Fundamentals of Computer Science
|
|
- Clarence Kelly
- 5 years ago
- Views:
Transcription
1 Fundamentals of Computer Science Chapter 8: Turing machines Henrik Björklund Umeå University February 17, 2014
2 The power of automata Finite automata have only finite memory. They recognize the regular languages.
3 The power of automata Finite automata have only finite memory. They recognize the regular languages. Pushdown automata have infinite memory, but it is restricted to stack access. They recognize the context-free languages.
4 The power of automata Finite automata have only finite memory. They recognize the regular languages. Pushdown automata have infinite memory, but it is restricted to stack access. They recognize the context-free languages. What is a reasonable extension of these automata that capture all algorithmic computation? This is the question that Alan Turing asked.
5 Alan Turing Alan Turing ( ) was a British mathematician, cryptoanalyst, and computer scientist. After undergraduate studies in Cambridge, he went to Princeton, where he studied under Alonzo Church and recieved his PhD. Turing is one of the most influential persons in the history of computer science.
6 Turing machines The model Turing came up with has come to be known as the Turing machines (TM). This invention was acknowledged by John von Neumann as the central notion leading to the development of the modern computer.
7 Turing machines The model Turing came up with has come to be known as the Turing machines (TM). This invention was acknowledged by John von Neumann as the central notion leading to the development of the modern computer. A Turing machine, like a PDA, has unlimited memory. The difference is that it is not limited to stack discipline, but can move freely along its storage, which is usually referred to as a tape.
8 #
9 #
10 #
11 #
12 #
13 #
14 #
15 #
16 #
17 #
18 #
19 #
20 #
21 # $
22 # $
23 # $
24 # $
25 # $
26 # $
27 $ # $
28 $ # $
29 $ # $
30 $ # $
31 $ # $
32 $ # $
33 $ # $
34 $ # $ $
35 $ # $ $
36 $ # $ $
37 $ # $ $
38 $ # $ $
39 $ # $ $
40 1 0 1 $ $ # $ $
41 1 0 1 $ $ # $ $
42 1 0 1 $ $ # $ $
43 1 0 1 $ $ # $ $
44 1 0 1 $ $ # $ $
45 1 0 1 $ $ # $ $
46 1 0 1 $ $ # $ $
47 1 0 1 $ $ # 1 1 $ $ $
48 1 0 1 $ $ # 1 1 $ $ $
49 1 0 1 $ $ # 1 1 $ $ $
50 1 0 1 $ $ # 1 1 $ $ $
51 1 0 1 $ $ # 1 1 $ $ $
52 1 0 1 $ $ # 1 1 $ $ $
53 1 0 $ $ $ # 1 1 $ $ $
54 1 0 $ $ $ # 1 1 $ $ $
55 1 0 $ $ $ # 1 1 $ $ $
56 1 0 $ $ $ # 1 1 $ $ $
57 1 0 $ $ $ # 1 1 $ $ $
58 1 0 $ $ $ # 1 1 $ $ $
59 1 0 $ $ $ # 1 1 $ $ $
60 1 0 $ $ $ # 1 $ $ $ $
61 1 0 $ $ $ # 1 $ $ $ $
62 1 0 $ $ $ # 1 $ $ $ $
63 1 0 $ $ $ # 1 $ $ $ $
64 1 0 $ $ $ # 1 $ $ $ $
65 1 0 $ $ $ # 1 $ $ $ $
66 1 $ $ $ $ # 1 $ $ $ $
67 1 $ $ $ $ # 1 $ $ $ $
68 1 $ $ $ $ # 1 $ $ $ $
69 1 $ $ $ $ # 1 $ $ $ $
70 1 $ $ $ $ # 1 $ $ $ $
71 1 $ $ $ $ # 1 $ $ $ $
72 1 $ $ $ $ # 1 $ $ $ $
73 1 $ $ $ $ # $ $ $ $ $
74 1 $ $ $ $ # $ $ $ $ $
75 1 $ $ $ $ # $ $ $ $ $
76 1 $ $ $ $ # $ $ $ $ $
77 1 $ $ $ $ # $ $ $ $ $
78 1 $ $ $ $ # $ $ $ $ $
79 $ $ $ $ $ # $ $ $ $ $
80 Formal definition A Turing machine is a tuple where Q is a finite set of states, Σ is the input alphabet, Γ Σ is the tape alphabet, M = (Q, Σ, Γ, δ, q 0,, F), δ : Q Γ Q Γ {L, R} is the transition function, q 0 Q is the initial state, (Γ \ Σ) is the blank tape symbol, and F Q is the set of accepting states. The machine halts when it cannot perform a transition. In particular, we assume that there are no transitions from states in F.
