Chapter3Section1_F11

# Chapter3Section1_F11 - 91.304 Foundations of (Theoretical)...

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91.304 Foundations of (Theoretical) Computer Science Chapter 3 Lecture Notes (Section 3.1: Turing Machines) David Martin dm@cs.uml.edu With some modifications by Prof. Karen Daniels, Fall 2011 This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by- 1 sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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“Manners are not ught in lessons ” taught in lessons, said Alice. “Lessons teach you to do sums, and things of at sort ” that sort. “And you do Addition?” the White Queen asked. “What's one and one and one and one and one and one and one and one and one and one?” “ don't know ” said I don t know, said Alice. “I lost count.” “She can't do 2 Addition,” the Red Queen interrupted. Excerpt: Through the Looking Glass, Lewis Carroll
a | b | a | b | a | b | t | t | t | t | t | t | t | L control tape Turing machine syntax efinition Turing Machine is an automaton ± Definition A Turing Machine is an automaton M=(Q, Σ , Γ , δ ,q 0 ,q acc ,q rej ) where 1. Q is a finite set of states . an input alphabet that does ot clude " " the 2. Σ is an input alphabet that does not include t , the special blank character 3. Γ is a tape alphabet satisfying . 1. t Γ 2. Σ ⊂ Γ 4. δ :Q × Γ Q × Γ × {L,R} is the transition function “ taying put” is not an option except at left end of tape 1. staying put is not an option, except at left end of tape 5. q 0 is the initial state 6. q acc is the single accepting state 3 7. q rej is the single rejecting state Alan Turing proposed the Turing Machine in 1936!

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Differences from Finite Automata uring machine ± Turing machine ² Can both read from and write onto tape. o LIFO access restriction as in PDA’s stack ± No LIFO access restriction as in PDA s stack ² Read/write head (control) can move both left and right. ² Tape is infinite. ² Special states for rejecting and accepting take effect immediately. ² In some cases machine can fail to halt 4 a | b | a | b | a | b | t | t | t | t | t | t | t | L control tape
Differences in input mechanism TM has a "tape head" that points to exactly one cell ± A TM has a tape head that points to exactly one cell on its tape, which extends infinitely to the right ² At each transition, the TM looks at the current state and the current cell, and decides what new state to ove to what to write on the current cell and move to, what to write on the current cell, and whether to move one cell to the left or one cell to the right (or stay put at left end of tape) ² Hence the transition function δ :Q × Γ Q × Γ × {L,R} ± Each tape cell initially contains the blank character t ± Our previous automata (DFAs, NFAs, PDAs) all had a separate read-only input stream ± But in a TM, the input is given all at once and just written onto the left end of the tape — overwriting the blanks there | b | a | b | a | b | L 5 a | b | a | b | a | b | t | t | t | t | t | t | t | in state q 7

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## This note was uploaded on 02/13/2012 for the course CS 91.304 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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Chapter3Section1_F11 - 91.304 Foundations of (Theoretical)...

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