Busch Complexity Lectures: Turing s Thesis. Costas Busch  LSU 1


 Lydia Carr
 1 years ago
 Views:
Transcription
1 Busch Complexity Lectures: Turing s Thesis Costas Busch  LSU 1
2 Turing s thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine Costas Busch  LSU 2
3 Algorithm: An algorithm for a problem is a Turing Machine which solves the problem The algorithm describes the steps of the mechanical means This is easily translated to computation steps of a Turing machine Costas Busch  LSU 3
4 When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm Costas Busch  LSU 4
5 Variations of the Turing Machine Costas Busch  LSU 5
6 The Standard Model Infinite Tape a a b a b b c a c a ReadWrite Head (Left or Right) Control Unit Deterministic Costas Busch  LSU 6
7 Variations of the Standard Model Turing machines with: StayOption SemiInfinite Tape Multitape Multidimensional Nondeterministic Different Turing Machine Classes Costas Busch  LSU 7
8 Same Power of two machine classes: both classes accept the same set of languages We will prove: each new class has the same power with Standard Turing Machine (accept TuringRecognizable Languages) Costas Busch  LSU 8
9 Same Power of two classes means: for any machine M 1 of first class there is a machine M 2 of second class such that: L ( M1) = L( M2) and viceversa Costas Busch  LSU 9
10 Simulation: A technique to prove same power. Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine M 2 M1 M1 simulates M1 Costas Busch  LSU 10
11 Configurations in the Original Machine have corresponding configurations in the Simulation Machine M 2 M 1 M 1 Original Machine: d 0 d 1 " d n Simulation Machine: dʹ 0 d 1 ʹ dn Costas Busch  LSU 11 M 2 " ʹ
12 Accepting Configuration Original Machine: d f Simulation Machine: dʹf the Simulation Machine and the Original Machine accept the same strings L ( M1) = L( M2) Costas Busch  LSU 12
13 Turing Machines with StayOption The head can stay in the same position a a b a b b c a c a Left, Right, Stay L,R,S: possible head moves Costas Busch  LSU 13
14 Example: Time 1 a a b a b b c a c a q 1 Time 2 b a b a b b c a c a q 2 a b, S q1 q2 Costas Busch  LSU 14
15 Theorem: StayOption machines have the same power with Standard Turing machines Proof: 1. StayOption Machines simulate Standard Turing machines 2. Standard Turing machines simulate StayOption machines Costas Busch  LSU 15
16 1. StayOption Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a StayOption machine Costas Busch  LSU 16
17 2. Standard Turing machines simulate StayOption machines We need to simulate the stay head option with two head moves, one left and one right Costas Busch  LSU 17
18 StayOption Machine a b, S q1 q2 Simulation in Standard Machine q 1 a b, L x x, R q 2 For every possible tape symbol x Costas Busch  LSU 18
19 For other transitions nothing changes StayOption Machine a b, L q1 q2 Simulation in Standard Machine a b, L q1 q2 Similar for Right moves Costas Busch  LSU 19
20 example of simulation StayOption Machine: q a b, S 1 q2 a a b a 1 2 q 1 b a b a q 2 1 Simulation in Standard Machine: a a b a q 1 2 b a b a q 2 3 b a b a q 3 END OF PROOF Costas Busch  LSU 20
21 A useful trick: Multiple Track Tape helps for more complicated simulations One Tape a b b a a c b d track 1 track 2 One head One symbol ( a, b) It is a standard Turing machine, but each tape alphabet symbol describes a pair of actual useful symbols Costas Busch  LSU 21
22 a b b a a c b d track 1 track 2 q 1 a c a b track 1 b d c d track 2 q 2 ( b, a) ( c, d), L q1 q2 Costas Busch  LSU 22
23 SemiInfinite Tape The head extends infinitely only to the right a b a c... Initial position is the leftmost cell When the head moves left from the border, it returns back to leftmost position Costas Busch  LSU 23
24 Theorem: SemiInfinite machines have the same power with Standard Turing machines Proof: 1. Standard Turing machines simulate SemiInfinite machines 2. SemiInfinite Machines simulate Standard Turing machines Costas Busch  LSU 24
