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Машина Тьюринга — различия между версиями

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Строка 1: Строка 1:
Неформально, ''Машина Тьюринга'' (далее '''МТ''')
+
#REDIRECT [[discopal:{{PAGENAME}}]]
представляет собой автомат с конечным числом состояний и неограниченной памятью,
+
представленной набором одной или более ''лент'', бесконечных в обоих направлениях.
+
Ленты поделены на соответственно бесконечное число ячеек, и на каждой ленте
+
выделена стартовая (нулевая) ячейка. В каждой ячейке может быть записан
+
только один символ из некоторого конечного алфавита <math>\Sigma</math>, где
+
предусмотрен символ «*» для обозначения пустой ячейки.
+
 
+
На каждой ленте имеется головка чтения-записи, и все они
+
подсоединены к «управляющему модулю» МТ — автомату с конечным множеством состояний <math>\Gamma</math>.
+
 
+
Имеется выделенное стартовое состояние (например, «START») и состояние (или набор состояний) завершения (например «STOP»).
+
 
+
Перед запуском МТ находится в состоянии «START», а все головки позиционированы
+
на нулевые ячейки соответствующих лент. На каждом шаге все головки считывают
+
информацию из своих текущих ячеек и посылают ее управляющему модулю МТ.
+
В зависимости от этих символов и собственного состояния управляющий модуль
+
производит следующие операции:
+
 
+
# Посылает каждой головке символ для записи в текущую ячейку каждой ленты;
+
# Посылает каждой головке одну из команд «LEFT»,"RIGHT","STAY";
+
# Выполняет переход в новое состояние (которое, впрочем, может совпадать с предыдущим).
+
 
+
Теперь то же самое более формально.
+
   
+
''Машина Тьюринга'' это набор
+
<math>T= \langle k,\Sigma,\Gamma,\alpha,\beta,\gamma \rangle</math>,
+
где
+
 
+
;k: натуральное число,
+
;<math>\Sigma,\Gamma</math>: конечные множества входного алфавита и состояний соответственно,
+
;<math>\star \in \Sigma</math>: символ-пробел,
+
;<math>START,STOP \in \Gamma</math>: выделенные состояния,
+
;<math>\alpha,\beta,\gamma</math>: произвольные отображения:
+
:*<math>\alpha: \Gamma \times \Sigma^k \rightarrow \Gamma  </math> (задает новое состояние);
+
:*<math>\beta : \Gamma \times \Sigma^k \rightarrow \Sigma^k  </math> (задает символы для записи на ленты);
+
:*<math>\gamma: \Gamma \times \Sigma^k \rightarrow \{-1,0,1\}^k </math> (определяет, как двигать головки).
+
 
+
Удобно считать, что алфавит <math>\Sigma</math> содержит кроме «пробела» («*»)
+
два выделенных символа, «0» и «1» (Обычно вовсе ограничиваются <math>\Sigma \equiv \{\star,0,1\}</math>).
+
 
+
Под ''входом'' для МТ подразумевается набор из ''k'' слов (''k''-кортеж слов из <math>\Sigma^*</math>),
+
записанных на ''k'' лентах начиная с нулевых позиций.
+
Обычно, на входные данные записывают только на первую ленту, и под входом ''x''
+
подразумевают ''k''-кортеж <math>\langle x,0,\ldots,0 \rangle</math>.
+
 
+
{\it Результатом} работы МТ на некоем входе является также
+
''k''-кортеж слов из <math>\Sigma^*</math>, оставшихся на лентах. Для простоты
+
также удобно считать, что ''результатом'' является только слово на последней
+
ленте, а все остальное — просто мусор.
+
 
+
Также считается, что входное слово не содержит пробелов — действительно, иначе было бы невозможно определить, где кончается входное слово
+
(Можно конечно разрешить пробелы, но тогда придется зарезервировать еще один символ — «конец ввода»).
+
 
+
Существуют следующие обобщения машины Тьюринга:
+
* [[Недетерминированная машина Тьюринга]]
+
* [[Вероятностная машина Тьюринга]]
+
 
+
 
+
==Примеры==
+
Если что-то осталось непонятным, можно рассмотреть симулятор работы МТ на языке [[Python]], который мы будем использовать, чтобы проиллюстрировать процесс работы машин Тьюринга.
+
 
+
<code-python>
+
def execute_MT(MT,input):
+
    T=MT["program"]
+
    tape=["*"]+input
+
    state=MT["start"]
+
    position=1
+
    history=[{"state":state, "position":position, "tape": []+tape}]
+
    step=0
+
    while 1:
+
      step=step+1
+
      if position>=len(tape):
+
          tape.append("*")
+
      symbol_under_head=tape[position]
+
      action=T[(state,(symbol_under_head))]
+
      state=action[0]
+
      symbol_to_write=action[1][0]
+
      tape[position]=symbol_to_write
+
      move=action[1][1]
+
      if move=="L": position=position-1
+
      if move=="R": position=position+1
+
      history.append({"state":state, "position":position, "tape": []+tape})
+
      if state==MT["stop"] or step>1000:
+
          break
+
    return history
+
</code-python>
+
 
+
 
+
Он принимает на вход описание машины Тьюринга в виде хэш-таблицы — рассмотрим пример машины тьюринга для задачи удвоения входной строки (алфавит состоит только из «1»).
+
Табличное описание МТ:
+
 
