Машина Тьюринга

Материал из свободной русской энциклопедии «Традиция»
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Неформально, Машина Тьюринга (далее МТ) 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 \(\Sigma\), 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 \(\Gamma\).

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:

  1. Посылает каждой головке символ для записи в текущую ячейку каждой ленты;
  2. Посылает каждой головке одну из команд «LEFT»,"RIGHT","STAY";
  3. Выполняет переход в новое состояние (которое, впрочем, может совпадать с предыдущим).

Теперь то же самое более формально.

Машина Тьюринга это набор \(T= \langle k,\Sigma,\Gamma,\alpha,\beta,\gamma \rangle\), где

k
натуральное число,
\(\Sigma,\Gamma\)
конечные множества входного алфавита и состояний соответственно,
\(\star \in \Sigma\)
символ-пробел,
\(START,STOP \in \Gamma\)
выделенные состояния,
\(\alpha,\beta,\gamma\)
произвольные отображения:
  • \(\alpha: \Gamma \times \Sigma^k \rightarrow \Gamma \) (задает новое состояние);
  • \(\beta : \Gamma \times \Sigma^k \rightarrow \Sigma^k \) (задает символы для записи на ленты);
  • \(\gamma: \Gamma \times \Sigma^k \rightarrow \{-1,0,1\}^k \) (определяет, как двигать головки).

Удобно считать, что алфавит \(\Sigma\) содержит кроме «пробела» («*») два выделенных символа, «0» и «1» (Обычно вовсе ограничиваются \(\Sigma \equiv \{\star,0,1\}\)).

Под входом для МТ подразумевается набор из k слов (k-кортеж слов из \(\Sigma^*\)), записанных на k лентах начиная с нулевых позиций. Обычно, на входные данные записывают только на первую ленту, и под входом x подразумевают k-кортеж \(\langle x,0,\ldots,0 \rangle\).

{\it Результатом} работы МТ на некоем входе является также k-кортеж слов из \(\Sigma^*\), 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:


Примеры[править]

Если что-то осталось непонятным, можно рассмотреть симулятор работы МТ на языке Python, который мы будем использовать, чтобы проиллюстрировать процесс работы машин Тьюринга.

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


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:

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',  (('*','')))
 }
}

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 image creation requires permission to upload.

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»:

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style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="120px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="120px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="120px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="120px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="150.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="140px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="140px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="140px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect 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fontsize="10px" fill="black" text-anchor="middle" y="95px" > s4</text> <rect width="20px" height="18.0px" x="120px" y="80px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="95px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="115px" > s5</text> <rect width="20px" height="18.0px" x="100px" y="100px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="115px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > s1</text> <rect width="20px" height="18.0px" x="120px" y="120px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" 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Можно рассмотреть другие примеры МТ:

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