Текст:Машина Тьюринга: Унарное сложение

Приведем пример машины Тьюринга, выполняющей унарное сложение двух операндов. Операнды представляются в унарной системе, т. к. целое неотрицательное число n представляется n+1 единицей.

Табличное представление МТ

<latex> \begin{tabular}{|cc|c|ccc|} \hline

<\textcolor{blue}{1pass}> & * & $\Rightarrow$ & 2pass & 1 & R \\ \hline
<\textcolor{blue}{1pass}> & 1 & $\Rightarrow$ & <\textcolor{blue}{1pass}> & 1 & R \\ \hline
2pass & * & $\Rightarrow$ & del1 & * & L \\ \hline
2pass & 1 & $\Rightarrow$ & 2pass & 1 & R \\ \hline
del1 & 1 & $\Rightarrow$ & del2 & * & L \\ \hline
del2 & 1 & $\Rightarrow$ & rewind & * & L \\ \hline
rewind & * & $\Rightarrow$ & [\textcolor{red}{q}] & * & R \\ \hline
rewind & 1 & $\Rightarrow$ & rewind & 1 & L \\ \hline

\end{tabular} </latex> <code-python> MT={

'k':1, 
'start': '1pass',
'stop': 'q',
'program': {
#(Состояние, символы на лентах) -> (новое состояние, (действия по каждой ленте))
 ('1pass', ('1')):         ('1pass', (('1','R'))), # проходим первое число
 ('1pass', ('*')):         ('2pass', (('1','R'))), # меняем разделитель
 ('2pass', ('1')):         ('2pass', (('1','R'))), # проходим второе число 
 ('2pass', ('*')):         ('del1',  (('*','L'))), # конец второго числа 
 ('del1',  ('1')):         ('del2',  (('*','L'))), # удаляем первую лишнюю 1
 ('del2',  ('1')):         ('rewind',(('*','L'))), # удаляем вторую лишнюю 1
 ('rewind',('1')):         ('rewind',(('1','L'))), # перематываем к началу. 
 ('rewind',('*')):         ('q',     (('*','R')))  # конец.
}

} </code-python>

Графовое представление МТ

Примеры выполнения МТ

«1» + «1»

<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="200px" height="200px" > <ellipse cx="50.0px" cy="30.0px" rx="28px" 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" x="100px" y="20px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="20px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="20px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="20px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="50.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="40px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="40px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="40px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="40px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="40px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="70.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="60px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="60px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="60px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="60px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="60px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="90.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="80px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="80px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="80px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="80px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="80px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="110.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="100px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="100px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="100px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="100px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="100px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="130.0px" rx="28px" ry="8px" 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;" /> <rect width="20px" height="18.0px" x="160px" 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 width="20px" height="18.0px" x="140px" y="140px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="140px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="170.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="160px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="160px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="160px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="160px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="160px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <ellipse cx="50.0px" cy="190.0px" rx="28px" ry="8px" style="fill:yellow;stroke:black;stroke-width:1" /> <g> <rect width="20px" height="18.0px" x="80px" y="180px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="100px" y="180px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="120px" y="180px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="140px" y="180px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="180px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1pass</text> <rect width="20px" height="18.0px" x="100px" y="20px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1</text> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="55px" > 1pass</text> <rect width="20px" height="18.0px" x="120px" y="40px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="55px" > 1</text> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="55px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="75px" > 2pass</text> <rect width="20px" height="18.0px" x="140px" y="60px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="75px" > 1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="75px" > 1</text> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="75px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="95px" > 2pass</text> <rect width="20px" height="18.0px" x="160px" y="80px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="95px" > 1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="95px" > 1</text> <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" > del1</text> <rect width="20px" height="18.0px" x="140px" y="100px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="115px" > 1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="115px" > 1</text> <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" > del2</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" text-anchor="middle" y="135px" > 1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="155px" > rewind</text> <rect width="20px" height="18.0px" x="100px" y="140px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="155px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="175px" > rewind</text> <rect width="20px" height="18.0px" x="80px" y="160px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="175px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="195px" > q</text> <rect width="20px" height="18.0px" x="100px" y="180px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="195px" > 1</text> </svg> </pic-svg>

«111» + «1»

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style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="160px" y="260px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="180px" y="260px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> <rect width="20px" height="18.0px" x="200px" y="260px" style="fill:none;stroke:#0a14e4;stroke-opacity:1.0000000;" /> </g> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1pass</text> <rect width="20px" height="18.0px" x="100px" y="20px" style="fill:none;stroke:darkblue;stroke-opacity:1.0000000;stroke-width:2px;" /> <text x="110.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1</text> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="35px" > 1</text> <text 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<text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > 2pass</text> <rect width="20px" height="18.0px" x="200px" 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" text-anchor="middle" y="135px" > 1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > 1</text> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > 1</text> <text x="170.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > 1</text> <text x="190.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="135px" > 1</text> <text x="50.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="155px" > del1</text> <rect width="20px" height="18.0px" x="180px" y="140px" 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1</text> <text x="130.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="275px" > 1</text> <text x="150.0px" font-family="Tahoma" fontsize="10px" fill="black" text-anchor="middle" y="275px" > 1</text> </svg> </pic-svg>

«11» + «111»

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По крайней мере часть этого текста взята с ресурса http://lib.custis.ru/ под лицензией GDFL.Список авторов доступен на этом ресурсе в статье под тем же названием.