+
<latex>
+
\begin{tabular}{|cc|c|ccc|}
+
\hline
+
<\textcolor{blue}{s1}> & * & $\Rightarrow$ & [\textcolor{red}{q}] & * &  \\ \hline
+
<\textcolor{blue}{s1}> & 1 & $\Rightarrow$ & s2 & * & R \\ \hline
+
s2 & * & $\Rightarrow$ & s3 & * & R \\ \hline
+
s2 & 1 & $\Rightarrow$ & s2 & 1 & R \\ \hline
+
s3 & * & $\Rightarrow$ & s4 & 1 & L \\ \hline
+
s3 & 1 & $\Rightarrow$ & s3 & 1 & R \\ \hline
+
s4 & * & $\Rightarrow$ & s5 & * & L \\ \hline
+
s4 & 1 & $\Rightarrow$ & s4 & 1 & L \\ \hline
+
s5 & * & $\Rightarrow$ & <\textcolor{blue}{s1}> & 1 & R \\ \hline
+
s5 & 1 & $\Rightarrow$ & s5 & 1 & L \\ \hline
+
\end{tabular}
+
</latex>
+
 
+
Таблица МТ в виде [[Python]]-структуры для вышеприведенного симулятора:
+
<code-python>
+
MT={
+
'k': 1,
+
'start': 's1',
+
'stop': 'q',
+
'program': {
+
#(Состояние, символы на лентах) -> (новое состояние, (действия по каждой ленте))
+
  ('s1', ('1')): ('s2', (('*','R'))),
+
  ('s2', ('1')): ('s2', (('1','R'))),
+
  ('s2', ('*')): ('s3', (('*','R'))),
+
  ('s3', ('*')): ('s4', (('1','L'))),
+
  ('s3', ('1')): ('s3', (('1','R'))),
+
  ('s4', ('1')): ('s4', (('1','L'))),
+
  ('s4', ('*')): ('s5', (('*','L'))),
+
  ('s5', ('1')): ('s5', (('1','L'))),
+
  ('s5', ('*')): ('s1', (('1','R'))),
+
  ('s1', ('*')): ('q',  (('*','')))
+
}
+
+
</code-python>
+
 
+
Обратите также внимание на альтернативное представление машины Тьюринга в виде ориентированных графа, где вершинами являются состояниями, возможные переходы между ними — ребрами, причем начало ребра помечено символом, который должен быть на ленте для активации перехода, а конец ребра помечен символом, который пишется на ленту, и командой перемещения головки («L», «R», «_» ):
+
 
+
<graph>
+
digraph G{rankdir=LR; edge[fontcolor="blue", fontsize="10", arrowhead="empty"];
+
"s1"[shape="diamond"];
+
"q"[shape="box"];"s5"->"s1" [headlabel="1R",taillabel="*"];
+
"s1"->"q" [headlabel="*",taillabel="*"];
+
"s5"->"s5" [headlabel="1L",taillabel="1"];
+
"s1"->"s2" [headlabel="*R",taillabel="1"];
+
"s3"->"s4" [headlabel="1L",taillabel="*"];
+
"s4"->"s5" [headlabel="*L",taillabel="*"];
+
"s2"->"s2" [headlabel="1R",taillabel="1"];
+
"s2"->"s3" [headlabel="*R",taillabel="*"];
+
"s3"->"s3" [headlabel="1R",taillabel="1"];
+
"s4"->"s4" [headlabel="1L",taillabel="1"];
+
}
+
</graph>
+
 
+
Симулятор возвращает историю выполнения — последовательность конфигураций (состояние, лента, положение головки) — МТ на данном входе.
+
 
+
Например, вот три запуска симулируемой «удваивающей» МТ на строках «1», «11», «111»:
+
 
+
{|
+
|
+
<pic-svg>
+
<svg
+
xmlns:svg="http://www.w3.org/2000/svg"
+
xmlns="http://www.w3.org/2000/svg"
+
xmlns:xlink="http://www.w3.org/1999/xlink"
+
version="1.0"
+
width="180px"
+
height="160px"
+
>
+
<ellipse
+
cx="50.0px"
+
cy="30.0px"
+
rx="28px"
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ry="8px"
+
style="fill:yellow;stroke:black;stroke-width:1"
+
/>
+
<g>
+
<rect
+
width="20px"
+
height="18.0px"
+
x="80px"
+
y="20px"
+
style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;"
+
/>
+
<rect
+
width="20px"
+
height="18.0px"
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x="100px"
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y="20px"
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style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;"
+
/>
+
<rect
+
width="20px"
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height="18.0px"
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x="120px"
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y="20px"
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style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;"
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<rect
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width="20px"
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/>
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</g>
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<ellipse
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cy="50.0px"
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<g>
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<rect
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width="20px"
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height="18.0px"
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<rect
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<text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" >
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s1</text>
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+
/>
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<text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" >
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<text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
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1</text>
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<text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
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1</text>
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1</text>
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<text x="190.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
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1</text>
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<text x="210.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
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1</text>
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<text x="230.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
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1</text>
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</pic-svg>
+
|}
+
 
+
 
+
Можно рассмотреть другие примеры МТ:
+
* [[Машина Тьюринга:Распознавание строки, с одинаковым количеством 0 и 1]]
+
* [[Машина Тьюринга:Унарное сложение]]
+
* [[Машина Тьюринга:Распознавание четности]]
+
 
+
 
+
[[Category:Теория сложности]]
+
{{replicate-from-custiswiki-to-lib}}
+

Текущая версия на 20:32, 13 июня 2011

  1. REDIRECT discopal:Машина Тьюринга