CS154. Turing Machines. Turing Machine. Turing Machines versus DFAs FINITE STATE CONTROL AI N P U T INFINITE TAPE. read write move.


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1 CS54 Turing Machines Turing Machine q 0 AI N P U T IN TAPE read write move read write move Language = {0} q This Turing machine recognizes the language {0} Turing Machines versus DFAs TM can both write to and read from the tape The head can move left and right The input doesn t have to be read entirely, and the computation can continue further (even, forever) after all input has been read Accept and Reject take immediate effect Deciding the language L = { w#w w {0,}* } q 0, FIND q, # FIND # q #, FIND q 0, FIND q, FIND q GO LEFT X X # 0# 0X. If there s no # on the tape,. 2. While there is a bit to the left of #, Replace the first bit with X, and check if the first bit to the right of the # is identical. (If not,.) Replace that bit with an X too. 3. If there s a bit to the right of #, then else accept
2 Definition: A Turing Machine is a 7tuple T = (Q, Σ, Γ, δ, q 0,, q ), where: Q is a finite set of s Σ is the input alphabet, where Σ Γ is the tape alphabet, where Γand Σ Γ δ : Q Γ Q Γ {L, R} q 0 Q is the start Q is the accept q Q is the, and q Turing Machine Configurations q corresponds to the configuration: 00q Defining Acceptance and Rejection Let C and C 2 be configurations of M We say C yields C 2 if, after running M in C for one step, M is then in configuration C 2 Suppose δ(q, b) = (q 2, c, L) Then aaq bb yields aq 2 acb Suppose δ(q, a) = (q 2, c, R) Then cabq a yields cabcq 2 Let w Σ* and M be a Turing machine M accepts w if there are configs C 0, C,..., C k, s.t. C 0 = q 0 w C i yields C i+ for i = 0,..., k, and C k contains the accepting A TM M recognizes a language L iff M accepts all and only those strings in L A language L is called recognizable or recursively enumerable (r.e.) iff some TM recognizes L A TM M decides a language L iff M accepts all strings in L and s all strings not in L A language L is called decidable or recursive iff some TM decides L A language is called recognizable (recursively enumerable) if some TM recognizes it A language is called decidable (recursive) if some TM decides it recognizable languages decidable languages { 02 n n 0 } PSEUDOCODE:. Sweep from left to right, cross out every other 0 2. If in stage, the tape had only one 0, accept 3. If in stage, the tape had an odd number of 0 s, 4. Move the head back to the first input symbol. 5. Go to stage. Why does this work? Idea: Every time we return to stage, the number of 0 s on the tape is halved. 2
3 2 { 0 n n 0 } q 0 q 0, R q q 2 0 x, R 0 x, R x x, L 0 0, L q 3 q 4 C = {a i b j c k k = i*j, and i, j, k } PSEUDOCODE:. If the input doesn t match a*b*c*,. 2. Move the head back to the leftmost symbol. 3. Cross off an a, scan to the right until b. Sweep between b s and c s, crossing off one of each until all b s are crossed off. If all c s get crossed off while doing this,. 4. Uncross all the b s. If there s another a left, then repeat stage 3. If all a s are crossed out, Check if all c s are crossed off. If yes, then accept, else. C = {a i b j c k k = i*j, and i, j, k } aabbbcccccc xabbbcccccc xayyyzzzccc xabbbzzzccc xxyyyzzzzzz Multitape Turing Machines δ : Q Γ k Q Γ k {L,R} k k # # # # # # 3
4 # # # # # # Nondeterministic TMs Theorem: Every nondeterministic Turing machine N can be transformed into a single tape Turing Machine M that recognizes the same language. Proof Idea (more details in Sipser): M(w): For all strings C {Q Γ #}* in lex. order, Check if C = C 0 # #C k where C 0,,C k is some accepting computation history for N on w. If so, accept. We can encode a TM as a bit string. n s 0 n 0 m 0 k 0 s 0 t 0 r 0 u m tape symbols (first k are input symbols) start accept blank symbol ( (p, a), (q, b, L) ) = 0 p 0 a 0 q 0 b 0 ( (p, a), (q, b, R) ) = 0 p 0 a 0 q 0 b Similarly, we can encode DFAs and NFAs as bit strings. So we can define the following languages: A DFA = { (B, w) B is a DFA that accepts string w } A NFA = { (B, w) B is an NFA that accepts string w } A TM = { (C, w) C is a TM that accepts string w } (Can define (x, y) := 0 x xy) A DFA = { (D, w) D is a DFA that accepts string w } Theorem: A DFA is decidable Proof Idea: Directly simulate D on w! A NFA = { (N, w) N is an NFA that accepts string w } Theorem: A NFA is decidable. (Why?) A TM = { (M, w) M is a TM that accepts string w } A TM is recognizable but not decidable 4
5 Universality of Turing Machines Theorem: There is a universal Turing machine U that can take as input  the code of an arbitrary TM M  an input string w, such that U(M, w) accepts iff M(w) accepts. This is a fundamental property: Turing machines can run their own code! Note that DFAs/NFAs do not have this property. That is, A DFA is not regular. The ChurchTuring Thesis Everyone s Intuitive Notion = Turing Machines of Algorithms This thesis is tested every time you write a program that does something! A TM recognizes a language if it accepts all and only those strings in the language A language is called recognizable or recursively enumerable, (or r.e.) if some TM recognizes it w Σ* TM w L? yes no w Σ* TM w L? yes no A TM decides a language if it accepts all strings in the language and s all strings not in the language A language is called decidable (or recursive) if some TM decides it accept L is decidable (recursive) accept or no output L is recognizable (recursively enumerable) Theorem: L is decidable if both L and L are recursively enumerable Recall: Given L Σ*, define L := Σ*  L Theorem: L is decidable iff both L and L are recognizable Given: a TM M that recognizes A and a TM M2 that recognizes A, we can build a new machine M that decides A How? Any ideas? M always accepts x, when x is in A M2 always accepts x, when x is not in A There are languages over {0,} that are not decidable Assuming the ChurchTuring Thesis, this means there are problems that NO computing device can solve We can prove this using a counting argument. We will show there is no onto function from the set of all Turing Machines to the set of all languages over {0,}. (But it works for any Σ) That is, every mapping from Turing machines to languages must miss some language. 5
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