CS154. Turing Machines. Turing Machine. Turing Machines versus DFAs FINITE STATE CONTROL AI N P U T INFINITE TAPE. read write move.


 Randall Holland
 1 years ago
 Views:
Transcription
1 CS54 Turing Machines Turing Machine q 0 AI N P U T IN TAPE read write move read write move Language = {0} q This Turing machine recognizes the language {0} Turing Machines versus DFAs TM can both write to and read from the tape The head can move left and right The input doesn t have to be read entirely, and the computation can continue further (even, forever) after all input has been read Accept and Reject take immediate effect Deciding the language L = { w#w w {0,}* } q 0, FIND q, # FIND # q #, FIND q 0, FIND q, FIND q GO LEFT X X # 0# 0X. If there s no # on the tape,. 2. While there is a bit to the left of #, Replace the first bit with X, and check if the first bit to the right of the # is identical. (If not,.) Replace that bit with an X too. 3. If there s a bit to the right of #, then else accept
2 Definition: A Turing Machine is a 7tuple T = (Q, Σ, Γ, δ, q 0,, q ), where: Q is a finite set of s Σ is the input alphabet, where Σ Γ is the tape alphabet, where Γand Σ Γ δ : Q Γ Q Γ {L, R} q 0 Q is the start Q is the accept q Q is the, and q Turing Machine Configurations q corresponds to the configuration: 00q Defining Acceptance and Rejection Let C and C 2 be configurations of M We say C yields C 2 if, after running M in C for one step, M is then in configuration C 2 Suppose δ(q, b) = (q 2, c, L) Then aaq bb yields aq 2 acb Suppose δ(q, a) = (q 2, c, R) Then cabq a yields cabcq 2 Let w Σ* and M be a Turing machine M accepts w if there are configs C 0, C,..., C k, s.t. C 0 = q 0 w C i yields C i+ for i = 0,..., k, and C k contains the accepting A TM M recognizes a language L iff M accepts all and only those strings in L A language L is called recognizable or recursively enumerable (r.e.) iff some TM recognizes L A TM M decides a language L iff M accepts all strings in L and s all strings not in L A language L is called decidable or recursive iff some TM decides L A language is called recognizable (recursively enumerable) if some TM recognizes it A language is called decidable (recursive) if some TM decides it recognizable languages decidable languages { 02 n n 0 } PSEUDOCODE:. Sweep from left to right, cross out every other 0 2. If in stage, the tape had only one 0, accept 3. If in stage, the tape had an odd number of 0 s, 4. Move the head back to the first input symbol. 5. Go to stage. Why does this work? Idea: Every time we return to stage, the number of 0 s on the tape is halved. 2
3 2 { 0 n n 0 } q 0 q 0, R q q 2 0 x, R 0 x, R x x, L 0 0, L q 3 q 4 C = {a i b j c k k = i*j, and i, j, k } PSEUDOCODE:. If the input doesn t match a*b*c*,. 2. Move the head back to the leftmost symbol. 3. Cross off an a, scan to the right until b. Sweep between b s and c s, crossing off one of each until all b s are crossed off. If all c s get crossed off while doing this,. 4. Uncross all the b s. If there s another a left, then repeat stage 3. If all a s are crossed out, Check if all c s are crossed off. If yes, then accept, else. C = {a i b j c k k = i*j, and i, j, k } aabbbcccccc xabbbcccccc xayyyzzzccc xabbbzzzccc xxyyyzzzzzz Multitape Turing Machines δ : Q Γ k Q Γ k {L,R} k k # # # # # # 3
4 # # # # # # Nondeterministic TMs Theorem: Every nondeterministic Turing machine N can be transformed into a single tape Turing Machine M that recognizes the same language. Proof Idea (more details in Sipser): M(w): For all strings C {Q Γ #}* in lex. order, Check if C = C 0 # #C k where C 0,,C k is some accepting computation history for N on w. If so, accept. We can encode a TM as a bit string. n s 0 n 0 m 0 k 0 s 0 t 0 r 0 u m tape symbols (first k are input symbols) start accept blank symbol ( (p, a), (q, b, L) ) = 0 p 0 a 0 q 0 b 0 ( (p, a), (q, b, R) ) = 0 p 0 a 0 q 0 b Similarly, we can encode DFAs and NFAs as bit strings. So we can define the following languages: A DFA = { (B, w) B is a DFA that accepts string w } A NFA = { (B, w) B is an NFA that accepts string w } A TM = { (C, w) C is a TM that accepts string w } (Can define (x, y) := 0 x xy) A DFA = { (D, w) D is a DFA that accepts string w } Theorem: A DFA is decidable Proof Idea: Directly simulate D on w! A NFA = { (N, w) N is an NFA that accepts string w } Theorem: A NFA is decidable. (Why?) A TM = { (M, w) M is a TM that accepts string w } A TM is recognizable but not decidable 4
