Turing Machines

(チューリング機械)

12rd lecture, June 23, 2017

Language Theory and Compilers

http://www.sw.it.aoyama.ac.jp/2017/Compiler/lecture12.html

Martin J. Dürst

AGU

© 2005-17 Martin J. Dürst 青山学院大学

Today's Schedule

 

Summary of Previous Lecture

 

Example Solution for bison Homework

(paper only)

 

Formal Language Hierarchy

(Chomsky hierarchy)

Grammar Type Language (family) Automaton
phrase structure grammar (psg) 0 phrase structure language Turing machine
context-sensitive grammar (csg) 1 context-sensitive language linear bounded automaton
context-free grammar (cfg) 2 context-free language pushdown automaton
regular grammar (rg) 3 regular language finite state automaton

 

Historic Background

 

Automata Commonalities

 

Automata Differences

 

How a Turing Machine Works

 

Turing Machine Example

 

State Transition Table for Example

Current state Current tape symbol New tape symbol Movement direction Next state
→1 0 1 L 2
→1 1 0 L 1
→1 _ _ R 3*
2 0 0 L 2
2 1 1 L 2
2 _ _ R 3*

 

Turing Machine Definition

6-tuple:

 

Techniques and Tricks for Programming

 

Extensions

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

 

Universal Turing Machine

 

Computability is Everywhere

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

All these mechanisms can simulate each other and have the same power.

A mechanism (or programming language,...) is called Turing-complete if it can be shown to have computing power equivalent to a Turing machine

 

Church-Turing Thesis

It is unclear whether this applies to Physics in general.

 

Other Contributions

 

Bibliography

 

Homework

Deadline: July 6, 2017 (Thursday), 19:00

Where to submit: Box in front of room O-529 (building O, 5th floor)

Format: A4 single page (using both sides is okay; NO cover page, staple in top left corner if more than one page is necessary), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

For the Turing machine given by the following state transition table:

Current state Current tape symbol New tape symbol Movement direction Next state
→1 0 1 L 1
→1 1 0 L 2
→1 _ _ L 4*
2 0 0 L 2
2 1 1 L 2
2 _ _ R 3
3 0 _ R 3
3 1 1 L 4*
  1. Draw the state transition diagram for this machine
  2. Show in detail how this machine processes the input ..._1100100_...
  3. Guess and explain what kind of calculation this machine does if the tape contains only a single contiguous sequence of '0'es and '1'es with at least one '1'(surrounded by blanks)

(this Turing machine always starts on the rightmost non-blank symbol)

  

Glossary

commonalities
共通点
nondeterminism
非決定性
universal turing machine
万能チューリング機器