the alphabet (finite) δ : Q Σ {ε} 2 Q is the transition function Q Q is the set of start state F Q is the set of accept states L(N) = the language of machine N = set of all strings machine N accepts
49 Deterministic Computation Non-Deterministic Computation reject accept or reject accept
50 NFAs ARE SIMPLER THAN DFAs An NFA that recognizes the language {}: A DFA that recognizes the language {}:,,
51 FROM NFA TO DFA Theorem: Every NFA has an equivalent DFA N is equivalent to M if L(N) = L (M) Corollary: A language is regular iff it is recognized by an NFA Corollary: L is regular iff L R is regular
52 FROM NFA TO DFA Input: N = (Q, Σ, δ, Q, F) Output: M = (Q, Σ, δ, q, F ) reject To learn if NFA accepts, we could do the computation in parallel, maintaining the set of states where all threads are. Idea: Q = 2 Q accept
53 FROM NFA TO DFA Input: N = (Q, Σ, δ, Q, F) Output: M = (Q, Σ, δ, q, F ) Q = 2 Q δ : Q Σ Q ε ε ε δ (R,σ) = ε( δ(r,σ) ), r R q = ε(q ) F = { R Q f R for some f F } For R Q, the ε-closure of R, ε(r) = {q that can be reached from some r R by traveling along zero or more ε arrows},
54 FROM NFA TO DFA () + () +
55 Regular Languages Closure Under Concatenation DFA -> DFA2 NFA Given DFAs M and M 2, construct NFA by connecting all accept states in M to start states in M 2.
56 Regular Languages Closure Under Star Let L be a regular language and M be a DFA for L We construct an NFA N that recognizes L* ε, ε ε
57 THE REGULAR OPERATIONS Union: L L 2 = { w w L or w L 2 } Intersection: L L 2 = { w w L and w L 2 } Negation: L = { w Σ* w L } Reverse: L R = { w w k w k w L } Concatenation: L L 2 = { vw v L and w L 2 } Star: L* = { w w k k and each w i L }
58 Are all languages regular?
59 Consider the language L = { n n n > } No finite automaton accepts this language. Can you prove this?
60 Idea? n n is not regular. No machine has enough states to keep track of the number of s it might encounter
61 That is a fairly weak argument Consider the following example
62 L = { w w has equal number of occurrences of and } No machine has enough states to keep track of the number of s it might encounter.
63 THE PUMPING LEMMA Let L be a regular language with L = Then there exists a positive integer P such that Any x L, x P can be written as x = uvw where:. v > 2. uv P 3. uv i w L for any i
64 THE PUMPING LEMMA Assume x L is such that x P Let P be the number of states in M We show x = uvw where:. v > 2. uv P 3. uv i w L for any i PIGEONHOLE: There must be j > i such that q i = q j u v w q q i q j q x
65 THE PUMPING LEMMA Assume x L is such that x P Let P be the number of states in M We show x = uvw where:. v > 2. uv P 3. uv i w L for any i PIGEONHOLE: There must be j > i such that q i = q j u v w q q i q j q x
66 THE PUMPING LEMMA L = { n n n > } HINT: Assume L is regular, and try pumping. u v w q q i q j q x
67 THE PUMPING LEMMA L = { n n n > } HINT: Assume L is regular, and try pumping. u v w q q i q j q x
68 References: CMU FLAC: Formal Languages, Automata and Computation, Spring 24 Stanford CS54: Introduction to Automata and Complexity Theory, Spring 29 John E. Hopcroft, etc., Introduction to Automata Theory, Languages, and Computation(Third Edition). Pearson; 26
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 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 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 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 informationCS 154 Introduction to Automata and Complexity Theory
CS 154 Introduction to Automata and Complexity Theory cs154.stanford.edu 1 INSTRUCTORS & TAs Ryan Williams Cody Murray Lera Nikolaenko Sunny Rajan 2 Textbook 3 Homework / Problem Sets Homework will be
More informationEquivalence 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 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 informationFORMAL 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 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 informationGreat 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 informationTheory 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 informationCS 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 informationCS 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 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 informationFinite 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 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 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 informationCOM364 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 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 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 informationCSE 105 THEORY OF COMPUTATION. Spring 2018 review class
CSE 105 THEORY OF COMPUTATION Spring 2018 review class Today's learning goals Summarize key concepts, ideas, themes from CSE 105. Approach your final exam studying with confidence. Identify areas to focus
More informationIntroduction 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 informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Summarize key concepts, ideas, themes from CSE 105. Approach your final exam studying with
More informationName: Student ID: Instructions:
Instructions: Name: CSE 322 Autumn 2001: Midterm Exam (closed book, closed notes except for 1-page summary) Total: 100 points, 5 questions, 20 points each. Time: 50 minutes 1. Write your name and student