5 Universality of Turing Machines Theorem: There is a universal Turing machine U that can take as input  the code of an arbitrary TM M  an input string w, such that U(M, w) accepts iff M(w) accepts. This is a fundamental property: Turing machines can run their own code! Note that DFAs/NFAs do not have this property. That is, A DFA is not regular. The ChurchTuring Thesis Everyone s Intuitive Notion = Turing Machines of Algorithms This thesis is tested every time you write a program that does something! A TM recognizes a language if it accepts all and only those strings in the language A language is called recognizable or recursively enumerable, (or r.e.) if some TM recognizes it w Σ* TM w L? yes no w Σ* TM w L? yes no A TM decides a language if it accepts all strings in the language and s all strings not in the language A language is called decidable (or recursive) if some TM decides it accept L is decidable (recursive) accept or no output L is recognizable (recursively enumerable) Theorem: L is decidable if both L and L are recursively enumerable Recall: Given L Σ*, define L := Σ*  L Theorem: L is decidable iff both L and L are recognizable Given: a TM M that recognizes A and a TM M2 that recognizes A, we can build a new machine M that decides A How? Any ideas? M always accepts x, when x is in A M2 always accepts x, when x is not in A There are languages over {0,} that are not decidable Assuming the ChurchTuring Thesis, this means there are problems that NO computing device can solve We can prove this using a counting argument. We will show there is no onto function from the set of all Turing Machines to the set of all languages over {0,}. (But it works for any Σ) That is, every mapping from Turing machines to languages must miss some language. 5
3515ICT Theory of Computation Turing Machines
Griffith University 3515ICT Theory of Computation Turing Machines (Based loosely on slides by Harald Søndergaard of The University of Melbourne) 90 Overview Turing machines: a general model of computation
More informationTuring Machines: An Introduction
CIT 596 Theory of Computation 1 We have seen several abstract models of computing devices: Deterministic Finite Automata, Nondeterministic Finite Automata, Nondeterministic Finite Automata with ɛtransitions,
More informationCSE 135: Introduction to Theory of Computation Decidability and Recognizability
CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 0428, 302014 HighLevel Descriptions of Computation Instead of giving a Turing
More informationCS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 ChurchTuring thesis Let s recap how it all started. In 1990, Hilbert stated a
More informationAutomata and Computability. Solutions to Exercises
Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata
More informationComputability Theory
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 110.
More informationNotes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
More informationTheory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras
Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding
More informationComputational Models Lecture 8, Spring 2009
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance
More information6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch
6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 4 Nancy Lynch Today Two more models of computation: Nondeterministic Finite Automata (NFAs)
More informationTheory of Computation Chapter 2: Turing Machines
Theory of Computation Chapter 2: Turing Machines GuanShieng Huang Feb. 24, 2003 Feb. 19, 2006 00 Turing Machine δ K 0111000a 01bb 1 Definition of TMs A Turing Machine is a quadruple M = (K, Σ, δ, s),
More informationCAs and Turing Machines. The Basis for Universal Computation
CAs and Turing Machines The Basis for Universal Computation What We Mean By Universal When we claim universal computation we mean that the CA is capable of calculating anything that could possibly be calculated*.
More information(IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.
3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem
More informationComputing Functions with Turing Machines
CS 30  Lecture 20 Combining Turing Machines and Turing s Thesis Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata
More informationIntroduction to Turing Machines
Automata Theory, Languages and Computation  Mírian HalfeldFerrari p. 1/2 Introduction to Turing Machines SITE : http://www.sir.blois.univtours.fr/ mirian/ Automata Theory, Languages and Computation
More informationTuring Machines, Part I
Turing Machines, Part I Languages The $64,000 Question What is a language? What is a class of languages? Computer Science Theory 2 1 Now our picture looks like Context Free Languages Deterministic Context
More informationWe give a basic overview of the mathematical background required for this course.