25 1. Standard Turing machines simulate SemiInfinite machines:... # a b a c... Standard Turing Machine SemiInfinite machine modifications a. insert special symbol # at left of input string b. Add a selfloop to every state (except states with no outgoing transitions) # #,R Costas Busch  LSU 25
26 2. SemiInfinite tape machines simulate Standard Turing machines:... Standard machine... SemiInfinite tape machine... Squeeze infinity of both directions to one direction Costas Busch  LSU 26
27 ... Standard machine a b c d e... reference point SemiInfinite tape machine with two tracks Right part Left part # # d e c b a... Costas Busch  LSU 27
28 Standard machine q 1 q 2 SemiInfinite tape machine Left part Right part L q 1 L q 2 R q 1 R q 2 Costas Busch  LSU 28
29 Standard machine a g, R q1 q2 SemiInfinite tape machine Right part R q 1 ( a, x) ( g, x), R R q 2 Left part ( x, a) ( x, g), L L q 1 For all tape symbols x L q 2 Costas Busch  LSU 29
30 Time 1... Standard machine a b c d e... q 1 Right part Left part SemiInfinite tape machine # d e # c b a L q 1... Costas Busch  LSU 30
31 Time 2... Standard machine g b c d e... q 2 Right part Left part SemiInfinite tape machine # d e # c b g L q 2... Costas Busch  LSU 31
32 At the border: SemiInfinite tape machine Right part R q 1 (#,#) (#,#),R L q 1 Left part L q 1 (#,#) (#,#),R R q 1 Costas Busch  LSU 32
33 Right part Left part SemiInfinite tape machine # # L q 1 c Time 1 d e b g... Right part Left part # # c Time 2 d e b g... R q 1 END OF PROOF Costas Busch  LSU 33
34 Multitape Turing Machines Control unit (state machine) Tape 1 Tape 2 a b c e f g Input string Input string appears on Tape 1 Costas Busch  LSU 34
35 Tape 1 Time 1 Tape 2 a b c e f g q1 q1 Tape 1 Time 2 Tape 2 a g c e d g q2 q2 ( b, f ) ( g, d), L, R q1 q2 Costas Busch  LSU 35
36 Theorem: Multitape machines have the same power with Standard Turing machines Proof: 1. Multitape machines simulate Standard Turing machines 2. Standard Turing machines simulate Multitape machines Costas Busch  LSU 36
37 1. Multitape machines simulate Standard Turing Machines: Trivial: Use one tape Costas Busch  LSU 37
38 2. Standard Turing machines simulate Multitape machines: Standard machine: Uses a multitrack tape to simulate the multiple tapes A tape of the Multitape machine corresponds to a pair of tracks Costas Busch  LSU 38
39 Multitape Machine Tape 1 Tape 2 a b c e f g h Standard machine with four track tape a b c e f g h 0 Tape 1 head position Tape 2 head position Costas Busch  LSU 39
40 Reference point # # # # a b c e f g h 0 Tape 1 head position Tape 2 head position Repeat for each Multitape state transition: 1. Return to reference point 2. Find current symbol in Track 1 and update 3. Return to reference point 4. Find current symbol in Tape 2 and update END OF PROOF Costas Busch  LSU 40
41 Same power doesn t imply same speed: n n L = { a b } Standard Turing machine: Go back and forth O( n 2 ) times to match the a s with the b s 2tape machine: O(n) time 1. Copy n b to tape 2 a n 2. Compare on tape 1 n and b tape 2 O( n 2 ) time ( O(n) steps) ( O(n) steps) Costas Busch  LSU 41
42 Multidimensional Turing Machines 2dimensional tape y c a b x MOVES: L,R,U,D U: up D: down HEAD Position: +2, 1 Costas Busch  LSU 42
43 Theorem: Multidimensional machines have the same power with Standard Turing machines Proof: 1. Multidimensional machines simulate Standard Turing machines 2. Standard Turing machines simulate MultiDimensional machines Costas Busch  LSU 43
44 1. Multidimensional machines simulate Standard Turing machines Trivial: Use one dimension Costas Busch  LSU 44
45 2. Standard Turing machines simulate Multidimensional machines Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2 Costas Busch  LSU 45
46 2dimensional machine y c a b x a b 1 # 1 # 2 # 1 q 1 Standard Machine c # 1 q 1 symbol coordinates Costas Busch  LSU 46
