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| − | 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 <m>\Sigma</m>, 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 <m>\Gamma</m>.
| + | |
| − | | + | |
| − | 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
| + | |
| − | <m>T= \langle k,\Sigma,\Gamma,\alpha,\beta,\gamma \rangle</m>,
| + | |
| − | где
| + | |
| − | | + | |
| − | ;k: натуральное число,
| + | |
| − | ;<m>\Sigma,\Gamma</m>: конечные множества входного алфавита и состояний соответственно,
| + | |
| − | ;<m>\star \in \Sigma</m>: символ-пробел,
| + | |
| − | ;<m>START,STOP \in \Gamma</m>: выделенные состояния,
| + | |
| − | ;<m>\alpha,\beta,\gamma</m>: произвольные отображения:
| + | |
| − | :*<m>\alpha: \Gamma \times \Sigma^k \rightarrow \Gamma </m> (задает новое состояние);
| + | |
| − | :*<m>\beta : \Gamma \times \Sigma^k \rightarrow \Sigma^k </m> (задает символы для записи на ленты);
| + | |
| − | :*<m>\gamma: \Gamma \times \Sigma^k \rightarrow \{-1,0,1\}^k </m> (определяет, как двигать головки).
| + | |
| − | | + | |
| − | Удобно считать, что алфавит <m>\Sigma</m> содержит кроме «пробела» («*»)
| + | |
| − | два выделенных символа, «0» и «1» (Обычно вовсе ограничиваются <m>\Sigma \equiv \{\star,0,1\}</m>).
| + | |
| − | | + | |
| − | Под ''входом'' для МТ подразумевается набор из ''k'' слов (''k''-кортеж слов из <m>\Sigma^*</m>),
| + | |
| − | записанных на ''k'' лентах начиная с нулевых позиций.
| + | |
| − | Обычно, на входные данные записывают только на первую ленту, и под входом ''x''
| + | |
| − | подразумевают ''k''-кортеж <m>\langle x,0,\ldots,0 \rangle</m>.
| + | |
| − | | + | |
| − | {\it Результатом} работы МТ на некоем входе является также
| + | |
| − | ''k''-кортеж слов из <m>\Sigma^*</m>, 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'))),
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| − | ('s2', ('1')): ('s2', (('1','R'))),
| + | |
| − | ('s2', ('*')): ('s3', (('*','R'))),
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| − | ('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»:
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| − | | + | |
| − | {|
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| − | |
| + | |
| − | <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"
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| − | version="1.0"
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| − | width="180px"
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| − | height="160px"
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| − | >
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| − | <ellipse
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| − | cx="50.0px"
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| − | cy="30.0px"
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| − | rx="28px"
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| − | ry="8px"
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| − | 1</text>
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| − | <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
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| − | <rect
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| − | style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;"
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| − | />
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| − | <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
| + | |
| − | 1</text>
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| − | <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
| + | |
| − | 1</text>
| + | |
| − | <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
| + | |
| − | 1</text>
| + | |
| − | <text x="190.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
| + | |
| − | 1</text>
| + | |
| − | <text x="210.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
| + | |
| − | 1</text>
| + | |
| − | <text x="230.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="595px" >
| + | |
| − | 1</text>
| + | |
| − | </svg>
| + | |
| − | </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}}
| + | |