1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The
More informationFinite Automata and Formal Languages
Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Lecture 14 May 19th 2011 Overview of today s lecture: Turing Machines Pushdown Automata Overview of the Course Undecidable and Intractable Problems
More informationPushdown Automata. place the input head on the leftmost input symbol. while symbol read = b and pile contains discs advance head remove disc from pile
Pushdown Automata In the last section we found that restricting the computational power of computing devices produced solvable decision problems for the class of sets accepted by finite automata. But along
More informationOutline. Database Theory. Perfect Query Optimization. Turing Machines. Question. Definition VU , SS Trakhtenbrot s Theorem
Database Theory Database Theory Outline Database Theory VU 181.140, SS 2016 4. Trakhtenbrot s Theorem Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 4.
More informationA TM. Recall the A TM language: A TM = { M, w M is a TM and M accepts w.} is undecidable. Theorem: The language. p.2/?
p.1/? Reducibility We say that a problem Q reduces to problem P if we can use P to solve Q. In the context of decidability we have the following templates : If A reduces to B and B is decidable, then so
More informationLecture 2: Universality
CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal
More information1 Definition of a Turing machine
Introduction to Algorithms Notes on Turing Machines CS 4820, Spring 2012 April 216, 2012 1 Definition of a Turing machine Turing machines are an abstract model of computation. They provide a precise,
More information24 Uses of Turing Machines
Formal Language and Automata Theory: CS2004 24 Uses of Turing Machines 24 Introduction We have previously covered the application of Turing Machine as a recognizer and decider In this lecture we will discuss
More informationFormal Languages and Automata Theory  Regular Expressions and Finite Automata 
Formal Languages and Automata Theory  Regular Expressions and Finite Automata  Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
More informationComputing basics. Ruurd Kuiper
Computing basics Ruurd Kuiper October 29, 2009 Overview (cf Schaum Chapter 1) Basic computing science is about using computers to do things for us. These things amount to processing data. The way a computer
More informationTuring Machine and the Conceptual Problems of Computational Theory
Research Inventy: International Journal of Engineering And Science Vol.4, Issue 4 (April 204), PP 5360 Issn (e): 2278472, Issn (p):2396483, www.researchinventy.com Turing Machine and the Conceptual
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTATION
FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15453 SPRING 2011 2 / 20 REDUCIBILITY
More informationHomework Five Solution CSE 355
Homework Five Solution CSE 355 Due: 17 April 01 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. Problem 1: Linz 8.1.13 and
More informationPractice Problems for Final Exam: Solutions CS 341: Foundations of Computer Science II Prof. Marvin K. Nakayama
Practice Problems for Final Exam: Solutions CS 341: Foundations of Computer Science II Prof. Marvin K. Nakayama 1. Short answers: (a) Define the following terms and concepts: i. Union, intersection, set
More informationDeterministic Finite Automata
1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function
More informationECS 120 Lesson 17 Church s Thesis, The Universal Turing Machine
ECS 120 Lesson 17 Church s Thesis, The Universal Turing Machine Oliver Kreylos Monday, May 7th, 2001 In the last lecture, we looked at the computation of Turing Machines, and also at some variants of Turing
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationImplementation of Recursively Enumerable Languages using Universal Turing Machine in JFLAP
International Journal of Information and Computation Technology. ISSN 09742239 Volume 4, Number 1 (2014), pp. 7984 International Research Publications House http://www. irphouse.com /ijict.htm Implementation
More informationCS 301 Course Information
CS 301: Languages and Automata January 9, 2009 CS 301 Course Information Prof. Robert H. Sloan Handout 1 Lecture: Tuesday Thursday, 2:00 3:15, LC A5 Weekly Problem Session: Wednesday, 4:00 4:50 p.m., LC
More informationThe Halting Problem is Undecidable
185 Corollary G = { M, w w L(M) } is not Turingrecognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine
More informationFinal Exam  Comments
University of Virginia  cs3102: Theory of Computation Spring 2010 Final Exam  Comments Problem 1: Definitions. (Average 7.2 / 10) For each term, provide a clear, concise, and precise definition from
More informationFinite Automata. Reading: Chapter 2
Finite Automata Reading: Chapter 2 1 Finite Automaton (FA) Informally, a state diagram that comprehensively captures all possible states and transitions that a machine can take while responding to a stream
More informationRegular Languages and Finite State Machines
Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries  some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection