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 informationIntroduction 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 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 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 Automata (contd)
Finite Automata (contd) CS 2800: Discrete Structures, Fall 2014 Sid Chaudhuri Recap: Deterministic Finite Automaton A DFA is a 5-tuple M = (Q, Σ, δ, q 0, F) Q is a fnite set of states Σ is a fnite input
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 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 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 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 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 informationTheory 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 informationCSE 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 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 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 informationHarvard 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 informationDeterministic 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 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 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 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 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 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 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 informationFurther discussion of Turing machines
Further discussion of Turing machines In this lecture we will discuss various aspects of decidable and Turing-recognizable languages that were not mentioned in previous lectures. In particular, we will
More informationFormal 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 informationFinal exam study sheet for CS3719 Turing machines and decidability.
Final exam study sheet for CS3719 Turing machines and decidability. A Turing machine is a finite automaton with an infinite memory (tape). Formally, a Turing machine is a 6-tuple M = (Q, Σ, Γ, δ, q 0,
More informationCS21 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 informationIntroduction to the Theory of Computing
Introduction to the Theory of Computing Lecture notes for CS 360 John Watrous School of Computer Science and Institute for Quantum Computing University of Waterloo June 27, 2017 This work is licensed under
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 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 informationCS: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 informationFORMAL 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 informationTheory of Computation
Theory of Computation COMP363/COMP6363 Prerequisites: COMP4 and COMP 6 (Foundations of Computing) Textbook: Introduction to Automata Theory, Languages and Computation John E. Hopcroft, Rajeev Motwani,
More informationSYLLABUS. Introduction to Finite Automata, Central Concepts of Automata Theory. CHAPTER - 3 : REGULAR EXPRESSIONS AND LANGUAGES
Contents i SYLLABUS UNIT - I CHAPTER - 1 : AUT UTOMA OMATA Introduction to Finite Automata, Central Concepts of Automata Theory. CHAPTER - 2 : FINITE AUT UTOMA OMATA An Informal Picture of Finite Automata,
More informationDM17. Beregnelighed. Jacob Aae Mikkelsen
DM17 Beregnelighed Jacob Aae Mikkelsen January 12, 2007 CONTENTS Contents 1 Introduction 2 1.1 Operations with languages...................... 2 2 Finite Automata 3 2.1 Regular expressions/languages....................
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 3.3, 4.1 State and use the Church-Turing thesis. Give examples of decidable problems.
More informationCSCE 551 Final Exam, Spring 2004 Answer Key
CSCE 551 Final Exam, Spring 2004 Answer Key 1. (10 points) Using any method you like (including intuition), give the unique minimal DFA equivalent to the following NFA: 0 1 2 0 5 1 3 4 If your answer is
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 informationWhat 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 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 informationThe Pumping Lemma and Closure Properties
The Pumping Lemma and Closure Properties Mridul Aanjaneya Stanford University July 5, 2012 Mridul Aanjaneya Automata Theory 1/ 27 Tentative Schedule HW #1: Out (07/03), Due (07/11) HW #2: Out (07/10),
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 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 informationTHEORY OF COMPUTATION (AUBER) EXAM CRIB SHEET
THEORY OF COMPUTATION (AUBER) EXAM CRIB SHEET Regular Languages and FA A language is a set of strings over a finite alphabet Σ. All languages are finite or countably infinite. The set of all languages
More informationLecture 24: Randomized Complexity, Course Summary
6.045 Lecture 24: Randomized Complexity, Course Summary 1 1/4 1/16 1/4 1/4 1/32 1/16 1/32 Probabilistic TMs 1/16 A probabilistic TM M is a nondeterministic TM where: Each nondeterministic step is called
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 informationComputational Models Lecture 2 1
Computational Models Lecture 2 1 Handout Mode Iftach Haitner. Tel Aviv University. October 30, 2017 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice Herlihy, Brown University.
