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Unformatted text preview: A Universal Turing Machine Turing Machines are “hardwired” they execute only one program A limitation of Turing Machines: Real Computers are reprogrammable Solution: Universal Turing Machine • Reprogrammable machine • Simulates any other Turing Machine Attributes: Universal Turing Machine simulates any Turing Machine M Input of Universal Turing Machine: Description of transitions of M Input string of M Universal Turing Machine M Description of Tape Contents of M State of M Three tapes Tape 2 Tape 3 Tape 1 We describe Turing machine as a string of symbols: We encode as a string of symbols M M Description of M Tape 1 Alphabet Encoding Symbols: a b c d Encoding: 1 11 111 1111 State Encoding States: 1 q 2 q 3 q 4 q Encoding: 1 11 111 1111 Head Move Encoding Move: Encoding: L R 1 11 Transition Encoding Transition: ) , , ( ) , ( 2 1 L b q a q = δ Encoding: 1 11 11 1 1 separator Turing Machine Encoding Transitions: ) , , ( ) , ( 2 1 L b q a q = δ Encoding: 1 11 11 1 1 ) , , ( ) , ( 3 2 R c q b q = δ 11 111 111 10 1 11 00 separator Tape 1 contents of Universal Turing Machine: binary encoding of the simulated machine M 1100 111 111 10 1 10011 11 11 1 1 Tape 1 A Turing Machine is described with a binary string of 0’s and 1’s The set of Turing machines forms a language: each string of this language is the binary encoding of a Turing Machine Therefore: Language of Turing Machines L = { 010100101, 00100100101111, 111010011110010101, …… } (Turing Machine 1) (Turing Machine 2) …… Countable Sets Infinite sets are either: Countable or Uncountable Countable set: There is a one to one correspondence of elements of the set to Natural numbers (Positive Integers) (every element of the set is mapped to a number such that no two elements are mapped to same number) Example: Even integers: (positive) , 6 , 4 , 2 , The set of even integers is countable Positive integers: Correspondence: , 4 , 3 , 2 , 1 n 2 corresponds to 1 + n Example: The set of rational numbers is countable Rational numbers: , 8 7 , 4 3 , 2 1 Naïve Approach...
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 Fall '11
 GraceGraham
 Turing Machines, S6, Zürich Hauptbahnhof

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