81 Turing machines In the definition we use here, the following features are noteworthy: The tape is unbounded in both directions. The machine is deterministic, i.e., given the current state and the current symbol under the read/write head, there is only one possible action. There is no special input source. The input is assumed to be written on the tape before the computation starts. The read/write head is assumed to point to the leftmost symbol of the input. The input is assumed to be preceeded and followed by infinite sequences of symbols.
82 Configurations A configuration of a TM must specify the current state, the current tape content, and the current position of the read/write head. If the machine is in state q, the non-blank part of the tape has the sequence a 1 a 2 a k 1 a k a k+1 a n of symbols, and the read/write head points at the tape cell with symbol a k, we write this as a 1 a 2 a k 1 qa k a k+1 a n.
83 Changing configuration We write C 1 C 2 if the machine can go from configuration C 1 to configuration C 2 with one transition. For example, if δ(q 1, a) = (q 2, b, R), then abqabb abbqbb.
84 Changing configuration We write C 1 C 2 if the machine can go from configuration C 1 to configuration C 2 with one transition. For example, if δ(q 1, a) = (q 2, b, R), then abqabb abbqbb. We write C 1 C 2 if the machine can go from C 1 to C 2 in zero or more steps.
85 Defining lanugages We can now use a TM M to define a language L(M) as follows: L(M) = {w Σ q 0 w uq f v with q f F u, v Γ }.
86 Defining lanugages We can now use a TM M to define a language L(M) as follows: L(M) = {w Σ q 0 w uq f v with q f F u, v Γ }. Note: If the input w does not belong to L(M), one of two things can happen: 1. The machines halts in a non-accpting state. 2. The machine enters an infinite loop.
87 TMs as transducers In general, a transducer is a device that transforms and input word into an output word.
88 TMs as transducers In general, a transducer is a device that transforms and input word into an output word. We can view a TM as a transducer that defines a function f : Σ Γ as follows: f (w) = w if and only if for some q f F. q 0 w q f w
89 Computable functions A function f with domain D is called Turing computable or just computable if there is a TM M such that for some q f F for all w D. q 0 w M q f f (w)
90 TMs as algorithms In Turing s day, the English word computer meant a human being who performed calculations with pen and paper.
91 TMs as algorithms In Turing s day, the English word computer meant a human being who performed calculations with pen and paper. Turing realized that such computers used algorithms, e.g., for multiplication, division, etc.
92 TMs as algorithms In Turing s day, the English word computer meant a human being who performed calculations with pen and paper. Turing realized that such computers used algorithms, e.g., for multiplication, division, etc. The Turing machine model was intended to capture the nature of such algorithms, i.e., for every such algorithm, there should be a TM that implemented it.
93 The Church-Turing thesis One possible statement: For every algorithm, there is a Turing machine that implements it.
94 The Church-Turing thesis One possible statement: For every algorithm, there is a Turing machine that implements it. Another possibility: Whatever can be computed by purely mechanical means can be computed by some Turing machine.
95 Alonzo Church Alonzo Church ( ) was an American mathematician who contributed greatly to the foundations of computer science. He recieved his PhD at Princeton. He is most well known as the inventor of the λ-calculus, proving that Hilbert s Entscheidungsproblem is undecidable, and the Church-Turing thesis. His doctoral students include Alan Turing, Michael O. Rabin, Dana Scott, and Stephen Kleene.
