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David Evans http://www.cs.virginia.edu/evans cs302: Theory of Computation University of Virginia Computer Science Lecture 13: Turing Machines

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2 The Story So Far Regular Languages Context-Free Languages Violates Pumping Lemma For RLs Violates Pumping Lemma For CFLs Described by DFA, NFA, RegExp, RegGram Described by CFG, NDPDA 0n1n0n1n 0n1n2n0n1n2n 0n0n w Deterministic CFLs LL(k) Languages Described by LL(k) Grammar Indexed Grammars

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3 Lecture 13: Turing Machines The Story So Far (Simplified) Regular Languages Context-Free Languages Violates Pumping Lemma For RLs Violates Pumping Lemma For CFLs Described by DFA, NFA, RegExp, RegGram Described by CFG, NDPDA 0n1n0n1n 0n0n w

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4 Lecture 13: Turing Machines Computability Story: This Week Languages recognizable by any mechanical computing machine

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5 Lecture 13: Turing Machines Computability Story: Next Week Undecidable Problems Decidable Problems Recognizable Languages

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6 Lecture 13: Turing Machines Computability Complexity (April) Decidable Problems Problems that can be solved by a computer (eventually). Problems that can be solved by a computer in a reasonable time. NP P Note: not known if P NP or P = NP

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7 Lecture 13: Turing Machines Exam 1

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8 Lecture 13: Turing Machines Exam 1 Problem 4c: Prove that the language {0 n 1 n 2 } is not context-free. Lengths of strings in L : n = 00 + 0 2 = 0 n = 11 + 1 2 = 2 n = 2 2 + 2 2 = 6 n = 33 + 3 2 = 12... n = kk + k 2 Pumping lemma for CFLs says there must be some way of picking s = uvxyz such that m = |v| + |y| > 0 and uv i xy i z in L for all i. So, increasing i by 1 adds m symbols to the string, which must produce a string of a length that is not the length of a string in L.

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9 Lecture 13: Turing Machines Recognizing {0 n 1 n 2 } DPDA with two stacks? DPDA with three stacks?... ?

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10 Lecture 13: Turing Machines 3-Stack DPDA Recognizing {0 n 1 n 2 } 0, ε/ε/ε $/$/$ 0, ε/ε/ε +/ ε /+ 1, $/$/$ ε/ε/ε 1, +/ε/ε ε/+/ε 1, $/+/+ $/ε/ε 1, ε/+/ε +/ε/ε Start Done Count 0s 1s b->r 1s r->b 1, ε/$/$ ε/ε/ε 1, + /$/+ ε /+/ε 1, $/ ε /$ ε/ε/ε

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11 Lecture 13: Turing Machines Can it be done with 2 Stacks?

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12 Lecture 13: Turing Machines Simulating 3-DPDA with 2-DPDA # # A B

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13 Lecture 13: Turing Machines Simulating 3-DPDA with 2-DPDA 3-DPDA # # pop green push B ($) push B (pop A ())... push B (pop A (#)) push B (pop A ())... push B (pop A (#)) res = pop A () push A (pop B (#)) push A (pop B ())... pop B ($) A B $ +

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14 Lecture 13: Turing Machines 2-DPDA + Forced ε -Transitions A B Need to do lots of stack manipulation to simulate 3-DPDA on one transition: need transitions with no input symbol (but not nondeterminism!) # # $

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15 Lecture 13: Turing Machines Impact of Forced ε -Transitions A B # # $ What is the impact of adding non-input consuming transitions? DPDA in length n input: runs for n steps DPDA+ ε in length n input: can run forever!

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16 Lecture 13: Turing Machines Is there any computing machine we can’t simulate with a 2-DPDA+?

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17 Lecture 13: Turing Machines What about an NDPDA? A B Use one stack to simulate the NDPDA’s stack. Use the other stack to keep track of nondeterminism points: copy of stack and decisions left to make.

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18 Lecture 13: Turing Machines Turing Machine? A B Tape Head

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19 Lecture 13: Turing Machines Turing Machine... Infinite tape: Γ* Tape head: read current square on tape, write into current square, move one square left or right FSM:like PDA, except: transitions also include direction (left/right) final accepting and rejecting states FSM

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20 Lecture 13: Turing Machines Turing Machine Formal Description... FSM 7-tuple: (Q, , Γ, δ, q 0, q accept, q reject ) Q : finite set of states : input alphabet (cannot include blank symbol, _) Γ : tape alphabet, includes and _ δ : transition function: Q Γ Q Γ {L, R} q 0 : start state, q 0 Q q accept : accepting state, q accept Q q reject : rejecting state, q reject Q (Sipser’s notation)

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21 Lecture 13: Turing Machines Turing Machine Computing Model... FSM q0q0 input ____ blanks Initial configuration: xxxxxxx x TM Configuration: Γ* Q Γ * tape contents left of head tape contents head and right current FSM state

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22 Lecture 13: Turing Machines TM Computing Model δ *: Γ* Q Γ * Γ* Q Γ * δ*(L, q accept, R) (L, q accept, R) δ*(L, q reject, R) (L, q reject, R) The q accept and q reject states are final:

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23 Lecture 13: Turing Machines TM Computing Model δ *: Γ* Q Γ * Γ* Q Γ *... FSM q a u, v Γ*, a, b Γ u b v δ*(ua, q, bv) = (uac, q r, v) if δ(q, b) = (q r, c, R) δ*(ua, q, bv) = (u, q r, acv) if δ(q, b) = (q r, c, L) Also: need a rule to cover what happens at left edge of tape

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24 Lecture 13: Turing Machines Thursday’s Class Robustness of TM model Church- Turing Thesis Read Chapter 3: It contains the most bogus sentence in the whole book. Identify it for +25 bonus points (only 1 guess allowed!)

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