47 Standard machine: Repeat for each transition followed in the 2dimensional machine: 1. Update current symbol 2. Compute coordinates of next position 3. Go to new position END OF PROOF Costas Busch  LSU 47
48 Nondeterministic Turing Machines a b, L q 2 Choice 1 q 1 a c, R q 3 Choice 2 Allows Non Deterministic Choices Costas Busch  LSU 48
49 Time 0 a b c Time 1 q 1 Choice 1 a b, L q 2 b b c q 2 q 1 Choice 2 a c, R q 3 c b c q 3 Costas Busch  LSU 49
50 Input string w there is a computation: q0w x q f y is accepted if Initial configuration Final Configuration Any accept state There is a computation: Costas Busch  LSU 50
51 Theorem: Nondeterministic machines have the same power with Standard Turing machines Proof: 1. Nondeterministic machines simulate Standard Turing machines 2. Standard Turing machines simulate Nondeterministic machines Costas Busch  LSU 51
52 1. Nondeterministic Machines simulate Standard (deterministic) Turing Machines Trivial: every deterministic machine is also nondeterministic Costas Busch  LSU 52
53 2. Standard (deterministic) Turing machines simulate Nondeterministic machines: Deterministic machine: Uses a 2dimensional tape (equivalent to standard Turing machine with one tape) Stores all possible computations of the nondeterministic machine on the 2dimensional tape Costas Busch  LSU 53
54 All possible computation paths Initial state Step 1 Step 2 reject accept infinite path Step i Step i+1 Costas Busch  LSU 54
55 The Deterministic Turing machine simulates all possible computation paths: simultaneously stepbystep with breadthfirst search depthfirst may result getting stuck at exploring an infinite path before discovering the accepting path Costas Busch  LSU 55
56 NonDeterministic machine a b, q 1 a c, L R q 2 a b c q 1 q 3 Deterministic machine # # # # # # # a b c # # q 1 # # # # # # Time 0 current configuration Costas Busch  LSU 56
57 a b, q 1 a c, L R NonDeterministic machine Time 1 q 2 q 2 q 3 q 3 Deterministic machine # # # # # # # b b c q # # 2 # # c b c # # q 3 # b b c c b c Computation 1 Costas Busch  LSU 57 Choice 1 Choice 2 Computation 2
58 Deterministic Turing machine Repeat For each configuration in current step of nondeterministic machine, if there are two or more choices: 1. Replicate configuration 2. Change the state in the replicas Until either the input string is accepted or rejected in all configurations Costas Busch  LSU 58
59 If the nondeterministic machine accepts the input string: The deterministic machine accepts and halts too The simulation takes in the worst case exponential time compared to the shortest length of an accepting path Costas Busch  LSU 59
60 If the nondeterministic machine does not accept the input string: 1. The simulation halts if all paths reach a halting state OR 2. The simulation never terminates if there is a neverending path (infinite loop) In either case the deterministic machine rejects too (1. by halting or 2. by simulating the infinite loop) END OF PROOF Costas Busch  LSU 60
Turing s Thesis. Fall Costas Busch  RPI!1
Turing s Thesis Costas Busch  RPI!1 Turing s thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine Costas Busch  RPI!2 Algorithm: An algorithm for a problem
More informationVariations of the Turing Machine
Variations of the Turing Machine 1 The Standard Model Infinite Tape a a b a b b c a c a ReadWrite Head (Left or Right) Control Unit Deterministic 2 Variations of the Standard Model Turing machines with:
More informationTuring s thesis: (1930) Any computation carried out by mechanical means can be performed by a Turing Machine
Turing s thesis: (1930) Any computation carried out by mechanical means can be performed by a Turing Machine There is no known model of computation more powerful than Turing Machines Definition of Algorithm:
More informationMore Turing Machines. CS154 Chris Pollett Mar 15, 2006.