Introduction to Turing Machines
Introduction to Turing Machines Deepak D Souza Department of Computer Science and Automation Indian Institute of Science, Bangalore. 12 November 2015 Outline 1 Turing Machines 2 Formal definitions 3 Computability
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 13 CHAPTER 4 TURING MACHINES 1. The definition of Turing machine 2. Computing with Turing machines 3. Extensions of Turing
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 15 Ana Bove May 17th 2018 Recap: Context-free Languages Chomsky hierarchy: Regular languages are also context-free; Pumping lemma
More informationCS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 8 January 24, 2018 Outline Turing Machines and variants multitape TMs nondeterministic TMs Church-Turing Thesis So far several models of computation finite automata
More informationChapter 3: The Church-Turing Thesis
Chapter 3: The Church-Turing Thesis 1 Turing Machine (TM) Control... Bi-direction Read/Write Turing machine is a much more powerful model, proposed by Alan Turing in 1936. 2 Church/Turing Thesis Anything
More informationTuring Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q0, qaccept, qreject), where Q, Σ, Γ are all finite
The Church-Turing Thesis CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Turing Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q 0, q accept, q reject ), where
More information1 Unrestricted Computation
1 Unrestricted Computation General Computing Machines Machines so far: DFAs, NFAs, PDAs Limitations on how much memory they can use: fixed amount of memory plus (for PDAs) a stack Limitations on what they
More informationECS 120 Lesson 15 Turing Machines, Pt. 1
ECS 120 Lesson 15 Turing Machines, Pt. 1 Oliver Kreylos Wednesday, May 2nd, 2001 Before we can start investigating the really interesting problems in theoretical computer science, we have to introduce
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 informationTuring Machines (TM) The Turing machine is the ultimate model of computation.
TURING MACHINES Turing Machines (TM) The Turing machine is the ultimate model of computation. Alan Turing (92 954), British mathematician/engineer and one of the most influential scientists of the last
More informationLecture 13: Turing Machine
Lecture 13: Turing Machine Instructor: Ketan Mulmuley Scriber: Yuan Li February 19, 2015 Turing machine is an abstract machine which in principle can simulate any computation in nature. Church-Turing Thesis:
More informationCSE 105 Theory of Computation
CSE 105 Theory of Computation http://www.jflap.org/jflaptmp/ Professor Jeanne Ferrante 1 Today s Agenda Quick Review of CFG s and PDA s Introduction to Turing Machines and their Languages Reminders and
More informationAutomata & languages. A primer on the Theory of Computation. Laurent Vanbever. ETH Zürich (D-ITET) October,
Automata & languages A primer on the Theory of Computation Laurent Vanbever www.vanbever.eu ETH Zürich (D-ITET) October, 19 2017 Part 5 out of 5 Last week was all about Context-Free Languages Context-Free
More informationCS 301. Lecture 17 Church Turing thesis. Stephen Checkoway. March 19, 2018
CS 301 Lecture 17 Church Turing thesis Stephen Checkoway March 19, 2018 1 / 17 An abridged modern history of formalizing algorithms An algorithm is a finite, unambiguous sequence of steps for solving a
More informationCS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)
CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM)? Proposed by Alan Turing in 936 finite-state control + infinitely long tape A
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/ Today's learning goals Sipser Ch 3 Trace the computation of a Turing machine using its transition function and configurations.
More informationTuring Machines Part II
Turing Machines Part II COMP2600 Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 2016 Slides created by Katya Lebedeva COMP 2600 Turing Machines 1 Why
More informationTuring machines Finite automaton no storage Pushdown automaton storage is a stack What if we give the automaton a more flexible storage?
