言語理論とコンパイラ

第十二回: Turing Machine

2015 年 6 月 26 日

http://www.sw.it.aoyama.ac.jp/2014/Compiler/lecture12.html

Martin J. Dürst

© 2005-15 Martin J. Dürst 青山学院大学

今日の予定

• 前回のまとめ
• bison の宿題の正解例
• Turing Machine

前回のまとめ

• 演算子の優先度と結合規則が文法の書き換え規則の形で決定
• 文法の書き換え規則ではパターンに注意
• 構文解析のエラー処理では要点が多く、対応が困難
• bison では error トークンを含む規則でエラーに対応可能

bison の宿題の正解例

(紙にて配布)

Formal Language Hierarchy

(Chomsky hierarchy)

Historic Background

• Alan Turing, British Mathematician, 1912-1954
• "On computable numbers, with an application to the Entscheidungsproblem" published in 1936
• Turing worked on en/decription during World War II
• Turing Prize: Most famous prize in Computer Science

Automata Commonalities

(共通点)

• (finite set of) states
• (finite set of) transitions between states
• Start state
• Accepting state(s)
• Deterministic or nondeterministic

Automata Differences

• Finite state automaton: No memory
• Pushdown automaton: Memory stack
• (Linear bounded automaton: Finite-length tape)
• Turing machine: Infinite length tape
  (can be simulated with two stacks)

How a Turing Machine Works

• 'Infinite' tape with symbols
• Special 'blank' symbol (␣ or _),
  used outside actual work area
• Start at first non-blank symbol from the right
  (or some other convenient position)
• Read/write head:
  • Reads a tape symbol from present position
  • Decides on symbol to write and next state
  • Writes symbol at present position
  • Moves to the right (R) or to the left (L)
  • Changes to new state
• Start state and accept state(s)

Turing Machine Example

• Adding 1 to a binary number
• Tape symbols: 0, 1 (+blank)
• Three states:
  • Adding/carry
  • Move to left
  • Accept

Turing Machine Definition

6-tuple:

• Finite, non-empty set of states Q
• Finite, non-empty set of tape symbols Σ
• Transition function
• Blank symbol (∈Σ)
• Initial state (∈Q)
• Set of final states (⊂Q)

Techniques and Tricks for Programming

• Interleave data fields and control fields
• Use special symbols as markers
• Use special states to move across tape to different locations

Extensions

• Nondeterminism (非決定性)
• Parallel tapes (平行なテープ)
• 2-dimensional tape (2次元テープ)
• Subroutines

It can be shown that all these extensions can be simulated on a plain Touring machine

Universal Turing Machine

(万能チューリング機器)

• It is possible to design a Turing machine that can simulate any Turing machine (even itself)
• Encode states (e.g. as binary or unary numbers)
• Encode tape symbols (e.g. as binary or unary numbers)
• Create different sections on tape for:
  • Data (encoded tape symbols)
  • State transition table (program)
  • Internal state
• Main problems:
  • Construction is tedious
  • Execution is very slow

Computability is Everywhere

It turns out that there are many other mechanisms that can simulate an (universal) Turing machine:

• Lambda calculus (everything is a function)
• Partial recursive functions
• SKI Combinator Calculus
• ι (iota) Calculus
• (Cyclic) tag systems
• Conway's game of life
• Wolfram's Rule 110 cellular automaton
• Wolfram's 2,3 Turing machine
  (Turing machine with only 2 states and three symbols)

Other Contributions

• Von Neumann style architecture: Current computer architecture closely follows Turing machine
  (main difference: Random Acccess Memory)
• Entscheidungsproblem: There are some Mathematical facts that cannot be proven
• Computable numbers: There are rationalreal numbers that cannot be computed
• Halting problem: There is no general way to decide whether a program will terminate (halt) or not

Bibliography

• The Annotated Turing, Charles Petzold, Wiley, 2008
• Understanding Computation, Tom Stuart, O'Reilly, 2013
  (also available in Japanese)
• A New Kind of Science, Stephen Wolfram, Wolfram Media, 2002

宿題

提出: 来週の木曜日 (7 月 2日) 19 時 00 分、O 棟 529 号室の前

あるチューリング機器の遷移表:

1. この機器の遷移表を書きなさい
2. この機器の ..._1100100_... に対する動作を示しなさい
3. テープの記号が 0, 1, _ のみでスタートする場合、どのような計算をするのか推測し、説明しなさい