Машина Тьюринга

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 Σ \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 Γ \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 = k , Σ , Γ , α , β , γ T= \langle k,\Sigma,\Gamma,\alpha,\beta,\gamma \rangle , где

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

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

Под входом для МТ подразумевается набор из k слов (k-кортеж слов из Σ \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 x , 0 , , 0 \langle x,0,\ldots,0 \rangle .

{\it Результатом} работы МТ на некоем входе является также k-кортеж слов из Σ \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:


ПримерыПравить

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», «_» ):

G s1 s1 q q s1->q * * s2 s2 s1->s2 *R 1 s5 s5 s5->s1 1R * s5->s5 1L 1 s2->s2 1R 1 s3 s3 s2->s3 *R * s3->s3 1R 1 s4 s4 s3->s4 1L * s4->s5 *L * s4->s4 1L 1

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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