Turing machines Finite automaton no storage Pushdown automaton storage is a stack What if we give the automaton a more flexible storage? What is the most powerful of automata? In this lecture we will introduce
More informationCS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)
CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM) Proposed by Alan Turing in 936 finite-state control + infinitely long tape A stronger
More informationComputability Theory. CS215, Lecture 6,
Computability Theory CS215, Lecture 6, 2000 1 The Birth of Turing Machines At the end of the 19th century, Gottlob Frege conjectured that mathematics could be built from fundamental logic In 1900 David
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 informationCSCE 551: Chin-Tser Huang. University of South Carolina
CSCE 551: Theory of Computation Chin-Tser Huang huangct@cse.sc.edu University of South Carolina Church-Turing Thesis The definition of the algorithm came in the 1936 papers of Alonzo Church h and Alan
More informationCOMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lec. 16 : Turing Machines
COMP-330 Theory of Computation Fall 2017 -- Prof. Claude Crépeau Lec. 16 : Turing Machines COMP 330 Fall 2017: Lectures Schedule 1-2. Introduction 1.5. Some basic mathematics 2-3. Deterministic finite
More informationCS4026 Formal Models of Computation
CS4026 Formal Models of Computation Turing Machines Turing Machines Abstract but accurate model of computers Proposed by Alan Turing in 1936 There weren t computers back then! Turing s motivation: find
More informationThe Church-Turing Thesis
The Church-Turing Thesis 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 informationDecidable Languages - relationship with other classes.
CSE2001, Fall 2006 1 Last time we saw some examples of decidable languages (or, solvable problems). Today we will start by looking at the relationship between the decidable languages, and the regular and
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 informationCOMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lec. 14 : Turing Machines
COMP-330 Theory of Computation Fall 2012 -- Prof. Claude Crépeau Lec. 14 : Turing Machines 1 COMP 330 Fall 2012: Lectures Schedule 1. Introduction 1.5. Some basic mathematics 2. Deterministic finite automata
More informationCSCC63 Worksheet Turing Machines
1 An Example CSCC63 Worksheet Turing Machines Goal. Design a turing machine, M that accepts only strings of the form {w#w w {0, 1} }. Idea. Describe in words how the machine would work. Read first symbol
More informationComputability and Complexity
Computability and Complexity Lecture 5 Reductions Undecidable problems from language theory Linear bounded automata given by Jiri Srba Lecture 5 Computability and Complexity 1/14 Reduction Informal Definition
More informationHarvard CS 121 and CSCI E-121 Lecture 14: Turing Machines and the Church Turing Thesis
Harvard CS 121 and CSCI E-121 Lecture 14: Turing Machines and the Church Turing Thesis Harry Lewis October 22, 2013 Reading: Sipser, 3.2, 3.3. The Basic Turing Machine The Basic Turing Machine a a b a
More informationCSE355 SUMMER 2018 LECTURES TURING MACHINES AND (UN)DECIDABILITY
CSE355 SUMMER 2018 LECTURES TURING MACHINES AND (UN)DECIDABILITY RYAN DOUGHERTY If we want to talk about a program running on a real computer, consider the following: when a program reads an instruction,
More informationIntroduction to Turing Machines. Reading: Chapters 8 & 9
Introduction to Turing Machines Reading: Chapters 8 & 9 1 Turing Machines (TM) Generalize the class of CFLs: Recursively Enumerable Languages Recursive Languages Context-Free Languages Regular Languages
More informationTuring Machines. 22c:135 Theory of Computation. Tape of a Turing Machine (TM) TM versus FA, PDA
Turing Machines A Turing machine is similar to a finite automaton with supply of unlimited memory. A Turing machine can do everything that any computing device can do. There exist problems that even a
More informationCSE 2001: Introduction to Theory of Computation Fall Suprakash Datta
CSE 2001: Introduction to Theory of Computation Fall 2013 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cse.yorku.ca/course/2001 11/7/2013 CSE
More informationIntroduction: Computer Science is a cluster of related scientific and engineering disciplines concerned with the study and application of computations. These disciplines range from the pure and basic scientific
More informationCOMPARATIVE ANALYSIS ON TURING MACHINE AND QUANTUM TURING MACHINE
Volume 3, No. 5, May 2012 Journal of Global Research in Computer Science REVIEW ARTICLE Available Online at www.jgrcs.info COMPARATIVE ANALYSIS ON TURING MACHINE AND QUANTUM TURING MACHINE Tirtharaj Dash
More informationTuring machines COMS Ashley Montanaro 21 March Department of Computer Science, University of Bristol Bristol, UK
COMS11700 Turing machines Department of Computer Science, University of Bristol Bristol, UK 21 March 2014 COMS11700: Turing machines Slide 1/15 Introduction We have seen two models of computation: finite
More informationTheory of Computation
Theory of Computation Lecture #2 Sarmad Abbasi Virtual University Sarmad Abbasi (Virtual University) Theory of Computation 1 / 1 Lecture 2: Overview Recall some basic definitions from Automata Theory.