More Turing Machines CS154 Chris Pollett Mar 15, 2006. Outline Multitape Turing Machines Nondeterministic Turing Machines Enumerators Introduction There have been many different proposals for what it means
More informationChapter 3: The ChurchTuring Thesis
Chapter 3: The ChurchTuring Thesis 1 Turing Machine (TM) Control... Bidirection Read/Write Turing machine is a much more powerful model, proposed by Alan Turing in 1936. 2 Church/Turing Thesis Anything
More informationTuring Machines. The Language Hierarchy. ContextFree Languages. Regular Languages. Courtesy Costas Busch  RPI 1
Turing Machines a n b n c The anguage Hierarchy n? ww? ContextFree 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 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 finitestate 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 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 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? ContextFree anguages n b n a ww egular anguages a* a *b* Prof. Busch  SU 2 a n b anguages accepted
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 finitestate control + infinitely long tape A
More informationTuring Machine properties. Turing Machines. Alternate TM definitions. Alternate TM definitions. Alternate TM definitions. Alternate TM definitions
Turing Machine properties Turing Machines TM Variants and the Universal TM There are many ways to skin a cat And many ways to define a TM The book s Standard Turing Machines Tape unbounded on both sides
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 informationThe ChurchTuring Thesis
1 The ChurchTuring Thesis Foundations of Computing Science Pallab Dasgupta Professor, Dept. of Computer Sc & Engg 2 Turing Machines A Turing Machine is a 7tuple, (Q, Σ, Γ, δ, q 0, q accept, q reject
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 informationTuring Machines A Turing Machine is a 7tuple, (Q, Σ, Γ, δ, q0, qaccept, qreject), where Q, Σ, Γ are all finite
The ChurchTuring Thesis CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Turing Machines A Turing Machine is a 7tuple, (Q, Σ, Γ, δ, q 0, q accept, q reject ), where
More informationHarvard CS 121 and CSCI E121 Lecture 14: Turing Machines and the Church Turing Thesis
Harvard CS 121 and CSCI E121 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 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 Twoway, infinite tape, broken into
More information1 Showing Recognizability
CSCC63 Worksheet Recognizability and Decidability 1 1 Showing Recognizability 1.1 An Example  take 1 Let Σ be an alphabet. L = { M M is a T M and L(M) }, i.e., that M accepts some string from Σ. Prove
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105abc/ Today's learning goals Sipser Ch 3 Trace the computation of a Turing machine using its transition function and configurations.
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 informationCS5371 Theory of Computation. Lecture 12: Computability III (Decidable Languages relating to DFA, NFA, and CFG)
CS5371 Theory of Computation Lecture 12: Computability III (Decidable Languages relating to DFA, NFA, and CFG) Objectives Recall that decidable languages are languages that can be decided by TM (that means,
More informationHomework. Turing Machines. Announcements. Plan for today. Now our picture looks like. Languages
Homework s TM Variants and the Universal TM Homework #6 returned Homework #7 due today Homework #8 (the LAST homework!) Page 262  Exercise 10 (build with JFLAP) Page 270  Exercise 2 Page 282  Exercise
More informationTuring s original paper (1936)
Turing s original paper (1936) Turing s original paper (1936) There exist problems that cannot be solved mechanically Turing s original paper (1936) There exist problems that cannot be solved mechanically
More informationThe ChurchTuring Thesis
The ChurchTuring 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 informationFoundations of
91.304 Foundations of (Theoretical) Computer Science Chapter 3 Lecture Notes (Section 3.2: Variants of Turing Machines) David Martin dm@cs.uml.edu With some modifications by Prof. Karen Daniels, Fall 2012
More informationChapter 8. Turing Machine (TMs)
Chapter 8 Turing Machine (TMs) Turing Machines (TMs) Accepts the languages that can be generated by unrestricted (phrasestructured) grammars No computational machine (i.e., computational language recognition
More informationHomework 5 Solution CSCI 2670 October 6, c) New text q 1 1##1 xq 3 ##1 x#q 5 #1 q reject. Old text. ~q 3 ##1 ~#q 5 #1
Homework 5 Solution CSCI 2670 October 6, 2005 3.2 c) New text q 1 1##1 xq 3 ##1 x#q 5 #1 q reject Old text q 1 1##1 ~q 3 ##1 ~#q 5 #1 q reject 3.2 e) New text q 1 10#10 xq 3 0#10 x0q 3 #10 x0#q 5 10 x0q
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 informationAsymptotic notation : bigo and smallo
Time complexity Here we will consider elements of computational complexity theory an investigation of the time (or other resources) required for solving computational problems. We introduce a way of measuring
More informationCSE 2001: Introduction to Theory of Computation Fall Suprakash Datta
CSE 2001: Introduction to Theory of Computation Fall 2012 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 4167362100 ext 77875 Course page: http://www.cs.yorku.ca/course/2001 11/13/2012 CSE
More informationVariants of Turing Machine (intro)