More informationthe lemma. Keep in mind the following facts about regular languages:
CPS 2: Discrete Mathematics Instructor: Bruce Maggs Assignment Due: Wednesday September 2, 27 A Tool for Proving Irregularity (25 points) When proving that a language isn t regular, a tool that is often
More informationNPCompleteness and Cook s Theorem
NPCompleteness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:
More informationTHE TURING DEGREES AND THEIR LACK OF LINEAR ORDER
THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More informationLecture 36: Turing Machines [Fa 14]
Caveat lector: This is the zeroth (draft) edition of this lecture note. In particular, some topics still need to be written. Please send bug reports and suggestions to jeffe@illinois.edu. Think globally,
More information6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationHonors Class (Foundations of) Informatics. Tom Verhoeff. Department of Mathematics & Computer Science Software Engineering & Technology
Honors Class (Foundations of) Informatics Tom Verhoeff Department of Mathematics & Computer Science Software Engineering & Technology www.win.tue.nl/~wstomv/edu/hci c 2011, T. Verhoeff @ TUE.NL 1/20 Information
More informationC H A P T E R Regular Expressions regular expression
7 CHAPTER Regular Expressions Most programmers and other powerusers of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun
More informationWelcome to... Problem Analysis and Complexity Theory 716.054, 3 VU
Welcome to... Problem Analysis and Complexity Theory 716.054, 3 VU Birgit Vogtenhuber Institute for Software Technology email: bvogt@ist.tugraz.at office hour: Tuesday 10:30 11:30 slides: http://www.ist.tugraz.at/pact.html
More information4.6 The Primitive Recursive Functions
4.6. THE PRIMITIVE RECURSIVE FUNCTIONS 309 4.6 The Primitive Recursive Functions The class of primitive recursive functions is defined in terms of base functions and closure operations. Definition 4.6.1
More informationAutomata Theory CS F15 Undecidiability
Automata Theory CS4112015F15 Undecidiability David Galles Department of Computer Science University of San Francisco 150: Universal TM Turing Machines are Hard Wired Addition machine only adds 0 n 1
More informationAutomata and Languages
Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful
More informationNotes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
More informationStructural properties of oracle classes
Structural properties of oracle classes Stanislav Živný Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK Abstract Denote by C the class of oracles relative
More informationChapter 7 Uncomputability
Chapter 7 Uncomputability 190 7.1 Introduction Undecidability of concrete problems. First undecidable problem obtained by diagonalisation. Other undecidable problems obtained by means of the reduction
More informationCS 341 Homework 9 Languages That Are and Are Not Regular
CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w
More informationAutomata and Formal Languages
Automata and Formal Languages Winter 20092010 Yacov HelOr 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,
More informationPushdown automata. Informatics 2A: Lecture 9. Alex Simpson. 3 October, 2014. School of Informatics University of Edinburgh als@inf.ed.ac.
Pushdown automata Informatics 2A: Lecture 9 Alex Simpson School of Informatics University of Edinburgh als@inf.ed.ac.uk 3 October, 2014 1 / 17 Recap of lecture 8 Contextfree languages are defined by contextfree
More informationMarkov Algorithm. CHEN Yuanmi December 18, 2007
Markov Algorithm CHEN Yuanmi December 18, 2007 1 Abstract Markov Algorithm can be understood as a priority string rewriting system. In this short paper we give the definition of Markov algorithm and also
More informationHomework Four Solution CSE 355
Homework Four Solution CSE 355 Due: 29 March 2012 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. Problem 1: Linz 5.1.23
More informationFinite Automata. Reading: Chapter 2
Finite Automata Reading: Chapter 2 1 Finite Automata Informally, a state machine that comprehensively captures all possible states and transitions that a machine can take while responding to a stream (or
More informationRegular Expressions and Automata using Haskell
Regular Expressions and Automata using Haskell Simon Thompson Computing Laboratory University of Kent at Canterbury January 2000 Contents 1 Introduction 2 2 Regular Expressions 2 3 Matching regular expressions
More informationTuring Machines and Understanding Computational Complexity
Turing Machines and Understanding Computational Complexity Paul M.B.Vitányi CWI, Science Park 123, 1098XG Amsterdam The Netherlands 1. Introduction A Turing machine refers to a hypothetical machine proposed
More information6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationChapter 2. Finite Automata. 2.1 The Basic Model
Chapter 2 Finite Automata 2.1 The Basic Model Finite automata model very simple computational devices or processes. These devices have a constant amount of memory and process their input in an online manner.