More informationCSci 311, Models of Computation Chapter 9 Turing Machines
CSci 311, Models of Computation Chapter 9 Turing Machines H. Conrad Cunningham 29 December 2015 Contents Introduction................................. 1 9.1 The Standard Turing Machine...................
More informationBefore We Start. Turing Machines. Languages. Now our picture looks like. Theory Hall of Fame. The Turing Machine. Any questions? The $64,000 Question
Before We Start s Any questions? Languages The $64,000 Question What is a language? What is a class of languages? Now our picture looks like Context Free Languages Deterministic Context Free Languages
More informationThe Turing machine model of computation
The Turing machine model of computation For most of the remainder of the course we will study the Turing machine model of computation, named after Alan Turing (1912 1954) who proposed the model in 1936.
More informationjflap demo Regular expressions Pumping lemma Turing Machines Sections 12.4 and 12.5 in the text
On the menu today jflap demo Regular expressions Pumping lemma Turing Machines Sections 12.4 and 12.5 in the text 1 jflap Demo jflap: Useful tool for creating and testing abstract machines Finite automata,
More informationUNIT 1 TURING MACHINE
UNIT 1 TURING MACHINE Structure Page Nos. 1.0 Introduction 5 1.1 Objectives 7 1.2 Prelude to Formal Definition 7 1.3 : Formal Definition and Examples 9 1.4 Instantaneous Description and Transition Diagram
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 Design a PDA and a CFG for a given language Give informal description for a PDA,
More informationIV. Turing Machine. Yuxi Fu. BASICS, Shanghai Jiao Tong University
IV. Turing Machine Yuxi Fu BASICS, Shanghai Jiao Tong University Alan Turing Alan Turing (23Jun.1912-7Jun.1954), an English student of Church, introduced a machine model for effective calculation in On
More informationLecture 14: Recursive Languages
Lecture 14: Recursive Languages Instructor: Ketan Mulmuley Scriber: Yuan Li February 24, 2015 1 Recursive Languages Definition 1.1. A language L Σ is called recursively enumerable (r. e.) or computably
More informationAn example of a decidable language that is not a CFL Implementation-level description of a TM State diagram of TM
Turing Machines Review An example of a decidable language that is not a CFL Implementation-level description of a TM State diagram of TM Varieties of TMs Multi-Tape TMs Nondeterministic TMs String Enumerators
More informationPart I: Definitions and Properties
Turing Machines Part I: Definitions and Properties Finite State Automata Deterministic Automata (DFSA) M = {Q, Σ, δ, q 0, F} -- Σ = Symbols -- Q = States -- q 0 = Initial State -- F = Accepting States
More informationTheory of Computation (IX) Yijia Chen Fudan University
Theory of Computation (IX) Yijia Chen Fudan University Review The Definition of Algorithm Polynomials and their roots A polynomial is a sum of terms, where each term is a product of certain variables and
More informationTheory of Computation Turing Machine and Pushdown Automata
Theory of Computation Turing Machine and Pushdown Automata 1. What is a Turing Machine? A Turing Machine is an accepting device which accepts the languages (recursively enumerable set) generated by type
More informationThe Power of One-State Turing Machines
The Power of One-State Turing Machines Marzio De Biasi Jan 15, 2018 Abstract At first glance, one state Turing machines are very weak: the Halting problem for them is decidable, and, without memory, they
More informationCAPES MI. Turing Machines and decidable problems. Laure Gonnord
CAPES MI Turing Machines and decidable problems Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@univ-lyon1.fr Université Claude Bernard Lyon1 2017 Motivation 1 Motivation 2 Turing Machines
More informationComputation Histories
208 Computation Histories The computation history for a Turing machine on an input is simply the sequence of configurations that the machine goes through as it processes the input. An accepting computation
More informationComputational Models - Lecture 6
Computational Models - Lecture 6 Turing Machines Alternative Models of Computers Multitape TMs, RAMs, Non Deterministic TMs The Church-Turing Thesis The language classes R = RE core David Hilbert s Tenth