CHAPTER 3 The ChurchTuring Thesis Contents Turing Machines definitions, examples, Turingrecognizable and Turingdecidable languages Variants of Turing Machine Multitape Turing machines, nondeterministic
More informationCS 301. Lecture 18 Decidable languages. Stephen Checkoway. April 2, 2018
CS 301 Lecture 18 Decidable languages Stephen Checkoway April 2, 2018 1 / 26 Decidable language Recall, a language A is decidable if there is some TM M that 1 recognizes A (i.e., L(M) = A), and 2 halts
More informationTuring Machines and Time Complexity
Turing Machines and Time Complexity Turing Machines Turing Machines (Infinitely long) Tape of 1 s and 0 s Turing Machines (Infinitely long) Tape of 1 s and 0 s Able to read and write the tape, and move
More informationChapter 7 Turing Machines
Chapter 7 Turing Machines Copyright 2011 The McGrawHill Companies, Inc. Permission required for reproduction or display. 1 A General Model of Computation Both finite automata and pushdown automata are
More informationTuring Machines Part II
Turing Machines Part II Problem Set Set Five Five due due in in the the box box up up front front using using a late late day. day. Hello Hello Condensed Slide Slide Readers! Readers! This This lecture
More informationCS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 8 January 24, 2018 Outline Turing Machines and variants multitape TMs nondeterministic TMs ChurchTuring Thesis So far several models of computation finite automata
More informationMore About Turing Machines. Programming Tricks Restrictions Extensions Closure Properties
More About Turing Machines Programming Tricks Restrictions Extensions Closure Properties 1 Overview At first, the TM doesn t look very powerful. Can it really do anything a computer can? We ll discuss
More informationV Honors Theory of Computation
V22.0453001 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 informationLecture 4 Nondeterministic Finite Accepters
Lecture 4 Nondeterministic Finite Accepters COT 4420 Theory of Computation Section 2.2, 2.3 Nondeterminism A nondeterministic finite automaton can go to several states at once. Transitions from one state
More 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 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 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: 4167362100 ext 77875 Course page: http://www.cse.yorku.ca/course/2001 11/7/2013 CSE
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 informationTuring Machine Variants. Sipser Section 3.2 pages
Turing Machine Variants Sipser Section 3.2 pages 176182 Marking symbols It is often convenient to have the Turing machine leave a symbol on the tape but to mark it in some way 1 1 0 2 B B B B B B B State
More informationTuring Machine Variants. Sipser pages
Turing Machine Variants Sipser pages 148154 Marking symbols It is often convenient to have the Turing machine leave a symbol on the tape but to mark it in some way 1 1 0 2 B B B B B B B State = 1 1 1
More informationReview of Complexity Theory
Review of Complexity Theory Breno de Medeiros Department of Computer Science Florida State University Review of Complexity Theory p.1 Turing Machines A traditional way to model a computer mathematically
More informationCSCI3390Assignment 2 Solutions
CSCI3390Assignment 2 Solutions due February 3, 2016 1 TMs for Deciding Languages Write the specification of a Turing machine recognizing one of the following three languages. Do one of these problems.
More informationTuring Machines Part II
Turing Machines Part II Hello Hello Condensed Slide Slide Readers! Readers! This This lecture lecture is is almost almost entirely entirely animations that that show show how how each each Turing Turing
More informationNotes for Comp 497 (Comp 454) Week 12 4/19/05. Today we look at some variations on machines we have already seen. Chapter 21
Notes for Comp 497 (Comp 454) Week 12 4/19/05 Today we look at some variations on machines we have already seen. Errata (Chapter 21): None known Chapter 21 So far we have seen the equivalence of Post Machines
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 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 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. ChurchTuring Thesis:
More informationTuring machine recap. Universal Turing Machines and Undecidability. Alphabet size doesn t matter. Robustness of TM
Turing machine recap A Turing machine TM is a tuple M = (Γ, Q, δ) where Γ: set of symbols that TM s tapes can contain. Q: possible states TM can be in. qstart: the TM starts in this state qhalt: the TM
More informationDecidability: ChurchTuring Thesis
Decidability: ChurchTuring 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 informationHomework Assignment 6 Answers
Homework Assignment 6 Answers CSCI 2670 Introduction to Theory of Computing, Fall 2016 December 2, 2016 This homework assignment is about Turing machines, decidable languages, Turing recognizable languages,
More informationTuring Machines Part III
Turing Machines Part III Announcements Problem Set 6 due now. Problem Set 7 out, due Monday, March 4. Play around with Turing machines, their powers, and their limits. Some problems require Wednesday's
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105ab/ Today's learning goals Sipser Ch 3.3, 4.1 State and use the ChurchTuring thesis. Give examples of decidable problems.