More informationNondeterministic Finite Automata
Chapter, Part 2 Nondeterministic Finite Automata CSC527, Chapter, Part 2 c 202 Mitsunori Ogihara Fundamental set operations Let A and B be two languages. In addition to union and intersection, we consider
More informationLecture 1: Time Complexity
Computational Complexity Theory, Fall 2010 August 25 Lecture 1: Time Complexity Lecturer: Peter Bro Miltersen Scribe: Søren Valentin Haagerup 1 Introduction to the course The field of Computational Complexity
More informationIncreasing Interaction and Support in the Formal Languages and Automata Theory Course
Increasing Interaction and Support in the Formal Languages and Automata Theory Course [Extended Abstract] Susan H. Rodger rodger@cs.duke.edu Jinghui Lim Stephen Reading ABSTRACT The introduction of educational
More informationCS Markov Chains Additional Reading 1 and Homework problems
CS 252  Markov Chains Additional Reading 1 and Homework problems 1 Markov Chains The first model we discussed, the DFA, is deterministic it s transition function must be total and it allows for transition
More informationRegular Expressions. Languages. Recall. A language is a set of strings made up of symbols from a given alphabet. Computer Science Theory 2
Regular Expressions Languages Recall. A language is a set of strings made up of symbols from a given alphabet. Computer Science Theory 2 1 String Recognition Machine Given a string and a definition of
More informationAutomata and Formal Languages. Push Down Automata. Sipser pages 109114. Lecture 13. Tim Sheard 1
Automata and Formal Languages Push Down Automata Sipser pages 109114 Lecture 13 Tim Sheard 1 Push Down Automata Push Down Automata (PDAs) are εnfas with stack memory. Transitions are labeled by an input
More informationA Formalization of the Turing Test Evgeny Chutchev
A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant
More informationSRM UNIVERSITY FACULTY OF ENGINEERING & TECHNOLOGY SCHOOL OF COMPUTING DEPARTMENT OF SOFTWARE ENGINEERING COURSE PLAN
Course Code : CS0355 SRM UNIVERSITY FACULTY OF ENGINEERING & TECHNOLOGY SCHOOL OF COMPUTING DEPARTMENT OF SOFTWARE ENGINEERING COURSE PLAN Course Title : THEORY OF COMPUTATION Semester : VI Course : June
More informationGenetic programming with regular expressions
Genetic programming with regular expressions Børge Svingen Chief Technology Officer, Open AdExchange bsvingen@openadex.com 20090323 Pattern discovery Pattern discovery: Recognizing patterns that characterize
More informationThe Classes P and NP
The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the
More informationComplexity Theory. Jörg Kreiker. Summer term 2010. Chair for Theoretical Computer Science Prof. Esparza TU München
Complexity Theory Jörg Kreiker Chair for Theoretical Computer Science Prof. Esparza TU München Summer term 2010 Lecture 8 PSPACE 3 Intro Agenda Wrapup Ladner proof and time vs. space succinctness QBF
More informationContextFree Grammars (CFG)
ContextFree Grammars (CFG) SITE : http://www.sir.blois.univtours.fr/ mirian/ Automata Theory, Languages and Computation  Mírian HalfeldFerrari p. 1/2 An informal example Language of palindromes: L
More informationLecture I FINITE AUTOMATA
1. Regular Sets and DFA Lecture I Page 1 Lecture I FINITE AUTOMATA Lecture 1: Honors Theory, Spring 02, Yap We introduce finite automata (deterministic and nondeterministic) and regular languages. Some
More informationDiagonalization. Ahto Buldas. Lecture 3 of Complexity Theory October 8, 2009. Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach.
Diagonalization Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Background One basic goal in complexity theory is to separate interesting complexity
More informationThe Ergodic Theorem and randomness
The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction
More informationHomework Three Solution CSE 355
Homework Three Solution CSE 355 Due: 06 March 2012 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. Problem 1: Linz 4.1.13
More informationUniversal Turing Machine: A Model for all Computational Problems
Universal Turing Machine: A Model for all Computational Problems Edward E. Ogheneovo Lecturer I, Dept of Computer Science, University of Port Harcourt, Port Harcourt Nigeria. ABSTRACT: Turing machines
More informationMenu. cs3102: Theory of Computation. Class 17: Undecidable Languages. Alternate Proof Input. Alternate Proof Input. Contradiction! Contradiction!
