Chapter3Section2_F11

Chapter3Section2_F11 - 91.304 Foundations of (Theoretical)...

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Unformatted text preview: 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 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. Variants of Turing Machines Robustness : Invariance under certain changes What kinds of changes? Stay put! Multiple tapes Nondeterminism Enumerators (Abbreviate Turing Machine by TM.) 2 Stay Put! ransition function of the form: Transition function of the form: , , Does this really provide additional } S R, L, { : Q Q computational power? No! Can convert TM with stay put ature to one without it ow? feature to one without it. How? Theme: Show 2 models are equivalent by showing they can simulate each other . 3 Multi-Tape Turing Machines rdinary TM with several tapes Ordinary TM with several tapes. Each tape has its own head for reading and writing. Initially the input is on tape 1, with the other tapes blank. Transition function of the form: k k k Q Q } S R, L, { : ( k = number of tapes) ) L R, L, , , , , ( ) , , , ( 1 1 K K K k j k i b b q a a q = When TM is in state q i and heads 1 through k are reading symbols a 1 through a k , TM goes to state q j , writes symbols hrough and moves associated 4 tes sy bo s b 1 t o u g b k , a d o es assoc ated tape heads L, R, or S. Source: Sipser textbook Note: k tapes (each with own alphabet) but only 1 common set of states! Multi-Tape Turing Machines ulti pe Turing machines are of equal Multi-tape Turing machines are of equal computational power with ordinary Turing machines! Corollary 3.15 : A language is Turing- recognizable if and only if some multi-tape uring machine recognizes it Turing machine recognizes it....
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Chapter3Section2_F11 - 91.304 Foundations of (Theoretical)...

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