More informationTheory of Computer Science
Theory of Computer Science D1. Turing-Computability Malte Helmert University of Basel April 18, 2016 Overview: Course contents of this course: logic How can knowledge be represented? How can reasoning
More informationVariants of Turing Machine (intro)
CHAPTER 3 The Church-Turing Thesis Contents Turing Machines definitions, examples, Turing-recognizable and Turing-decidable languages Variants of Turing Machine Multi-tape Turing machines, non-deterministic
More informationUNRESTRICTED GRAMMARS
136 UNRESTRICTED GRAMMARS Context-free grammar allows to substitute only variables with strings In an unrestricted grammar (or a rewriting system) one may substitute any non-empty string (containing variables
More informationTheory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death
Theory of Computation CS3102 Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory
More informationTuring Machines. Lecture 8
Turing Machines Lecture 8 1 Course Trajectory We will see algorithms, what can be done. But what cannot be done? 2 Computation Problem: To compute a function F that maps each input (a string) to an output
More informationLecture 12: Mapping Reductions
Lecture 12: Mapping Reductions October 18, 2016 CS 1010 Theory of Computation Topics Covered 1. The Language EQ T M 2. Mapping Reducibility 3. The Post Correspondence Problem 1 The Language EQ T M The
More informationExam Computability and Complexity
Total number of points:... Number of extra sheets of paper:... Exam Computability and Complexity by Jiri Srba, January 2009 Student s full name CPR number Study number Before you start, fill in the three
More informationAutomata Theory (2A) Young Won Lim 5/31/18
Automata Theory (2A) Copyright (c) 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationAutomata Theory. Definition. Computational Complexity Theory. Computability Theory
Outline THEORY OF COMPUTATION CS363, SJTU What is Theory of Computation? History of Computation Branches and Development Xiaofeng Gao Dept. of Computer Science Shanghai Jiao Tong University 2 The Essential
More informationTuring Machine Variants
CS311 Computational Structures Turing Machine Variants Lecture 12 Andrew Black Andrew Tolmach 1 The Church-Turing Thesis The problems that can be decided by an algorithm are exactly those that can be decided
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 informationMidterm Exam 2 CS 341: Foundations of Computer Science II Fall 2018, face-to-face day section Prof. Marvin K. Nakayama
Midterm Exam 2 CS 341: Foundations of Computer Science II Fall 2018, face-to-face day section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand
More informationThe Turing Machine. Computability. The Church-Turing Thesis (1936) Theory Hall of Fame. Theory Hall of Fame. Undecidability
The Turing Machine Computability Motivating idea Build a theoretical a human computer Likened to a human with a paper and pencil that can solve problems in an algorithmic way The theoretical provides a
More informationCS151 Complexity Theory. Lecture 1 April 3, 2017
CS151 Complexity Theory Lecture 1 April 3, 2017 Complexity Theory Classify problems according to the computational resources required running time storage space parallelism randomness rounds of interaction,
More informationTuring Machines. The Language Hierarchy. Context-Free Languages. Regular Languages. Courtesy Costas Busch - RPI 1
Turing Machines a n b n c The anguage Hierarchy n? ww? Context-Free anguages a n b n egular anguages a * a *b* ww Courtesy Costas Busch - PI a n b n c n Turing Machines anguages accepted by Turing Machines
More informationTesting Emptiness of a CFL. Testing Finiteness of a CFL. Testing Membership in a CFL. CYK Algorithm
Testing Emptiness of a CFL As for regular languages, we really take a representation of some language and ask whether it represents φ Can use either CFG or PDA Our choice, since there are algorithms to
More informationCS187 - Science Gateway Seminar for CS and Math
CS187 - Science Gateway Seminar for CS and Math Fall 2013 Class 3 Sep. 10, 2013 What is (not) Computer Science? Network and system administration? Playing video games? Learning to use software packages?
