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TuringMach020211

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ADVENTURES IN LOGIC FOR UNDERGRADUATES by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Lecture 3. Turing Machines

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LECTURE 1. LOGICAL CONNECTIVES. Jan. 18, 2011 LECTURE 2. LOGICAL QUANTIFIERS. Jan. 25, 2011 LECTURE 3. TURING MACHINES. Feb. 1, 2011 LECTURE 4. GÖDEL’S BLESSING AND GÖDEL’S CURSE. Feb. 8, 2011 LECTURE 5. FOUNDATIONS OF MATHEMATICS Feb. 15, 2011 SAME TIME - 10:30AM SAME ROOM - Room 355 Jennings Hall WARNING: CHALLENGES RANGE FROM EASY, TO MAJOR PARTS OF COURSES
TURING MACHINES - GENERAL STRUCTURE Turing machines are the original theoretical model of computation, due to Alan Turing, 1937. It is extremely primitive. We follow the version by Emil Post, 1947. Remarkably, from the theoretical point of view, it is a “complete” model of computation - in various senses of “complete”. A TM consists of a “tape” which is infinite in both directions, divided into unit squares indexed by the integers. A TM also comes with a “reading head” which always hovers over a square of tape, and “reads” what is on that square of tape. Computation begins based on a “program” and an “input”. First, “initialization” takes place. At any stage of computation, the TM is in a “state”, each square of tape has a “symbol” written on it, and the reading head is “on” a square of tape.

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TURING MACHINES - GENERAL OPERATION TM, infinite two way infinite tape, reading head, program, input, initialization, state, symbol. The states (state symbols) are written q 0 ,q 1 ,... . The symbols (tape symbols) are written S 0 ,S 1 ,... . Initialization is based on the input. It causes the TM to go into state q 0 (the initial state), with the reading head at square 0. The squares of tape have symbols placed on them according to the input. Computation continues step by step, according to the program. The reading head may change symbols it is reading, and may move left or right one square. The TM state may change. Computation halts if and when no program instruction applies. The output is read off from the symbols on the tape. If computation never halts, then there is no output.
TURING MACHINES - INITIALIZATION TM, infinite two way tape, reading head, program, input, initialization, state, symbol. Reading changing symbols, moving left/right. Halting when no instruction applies. Halting yields output. No halting means no output. States q 0 ,q 1 ,... . Symbols S 0 ,S 1 ,... . The inputs for a TM are finite sequences of nonnegative integers. The outputs for a TM are nonnegative integers. (For some purposes, finite strings from a finite alphabet are used). Let n 1 ,...,n k N. S 0 serves as the “blank”, S 1 as the “1”. Initialize as follows: at square 0, put n

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