Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out

More Turing Machines - Mount Holyoke College

VIEWS: 0 PAGES: 13

									More Turing Machines

 Sipser 3.2 (pages 148-154)
    Multitape Turing Machines
                      0     1      0      1   0   ⨆



         Finite       a     a      a      ⨆
        control


                      b     a     ⨆



• Formally, we need only change the transition
  function to
  δ: Q×Γk → Q×Γk×{L,R}k
                         CS 311
                                                      2
                  Mount Holyoke College
     Seems more powerful, but…
• Theorem 3.13: Every multitape Turing machine
  has an equivalent single-tape Turing machine.
                            0     1     0      1   0   ⨆



           M                a     a     a     ⨆



                            b     a     ⨆



 S     #   0   1   0   1    0      #     a     a   a   #   b   a   #   ⨆

                                  CS 311
                                                                           3
                           Mount Holyoke College
 Turing-recognizable languages
• Corollary 3.15: A language is Turing-
  recognizable if and only if some multitape
  Turing machine recognizes it.




                         CS 311
                                               4
                  Mount Holyoke College
 Recognizing Composite Numbers
• Let L = {In | n is a composite number}
• Designing a Turing machine to accept L
  would seem to involve factoring n
• However, if we could guess ...




                        CS 311
                                           5
                 Mount Holyoke College
                   Guessing
• Design a machine M that on input In
  performs the following steps:
  1. Nondeterministically choose two numbers p,
     q > 1 and transform the input into #In#Ip#Iq#
  2. Multiply p by q to obtain #In#Ipq#
  3. Checks the number of I's before and after
     the middle # for equality
     • Accepts if equal, and rejects otherwise


                             CS 311
                                                     6
                      Mount Holyoke College
 Nondeterministic Turing machines
• Simply modify the transition function to
  satisfy:
  δ: Q×Γ → P(Q×Γ×{L,R})




                         CS 311
                                             7
                  Mount Holyoke College
Guessing doesn’t buy us anything!
• Theorem 3.16: Every nondeterministic Turing machine
  has an equivalent deterministic Turing machine.


                0   0    1     0     ⨆     Input tape



   D            x   x    #     0      1     x   ⨆   Simulation tape



                1   2    3      3     2     3   1       2   1   ⨆   Address tape




                               CS 311
                                                                             8
                        Mount Holyoke College
          Enumerator

                                         aa
                                        baba
                                        abba


control                             printer



                  0     0     1     0   ⨆   Work tape




                   CS 311
                                                        9
            Mount Holyoke College
 Equivalence with Turing machines
• Theorem 3.21: A language is Turing-
  recognizable if and only if some enumerator
  enumerates it.
• Proof
  (⇐) Suppose enumerator E enumerates L.
  Define M = "On input w:
  1. Run E. Every time that E outputs a string,
     compare it with w.
  2. If w ever appears in the output of E, accept."
                            CS 311
                                                      10
                     Mount Holyoke College
 Equivalence with Turing machines
• Theorem 3.21: A language is Turing-recognizable if
  and only if some enumerator enumerates it.
• Proof
  (⇒) Suppose TM M recognizes L. Build a
  lexicographic enumerator to generate the list of all
  possible strings s1,s2,... over Σ*.
  Define E = "Ignore input.
  1. Repeat the following for i = 1, 2, 3, ...
  2.  Run M for i steps on each input s1,s2,..,si.
  3.  If any computations accept, print out
        corresponding si."


                               CS 311
                                                     11
                        Mount Holyoke College
  TMs take their own sweet time...
• Recognizers, like enumerators may take a while to
  answer yes, ... and even longer to answer no
                                               aa
                                              baba
                                              abba


              control                     printer



                              0   0   1   0   ⨆      Work tape

• A TM that halts on all inputs is called a decider
• A decider that recognizes a languages is said to
  decide that language
• Call a language Turing-decidable if some Turing
  machine decides it

                               CS 311
                                                                 12
                        Mount Holyoke College
     Recognizers and Deciders
• Theorem: A language is Turing-decidable if and only if
  both it and its complement are Turing-recognizable
• Proof:
  (⇒) By definition.
                                #     w       #   ⨆



              M                 #     w       #   ⨆



  (⇐) Simulate, in parallel, M1 on tape 1 and M2 on tape
  2.
                             CS 311
                                                       13
                      Mount Holyoke College

								
To top