More informationTuring Machines Decidability
Turing Machines Decidability Master Informatique 2016 Some General Knowledge Alan Mathison Turing UK, 1912 1954 Mathematician, computer scientist, cryptanalyst Most famous works: Computation model («Turing
More informationEquivalence of TMs and Multitape TMs. Theorem 3.13 and Corollary 3.15 By: Joseph Lauman
Equivalence of TMs and Multitape TMs Theorem 3.13 and Corollary 3.15 By: Joseph Lauman Turing Machines First proposed by Alan Turing in 1936 Similar to finite automaton, but with an unlimited and unrestricted
More informationAn example of a decidable language that is not a CFL Implementation-level description of a TM State diagram of TM
Turing Machines Review An example of a decidable language that is not a CFL Implementation-level description of a TM State diagram of TM Varieties of TMs Multi-Tape TMs Nondeterministic TMs String Enumerators
More informationMidterm Exam 2 CS 341: Foundations of Computer Science II Fall 2016, face-to-face day section Prof. Marvin K. Nakayama
Midterm Exam 2 CS 341: Foundations of Computer Science II Fall 2016, face-to-face day section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand
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 informationTuring Machines. Fall The Chinese University of Hong Kong. CSCI 3130: Formal languages and automata theory
The Chinese University of Hong Kong Fall 2011 CSCI 3130: Formal languages and automata theory Turing Machines Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130 Turing Machines control head a
More informationThe decision problem (entscheidungsproblem), halting problem, & Turing machines. CS 350 Fall 2018 gilray.org/classes/fall2018/cs350/
The decision problem (entscheidungsproblem), halting problem, & Turing machines CS 350 Fall 2018 gilray.org/classes/fall2018/cs350/ 1 We must not believe those, who today, with philosophical bearing and
More informationAutomata Theory - Quiz II (Solutions)
Automata Theory - Quiz II (Solutions) K. Subramani LCSEE, West Virginia University, Morgantown, WV {ksmani@csee.wvu.edu} 1 Problems 1. Induction: Let L denote the language of balanced strings over Σ =
More informationTheory of Computation
Theory of Computation Lecture #6 Sarmad Abbasi Virtual University Sarmad Abbasi (Virtual University) Theory of Computation 1 / 39 Lecture 6: Overview Prove the equivalence of enumerators and TMs. Dovetailing
More informationENEE 459E/CMSC 498R In-class exercise February 10, 2015
ENEE 459E/CMSC 498R In-class exercise February 10, 2015 In this in-class exercise, we will explore what it means for a problem to be intractable (i.e. it cannot be solved by an efficient algorithm). There
More informationTuring Machines. Our most powerful model of a computer is the Turing Machine. This is an FA with an infinite tape for storage.