More informationCFGs and PDAs are Equivalent. We provide algorithms to convert a CFG to a PDA and vice versa.
CFGs and PDAs are Equivalent We provide algorithms to convert a CFG to a PDA and vice versa. CFGs and PDAs are Equivalent We now prove that a language is generated by some CFG if and only if it is accepted
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 informationwhere Q is a finite set of states
Space Complexity So far most of our theoretical investigation on the performances of the various algorithms considered has focused on time. Another important dynamic complexity measure that can be associated
More information2.6 Variations on Turing Machines
2.6 Variations on Turing Machines Before we proceed further with our exposition of Turing Machines as language acceptors, we will consider variations on the basic definition of Slide 10 and discuss, somewhat
More informationTURING MAHINES
15453 TURING MAHINES TURING MACHINE FINITE STATE q 10 CONTROL AI N P U T INFINITE TAPE read write move 0 0, R, R q accept, R q reject 0 0, R 0 0, R, L read write move 0 0, R, R q accept, R 0 0, R 0 0,
More informationCS 525 Proof of Theorems 3.13 and 3.15
Eric Rock CS 525 (Winter 2015) Presentation 1 (Week 4) Equivalence of SingleTape and MultiTape Turing Machines 1/30/2015 1 Turing Machine (Definition 3.3) Formal Definition: (Q, Σ, Γ, δ, q 0, q accept,
More informationDecidability (intro.)
CHAPTER 4 Decidability Contents Decidable Languages decidable problems concerning regular languages decidable problems concerning contextfree languages The Halting Problem The diagonalization method The
More informationA Note on Turing Machine Design
CS103 Handout 17 Fall 2013 November 11, 2013 Problem Set 7 This problem explores Turing machines, nondeterministic computation, properties of the RE and R languages, and the limits of RE and R languages.
More informationAn example of a decidable language that is not a CFL Implementationlevel description of a TM State diagram of TM
Turing Machines Review An example of a decidable language that is not a CFL Implementationlevel description of a TM State diagram of TM Varieties of TMs MultiTape TMs Nondeterministic TMs String Enumerators
More informationIntroduction 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 informationFORMAL LANGUAGES, AUTOMATA AND COMPUTATION
FORMAL LANGUAGES, AUTOMATA AND COMPUTATION DECIDABILITY ( LECTURE 15) SLIDES FOR 15453 SPRING 2011 1 / 34 TURING MACHINESSYNOPSIS The most general model of computation Computations of a TM are described
More informationEquivalence of Regular Expressions and FSMs
Equivalence of Regular Expressions and FSMs Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Regular Language Recall that a language
More informationTuring Machines Extensions
Turing Machines Extensions KR Chowdhary Professor & Head Email: kr.chowdhary@ieee.org Department of Computer Science and Engineering MBM Engineering College, Jodhpur October 6, 2010 Ways to Extend Turing
More informationChomsky Normal Form and TURING MACHINES. TUESDAY Feb 4
Chomsky Normal Form and TURING MACHINES TUESDAY Feb 4 CHOMSKY NORMAL FORM A contextfree grammar is in Chomsky normal form if every rule is of the form: A BC A a S ε B and C aren t start variables a is
More informationChapter 5. Finite Automata
Chapter 5 Finite Automata 5.1 Finite State Automata Capable of recognizing numerous symbol patterns, the class of regular languages Suitable for patternrecognition type applications, such as the lexical
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 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 informationTheory of Computation
Theory of Computation Lecture #10 Sarmad Abbasi Virtual University Sarmad Abbasi (Virtual University) Theory of Computation 1 / 43 Lecture 10: Overview Linear Bounded Automata Acceptance Problem for LBAs
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.19127Jun.1954), an English student of Church, introduced a machine model for effective calculation in On
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 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 informationCS 410/610, MTH 410/610 Theoretical Foundations of Computing
CS 410/610, MTH 410/610 Theoretical Foundations of Computing Fall Quarter 2010 Slides 2 Pascal Hitzler Kno.e.sis Center Wright State University, Dayton, OH http://www.knoesis.org/pascal/ CS410/610 MTH410/610
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
