Turing Machines

(チューリング機械)

12th lecture, July 5, 2019

Language Theory and Compilers

http://www.sw.it.aoyama.ac.jp/2019/Compiler/lecture12.html

Martin J. Dürst

© 2005-19 Martin J. Dürst 青山学院大学

Today's Schedule

• Summary of last lecture
• Example solution for bison homework
• Remaining schedule, final exam
• Turing Machines
  • How a Turing Machine works
  • Universal Turing Machine
  • Computability is everywhere

Summary of Previous Lecture

• Error processing in a parser has many requirements and difficulties
• In bison, errors can be caught with the error token
• The output of a parser includes the symbol table and an abstract parse tree as intermediate representations
• Semantic analysis is mainly concerned with type checking and type inference

Example Solution for bison Homework

(paper only)

Remaining Schedule

• July 5: Turing machines
• July 12: Code generation
• July 19: Optimization
• July 26: Executing environment: virtual machines, garbage collection,...
• August 2:, 11:10-12:35: Term final exam

Final Exam

• Past problems and example solutions (85'): 2018, 2017, 2016, 2015, 2014, 2013 (60'), 2012, 2011 (45'), 2010, 2009, 2008, 2007, 2006, 2005
• How to view example solutions:
  • Use buttons in upper left corner (blue area)
  • Use 's' and 'p' keys
• Solutions are only examples; other solutions may be possible; sometimes, solutions are missing
• Best way to use:
  
  • Simulate actual exam: Solve a full set of problems in 85'
  • Check solutions
  • Find weak areas and review content
  • Repeat
  • Concentrate on understanding, not rote learning

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 test: test whether a machine can 'think', 1950
• Turing Award: Most famous prize in Computer Science

Automata Commonalities

• (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: State Transition Table

Explanations for Example

• Function: Adding 1 to a binary number
• Tape symbols: 0, 1 (+blank)
• Three states:
  1. Add/carry
  2. Move back to right
  3. 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)

Using Turing Machines for Input Recognition

• Finite state automata and pushdown automata recognize input which is separate from overall machine state
• Turing machines can be used for input recognition
• Input is written in a particular area of the tape
• Input symbols and other symbols on the tape are not necessarily separate

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
• Build up functionality from primitives (e.g. plus 1 → addition → multiplication → exponentiation)

Extensions

• Nondeterminism
• Parallel tapes
• 2-dimensional tape
• Subroutines

It can be shown that all these extensions (and many more) can be simulated on a plain Turing machine.

Universal Turing Machine

• 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

Wolfram's 2,3 Turing Machine

• Turing machine with 2 states and 3 symbols
• Powerful (complicated) enough to simulate an Universal Turing Machine

See proof of universality and other details

Computability is Everywhere

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

• Wolfram's 2,3 Turing machine
• Lambda calculus (everything is a function; invented by Alonzo Church)
• Partial recursive functions
• SKI Combinator Calculus
• ι (iota) Calculus
• (Cyclic) tag systems
• Conway's game of life
• Wolfram's Rule 110 cellular automaton

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

All these mechanisms are very simple, but simulation is very slow.

Church-Turing Thesis

• Anything (any function on natural numbers) that can be calculated can be calculated by a Turing machine
• Anything that cannot be calculated by a Turing machine cannot be calculated

It is unclear whether this applies to Physics in general.

Turing Completeness

• A mechanism (or programming language,...) is called Turing complete if it can be shown to have computing power equivalent to a Turing machine
• Any serious programming language is Turing complete
• Future programming languages will not be substantially more powerful that current ones
• But future programming languages may be more convenient and/or faster

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 real 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
• To Mock a Mockingbird, Raymond M. Smullyan, Oxford University Press, 2000

Homework

Deadline: July 11, 2019 (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 (this Turing machine always starts on the rightmost non-blank symbol):

1. Draw the state transition diagram for this machine
2. Show in detail how this machine processes the input ..._1101000_...
3. Explain what kind of calculation this machine does if the tape contains only a single contiguous sequence of '0'es and '1'es (surrounded by blanks), and the leftmost non-blank symbol is a '1'

Glossary

rote learning

暗記

commonalities

共通点

nondeterminism

非決定性

universal turing machine

万能チューリング機械

mechanism

機構、仕組み

lambda calculus

ラムダ計算

partial recursive function

部分再帰関数

unary numbers

一進法 (1, 11, 111, 1111,...)

Entscheidungsproblem

ヒルベルトの決定問題

Turing complete(ness)

チューリング完全 (性)

computable number

計算可能数

halting problem

停止性問題