cs3102: Theory of Computation Class 17: Undecidable Languages Spring 2010 University of Virginia David Evans Menu Another SELFREJECTINGargument: diagonalization A language that is Turingrecognizable
More informationEnhancement of Turing Machine to Universal Turing Machine to Halt for Recursive Enumerable Language and its JFLAP Simulation
, pp.193202 http://dx.doi.org/10.14257/ijhit.2015.8.1.17 Enhancement of Turing Machine to Universal Turing Machine to Halt for Recursive Enumerable Language and its JFLAP Simulation Tribikram Pradhan
More informationCS5236 Advanced Automata Theory
CS5236 Advanced Automata Theory Frank Stephan Semester I, Academic Year 20122013 Advanced Automata Theory is a lecture which will first review the basics of formal languages and automata theory and then
More informationTuring Machines, Busy Beavers, and Big Questions about Computing
Turing Machines, usy eavers, and ig Questions about Computing My Research Group Computer Security: computing in the presence of adversaries Last summer student projects: Privacy in Social Networks (drienne
More informationCSC4510 AUTOMATA 2.1 Finite Automata: Examples and D efinitions Definitions
CSC45 AUTOMATA 2. Finite Automata: Examples and Definitions Finite Automata: Examples and Definitions A finite automaton is a simple type of computer. Itsoutputislimitedto yes to or no. It has very primitive
More informationModel 2.4 Faculty member + student
Model 2.4 Faculty member + student Course syllabus for Formal languages and Automata Theory. Faculty member information: Name of faculty member responsible for the course Office Hours Office Number Email
More informationThe P versus NP Solution
The P versus NP Solution Frank Vega To cite this version: Frank Vega. The P versus NP Solution. 2015. HAL Id: hal01143424 https://hal.archivesouvertes.fr/hal01143424 Submitted on 17 Apr
More information1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)
Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:
More informationIntroduction to Automata Theory. Reading: Chapter 1
Introduction to Automata Theory Reading: Chapter 1 1 What is Automata Theory? Study of abstract computing devices, or machines Automaton = an abstract computing device Note: A device need not even be a
More informationAutomata on Infinite Words and Trees
Automata on Infinite Words and Trees Course notes for the course Automata on Infinite Words and Trees given by Dr. Meghyn Bienvenu at Universität Bremen in the 20092010 winter semester Last modified:
More informationLecture summary for Theory of Computation
Lecture summary for Theory of Computation Sandeep Sen 1 January 8, 2015 1 Department of Computer Science and Engineering, IIT Delhi, New Delhi 110016, India. E mail:ssen@cse.iitd.ernet.in Contents 1 The
More informationUniversality in the theory of algorithms and computer science
Universality in the theory of algorithms and computer science Alexander Shen Computational models The notion of computable function was introduced in 1930ies. Simplifying (a rather interesting and puzzling)
More informationPlease note that there is more than one way to answer most of these questions. The following only represents a sample solution.
HOMEWORK THREE SOLUTION CSE 355 DUE: 10 MARCH 2011 Please note that there is more than one way to answer most of these questions. The following only represents a sample solution. (1) 2.4 and 2.5 (b),(c),(e),(f):
More informationIntroduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.
More informationOutline 2. 1 Turing Machines. 2 Coding and Universality. 3 The Busy Beaver Problem. 4 Wolfram Prize. 5 ChurchTuring Thesis.
Outline 2 CDM Turing Machines Turing Machines 2 Coding and Universality Klaus Sutner Carnegie Mellon University 3turingmach 205/8/30 9:46 3 The Busy Beaver Problem 4 Wolfram Prize 5 ChurchTuring Thesis
More informationTheory of Computation Lecture Notes
Theory of Computation Lecture Notes Abhijat Vichare August 2005 Contents 1 Introduction 2 What is Computation? 3 The λ Calculus 3.1 Conversions: 3.2 The calculus in use 3.3 Few Important Theorems 3.4 Worked
More informationOracle Turing machines faced with the verification problem
Oracle Turing machines faced with the verification problem 1 Introduction Alan Turing is widely known in logic and computer science to have devised the computing model today named Turing machine. In computer
More informationSimulationBased Security with Inexhaustible Interactive Turing Machines
SimulationBased Security with Inexhaustible Interactive Turing Machines Ralf Küsters Institut für Informatik ChristianAlbrechtsUniversität zu Kiel 24098 Kiel, Germany kuesters@ti.informatik.unikiel.de
More information3. Recurrence Recursive Definitions. To construct a recursively defined function:
3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a
More information