15453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY Chomsky Normal Form and TURING MACHINES TUESDAY Feb 4 CHOMSKY NORMAL FORM A contextfree grammar is in Chomsky normal form if every rule is of the form:
More informationFurther discussion of Turing machines
Further discussion of Turing machines In this lecture we will discuss various aspects of decidable and Turingrecognizable languages that were not mentioned in previous lectures. In particular, we will
More informationComplexity Theory Part I
Complexity Theory Part I Problem Problem Set Set 77 due due right right now now using using a late late period period The Limits of Computability EQ TM EQ TM core R RE L D ADD L D HALT A TM HALT A TM
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 informationUniversal Turing Machine. Lecture 20
Universal Turing Machine Lecture 20 1 Turing Machine move the head left or right by one cell read write sequentially accessed infinite memory finite memory (state) nextaction lookup table Variants don
More informationNonemptiness Testing for TMs
180 5. Reducibility The proof of unsolvability of the halting problem is an example of a reduction: a way of converting problem A to problem B in such a way that a solution to problem B can be used to
More informationAutomata Theory CS S12 Turing Machine Modifications
Automata Theory CS4112015S12 Turing Machine Modifications David Galles Department of Computer Science University of San Francisco 120: Extending Turing Machines When we added a stack to NFA to get a
More informationCSCE 551: ChinTser Huang. University of South Carolina
CSCE 551: Theory of Computation ChinTser Huang huangct@cse.sc.edu University of South Carolina ChurchTuring Thesis The definition of the algorithm came in the 1936 papers of Alonzo Church h and Alan
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 informationTime Complexity (1) CSCI Spring Original Slides were written by Dr. Frederick W Maier. CSCI 2670 Time Complexity (1)
Time Complexity (1) CSCI 2670 Original Slides were written by Dr. Frederick W Maier Spring 2014 Time Complexity So far we ve dealt with determining whether or not a problem is decidable. But even if it
More information1 Deterministic Turing Machines
Time and Space Classes Exposition by William Gasarch 1 Deterministic Turing Machines Turing machines are a model of computation. It is believed that anything that can be computed can be computed by a Turing
More informationThe purpose here is to classify computational problems according to their complexity. For that purpose we need first to agree on a computational
1 The purpose here is to classify computational problems according to their complexity. For that purpose we need first to agree on a computational model. We'll remind you what a Turing machine is  you
More informationAnnouncements. Problem Set 6 due next Monday, February 25, at 12:50PM. Midterm graded, will be returned at end of lecture.
Turing Machines Hello Hello Condensed Slide Slide Readers! Readers! This This lecture lecture is is almost almost entirely entirely animations that that show show how how each each Turing Turing machine
More informationCSCE 551: ChinTser Huang. University of South Carolina
CSCE 551: Theory of Computation ChinTser Huang huangct@cse.sc.edu University of South Carolina Computation History A computation history of a TM M is a sequence of its configurations C 1, C 2,, C l such
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 informationLecture Notes: The Halting Problem; Reductions
Lecture Notes: The Halting Problem; Reductions COMS W3261 Columbia University 20 Mar 2012 1 Review Key point. Turing machines can be encoded as strings, and other Turing machines can read those strings
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 informationUNITVIII COMPUTABILITY THEORY
CONTEXT SENSITIVE LANGUAGE UNITVIII COMPUTABILITY THEORY A Context Sensitive Grammar is a 4tuple, G = (N, Σ P, S) where: N Set of non terminal symbols Σ Set of terminal symbols S Start symbol of the
More informationTuring Machine Variants
CS311 Computational Structures Turing Machine Variants Lecture 12 Andrew Black Andrew Tolmach 1 The ChurchTuring Thesis The problems that can be decided by an algorithm are exactly those that can be decided
More informationLecture 1: Course Overview and Turing machine complexity
CSE 531: Computational Complexity I Winter 2016 Lecture 1: Course Overview and Turing machine complexity January 6, 2016 Lecturer: Paul Beame Scribe: Paul Beame 1 Course Outline 1. Basic properties of
More information