Turing Machines Our most powerful model of a computer is the Turing Machine. This is an FA with an infinite tape for storage. A Turing Machine A Turing Machine (TM) has three components: An infinite tape
More informationTuring machines and linear bounded automata
and linear bounded automata Informatics 2A: Lecture 29 John Longley School of Informatics University of Edinburgh jrl@inf.ed.ac.uk 25 November, 2011 1 / 13 1 The Chomsky hierarchy: summary 2 3 4 2 / 13
More informationHIS LEGACY. 100 Years Turing celebration. Gordana Dodig Crnkovic, IDT Open Seminar. Computer Science and Network Department Mälardalen University
IDT Open Seminar AAN TUING AND HIS EGACY 00 Years Turing celebration http://www.mrtc.mdh.se/~gdc/work/turingcentenary.pdf http://www.mrtc.mdh.se/ mdh se/~gdc/work/turingmachine.pdf Gordana Dodig Crnkovic,
More information7.2 Turing Machines as Language Acceptors 7.3 Turing Machines that Compute Partial Functions
CSC4510/6510 AUTOMATA 7.1 A General Model of Computation 7.2 Turing Machines as Language Acceptors 7.3 Turing Machines that Compute Partial Functions A General Model of Computation Both FA and PDA are
More informationTheory of Computation - Module 4
Theory of Computation - Module 4 Syllabus Turing Machines Formal definition Language acceptability by TM TM as acceptors, Transducers - designing of TM- Two way infinite TM- Multi tape TM - Universal Turing
More informationDecidability: Church-Turing Thesis
Decidability: Church-Turing Thesis While there are a countably infinite number of languages that are described by TMs over some alphabet Σ, there are an uncountably infinite number that are not Are there
More informationModels. Models of Computation, Turing Machines, and the Limits of Turing Computation. Effective Calculability. Motivation for Models of Computation
Turing Computation /0/ Models of Computation, Turing Machines, and the Limits of Turing Computation Bruce MacLennan Models A model is a tool intended to address a class of questions about some domain of
More informationTuring Machines. COMP2600 Formal Methods for Software Engineering. Katya Lebedeva. Australian National University Semester 2, 2014
Turing Machines COMP2600 Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 2014 Slides created by Jeremy Dawson and Ranald Clouston COMP 2600 Turing Machines
More informationAutomata Theory CS S-12 Turing Machine Modifications
Automata Theory CS411-2015S-12 Turing Machine Modifications David Galles Department of Computer Science University of San Francisco 12-0: Extending Turing Machines When we added a stack to NFA to get a
More informationGriffith University 3130CIT Theory of Computation (Based on slides by Harald Søndergaard of The University of Melbourne) Turing Machines 9-0
Griffith University 3130CIT Theory of Computation (Based on slides by Harald Søndergaard of The University of Melbourne) Turing Machines 9-0 Turing Machines Now for a machine model of much greater power.
More informationCMP 309: Automata Theory, Computability and Formal Languages. Adapted from the work of Andrej Bogdanov
CMP 309: Automata Theory, Computability and Formal Languages Adapted from the work of Andrej Bogdanov Course outline Introduction to Automata Theory Finite Automata Deterministic Finite state automata
More informationEquivalent Variations of Turing Machines
Equivalent Variations of Turing Machines Nondeterministic TM = deterministic TM npda = pushdown automata with n stacks 2PDA = npda = TM for all n 2 Turing machines with n tapes (n 2) and n tape heads has
More informationComputation. Some history...
Computation Motivating questions: What does computation mean? What are the similarities and differences between computation in computers and in natural systems? What are the limits of computation? Are
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 informationSCHEME FOR INTERNAL ASSESSMENT TEST 3
SCHEME FOR INTERNAL ASSESSMENT TEST 3 Max Marks: 40 Subject& Code: Automata Theory & Computability (15CS54) Sem: V ISE (A & B) Note: Answer any FIVE full questions, choosing one full question from each
More informationBusch Complexity Lectures: Turing Machines. Prof. Busch - LSU 1
Busch Complexity ectures: Turing Machines Prof. Busch - SU 1 The anguage Hierarchy a n b n c n? ww? Context-Free anguages n b n a ww egular anguages a* a *b* Prof. Busch - SU 2 a n b anguages accepted
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 informationUndecidable Problems and Reducibility
University of Georgia Fall 2014 Reducibility We show a problem decidable/undecidable by reducing it to another problem. One type of reduction: mapping reduction. Definition Let A, B be languages over Σ.
More information