Intro to Theory of Computation
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1 Intro to Theory of Computation LECTURE 13 Last time: Turing Machines and Variants Today Turing Machine Variants Church-Turing Thesis Sofya Raskhodnikova 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L13.1
2 TMs are equivalent to TMs are equivalent to multitape TMs (last time) TMs are equivalent to nondeterministic TMs (today) TMs are equivalent to doubly unbounded TMs (today) TMs are equivalent to enumerators. (today) TMs are equivalent to FIFO automata. (HW problem) TMs are equivalent to primitive recursive functions. TMs are equivalent to cellular automata. 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L13.2
3 I-clicker problem (frequency: AC) Which of these statements are valid descriptions of nondeterministic steps (in a PDA)? A. Nondeterministically read the input and push it onto the stack. B. Nondeterministically either read a and push it onto the stack or read b and pop b from the stack. C. Nondeterministically read the input character a and either push it onto the stack or pop b from the stack. D. Nondeterministically push one of positive integers onto the stack. E. None of the above.
4 NTMs are equivalent to TMs Theorem. A deterministic TM can simulate a nondeterministic TM. Proof idea: Consider an NTM N. Use a 3-tape TM. FINITE CONTROL w q 0 w # input to TM N (read-only tape) simulation tape (run N on w using nondeterministic choices from tape 3) address in the computation tree Let b be the largest # of nondeterministic choices N has in a step. Use alphabet {1,, b} for addresses. Do a BFS of the computation tree. 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper
5 Doubly unbounded TMs A TM with doubly unbounded tape is like an ordinary TM but Its tape is infinite on the left and on the right. Initially, only the input is written on the tape and the head is on the first nonblack symbol. 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper
6 I-clicker problem (frequency: AC) A TM can simulate a doubly unbounded TM U A. by marking the leftmost ``investigated square and using a nondeterministic step every time U moves to the left of it. B. by using 2 tapes: one for input + squares to the right; the other for squares to the left of the input. C. by using each square of the tape to keep two characters from U s tape alphabet (2 tracks on the tape). D. None of the above. E. More than one choice above works.
7 TM variant: enumerator FINITE CONTROL Starts with a blank tape Prints strings L(E) = set of strings that E eventually prints. Enumerator E enumerates language L(E). May never terminate even if the language is finite. May print the same string many times. tape printer 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper
8 TMs vs. enumerators Theorem. A language is Turing-recognizable some enumerator enumerates it. Proof: Start with an enumerator E that enumerates A. Give a TM that recognizes A. Start with a TM that recognizes A. Give an enumerator E that enumerates A. 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper
9 TMs are equivalent to TMs are equivalent to multitape TMs (last time) TMs are equivalent to nondeterministic TMs (on the board) TMs are equivalent to double unbounded TMs (on the board) TMs are equivalent to enumerators. (on the board) TMs are equivalent to FIFO automata. (HW problem) TMs are equivalent to primitive recursive functions. TMs are equivalent to cellular automata. 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L13.9
10 The Church-Turing Thesis (1936) L is recognized by a program for some computer* L is recognized by a TM 23 Hilbert s problems (1900) History stated at International Congress of Mathematicians 10 th problem: Give a procedure for determining if a polynomial in k variables has an integral root. * The computer must be reasonable 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L13.10
11 The Church-Turing Thesis is consistent with all known reasonable computers R1: R2: RAM: #1011# # #...# Programs for a computer have instructions like ADD R1, R2, R3; LOAD R1, R2; STORE R1,R2; MUL R1, R2, R2; BRANCH R1, X; 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L13.11
12 Programming languages Programming languages like Java, Python, Scheme, C, are equivalent to TMs We call such languages Turing-complete Corollary. If two programming languages are Turing-complete, then they can recognize exactly the same set of languages. 2/23/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L13.12
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