More informationLanguages. 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 informationRecap 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 informationComputational Models - Lecture 3
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Computational Models - Lecture 3 Equivalence of regular expressions and regular languages (lukewarm leftover
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 informationComputational Models - Lecture 4
Computational Models - Lecture 4 Regular languages: The Myhill-Nerode Theorem Context-free Grammars Chomsky Normal Form Pumping Lemma for context free languages Non context-free languages: Examples Push
More informationComputer 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 informationComputational Models Lecture 2 1
Computational Models Lecture 2 1 Handout Mode Ronitt Rubinfeld and Iftach Haitner. Tel Aviv University. March 16/18, 2015 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice
More informationRegular Languages. Problem Characterize those Languages recognized by Finite Automata.
Regular Expressions Regular Languages Fundamental Question -- Cardinality Alphabet = Σ is finite Strings = Σ is countable Languages = P(Σ ) is uncountable # Finite Automata is countable -- Q Σ +1 transition
More informationCritical 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 informationCPS 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 informationCS Lecture 29 P, NP, and NP-Completeness. k ) for all k. Fall The class P. The class NP
CS 301 - Lecture 29 P, NP, and NP-Completeness Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of
More informationCHAPTER 1 Regular Languages. Contents
Finite Automata (FA or DFA) CHAPTER Regular Languages Contents definitions, examples, designing, regular operations Non-deterministic Finite Automata (NFA) definitions, euivalence of NFAs and DFAs, closure
More informationFinite Automata. Mahesh Viswanathan
Finite Automata Mahesh Viswanathan In this lecture, we will consider different models of finite state machines and study their relative power. These notes assume that the reader is familiar with DFAs,
More informationAutomata: 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 informationAC68 FINITE AUTOMATA & FORMULA LANGUAGES DEC 2013
Q.2 a. Prove by mathematical induction n 4 4n 2 is divisible by 3 for n 0. Basic step: For n = 0, n 3 n = 0 which is divisible by 3. Induction hypothesis: Let p(n) = n 3 n is divisible by 3. Induction
More informationTheory 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 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 informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2018 http://cseweb.ucsd.edu/classes/sp18/cse105-ab/ Today's learning goals Sipser Ch 4.1 Explain what it means for a problem to be decidable. Justify the use of encoding.
More informationAutomata Theory, Computability and Complexity
Automata Theory, Computability and Complexity Mridul Aanjaneya Stanford University June 26, 22 Mridul Aanjaneya Automata Theory / 64 Course Staff Instructor: Mridul Aanjaneya Office Hours: 2:PM - 4:PM,
More informationGEETANJALI INSTITUTE OF TECHNICAL STUDIES, UDAIPUR I
GEETANJALI INSTITUTE OF TECHNICAL STUDIES, UDAIPUR I Internal Examination 2017-18 B.Tech III Year VI Semester Sub: Theory of Computation (6CS3A) Time: 1 Hour 30 min. Max Marks: 40 Note: Attempt all three
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Ch 4.1 Explain what it means for a problem to be decidable. Justify the use of encoding.
More informationDeterministic Finite Automaton (DFA)
1 Lecture Overview Deterministic Finite Automata (DFA) o accepting a string o defining a language Nondeterministic Finite Automata (NFA) o converting to DFA (subset construction) o constructed from a regular
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 informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 2 Define push down automata Trace the computation of a push down automaton Design
More informationV Honors Theory of Computation
V22.0453-001 Honors Theory of Computation Problem Set 3 Solutions Problem 1 Solution: The class of languages recognized by these machines is the exactly the class of regular languages, thus this TM variant
More informationFalse. They are the same language.
CS 3100 Models of Computation Fall 2010 Notes 8, Posted online: September 16, 2010 These problems will be helpful for Midterm-1. More solutions will be worked out. The midterm exam itself won t be this
More informationSeptember 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 informationMore 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 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 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 information