Dept Computer Science Korea Univ Intelligent Information System

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Automata and Computation Theory 제 7장 Turing machine & 제 8장 Undecidability Intelligent Information System Lab Computer Science Department Korea University Professor I. J. Jung 2021 -10 -26 1

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Language L⊆Σ* is called recursively enumerable (r. e) if there is a TM T such that L = L(T) if there is a TM T which halts for every input x∈ Σ* and has L=L(T), then L is called recursive. 2021 -10 -26 2

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. A TM T can be regarded as defining a (partial) mapping T: Σ*→ Γ* by taking T(w) to be the sequence extending through the last (i. e. right most) non-blank symbol when (and if) T halts after being started an w. 2021 -10 -26 3

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Example : TM computes strategy : note = m iff m 2≥n & (m-1)2<n, so compute 12, 22, 32, …, i 2 until first encounter i 2 ≥ n input 0 n 1 i 0 i i 2 0 i endmark A 2021 -10 -26 1 2 1 # # … 4

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. 0 q 0 0 q 1 R 0 i=1 B A q 0 1 f q 1 0 q 2 1 R if n=0 halts with result = 0 0 0 B L A q 1 1 q 2 0 0 Terminate i track 2 L B 2021 -10 -26 A 5

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. a (subroutine) TM which computes the sequence of the integer in track 2 placing result in track 3 does not change tracks 1, 2, 4 moves to endmark and transfers to q 3 q 2 Transitions Not specified 0 0 q 3 2021 -10 -26 0 B A 0 0 q 3 q 4 0 R B A 0 0 B B A A Compare i 2 and n For all X and Y R i 2<n and i = i 2 set i = i+1 for all Y 6

Dept. Computer Science, Korea Univ. q 3 Intelligent Information System Lab. 0 0 B q 3 B A A 0 0 0 B A q 4 0 B R i < i 2 < n R n < i 2 For all X≠ 1 and all Y A . . 2021 -10 -26 7

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Post correspondence Problem (PCP) An instance of this problem over alphabet ∑ consists of 2 tuples of strings A=<a 1, …, ak>, B=<b 1, …, bk>, where ai, bi ∈ ∑* The question is whether ∃ a sequence i 1, i 2, …, im with m≥ 1 of positive integer (soln) ∍: ai 1 ai 2. . . aim = bi 1 bi 2. . . Bim 2021 -10 -26 8

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Thm 8. 8 PCP is undecidable Thm 8. 9 It’s undecidable whether an arbitrary CFL is ambiguous. 2021 -10 -26 9

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Universal language Lu = { <T, w> | T accepts w } Clearly if we have a Turing accepts for Lu, we need no other TMs. To determine if T will accept w, just give code(T) w to the UM. 2021 -10 -26 10

Dept. Computer Science, Korea Univ. Intelligent Information System Lab. Thm 8. 4 Lu is r. e. (i. e. ∃ a UM) Thm 8. 5 Lu is not recursive. Assertion : The halting problem for TMs is undecidable (i. e. does T halt on w? ) Assertion : The membership problem for TM (i. e. is w∈L(T)? ) is undecidable. 2021 -10 -26 11