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Turing Machines. (At last!). Designing Universal Computational Devices Was Not The Only Contribution from Alan Turing…. Enter the year 1940:. The world is at war Nazi Germany has succeeded in conquering most of west Europe Britain is under siege British supply lines are threaten by German

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turing machines

Turing Machines

(At last!)

designing universal computational devices was not the only contribution from alan turing
Designing Universal Computational Devices Was Not The Only Contribution from Alan Turing…

Enter the year 1940:

  • The world is at war
  • Nazi Germany has succeeded in conquering most of west Europe
  • Britain is under siege
  • British supply lines are threaten by German
  • Germany used the Enigma Code, considered unbreakable
  • Alan Turing led a group of scientist that broke the enigma code

Enter the year 2006: The National Security Agency (NSA) needs people! (www.nsa.gov)

background

Pushdown Automata accepts context-free languages only

For example, {anbncn : w  *} is not context-free

Background
  • Automata accepts regular languages only

For example, {anbn : n = 0, 1, …} is not regular

  • We can easily write an algorithm (in C) recognizing if a sequence of characters have the form anbncn or not
the key question
The Key Question
  • Is it possible to design a formal model of a computational device that capture the capabilities of any algorithm?

Alan Turing, 1940’s: Yes!

basic idea
Basic Idea

Turing decomposed operations in the device as follows:

  • A pencil will be “observing” symbols from a paper
  • The pencil can move left and right
  • Computational steps erase a symbol observed by the pencil and write a new one in its place
  • The decision about which symbol should be written and which will be observed next depend on:
  • The current symbol being observed
  • The “state of mind” of the device

This device is called a Turing Machine

turing machine looks similar to finite automata
Turing Machine Looks Similar to Finite Automata
  • Common:
    • A “state of mind” sounds like the collection of states
    • A “pencil” sounds like the pointer for the next word

Main differences:

  • Turing machines can write on the “paper”
  • Turing machines can backup and read symbols again

This turns out to be a major difference!

in turing s own words
In Turing’s Own Words

... which can be made to do the work of any special-purpose machine, that is to say to carry out any piece of computing, if a tape bearing suitable "instructions" is inserted into it.

expressivity of turing machines tms
Expressivity of Turing Machines (TMs)
  • TMs can simulate any data structure
  • TMs can simulate major components of imperative languages: sequence, branching and loop
  • TMs can control branching and loops
  • Several extensions have been attempted: adding multiple heads (pencils), tapes, non-determinism, etc. But we can show that TMs can simulate each of these extensions
church turing thesis
Church-Turing Thesis

Every computer algorithm can be implemented as a Turing machine

Therefore, C, C++, Prolog, Lisp, Small talk, and Java programs can be simulated in Turing machines

Definition: a programming language is Turing-complete if it is equivalent to a Turing machine.

informal description

Control

head

Tape

cell

Informal Description

a1

a2

  • The head:
    • Reads the symbol from the cell it is pointing to, and
    • Either:
      • Writes a new symbol in the cell, or
      • Moves one cell to the left or right
informal description ii
Informal Description (II)
  • New cells can be added to the right of the tape as needed (similar to RAM memory)
  • These new cells contain the blank symbol, 
  • The tape is bounded to the left
preliminaries to formal description

Transitions can be described by (Case I):

((s,a),(q,b,))

If the machine is in state s and the current cell has an a then jump to state q and write b in the current cell and moves head to right

Preliminaries to Formal Description
  •  denotes the alphabet of characters in language as usual.
  •  denotes the set of symbols that can be written on tape
    • It contains  and all symbols in 
preliminaries to formal description ii

Transitions can be described by (Case II):

((s,a),(q,b,))

If the machine is in state s and the current cell has an a then jump to state q and write b in the current cell and moves head to left

Preliminaries to Formal Description (II)
formal definition
Formal Definition

Definition. A Turing machine is a 7-tuple (Q, , , , q0, qaccept, qreject),

where:

  • Q is a set of states
  •  is a set of symbols (the alphabet)
  •  is a set of symbols that can be written in tape,    and   
  • q0 Q is the initial state
  • qaccept is the accepting state
  • qreject is the rejecting state, qreject  qaccept
formal definition ii
Formal Definition (II)
  •  is a collection of transitions defined by the function:
  • : (Q  {qaccept, qreject})    Q    {, }
my first turing machine
My First Turing Machine
  • Construct a Turing machine such that:
    • input: a substring over the alphabet  ={a,b}, the substring contains at least 1 character b
    • the header is pointing to the first cell in tape
    • When the Turing machine halts, the header must be pointing to the first occurrence of b in the input

a

b

a

a

b

a

slide17
Discussion: Are you sure that such a simple model can simulate my C++ program for computing the Minimum Spanning Tree?
the eraser tm
The “Eraser” TM

Construct a Turing machine that receives as input a substring of a’s and replace each a for a blank space, 

a

a

a

configuration for turing machines

(q,h,LS,RS)

  • The current state, q
  • The symbol pointed by the head, h
  • The string left of the head, LS
  • The string right of the head, RS
Configuration for Turing Machines

Configuration:

configuration for turing machines 2
Configuration for Turing Machines (2)

Instead of writing: (q,h,LS,RS), we write:LSqhRS

If q = qaccept, then LSqhRS is called an accepting configuration

If q = qreject, then LSqhRS is called a rejecting configuration

Examples:

- Initial configuration with 3 a’s for the eraser TM:

eq0aaa

- Accepting configuration starting with eq0aaa:

qaccept

computation in turing machines

C yields C’, written C * C’:

There is a sequence of configurations in one step:

C C0 C1 … C’

Computation in Turing Machines
  • A configuration C1 = LS1q1h1RS1yields a configuration C2 = LS2h2q2RS2 in one step, written C1 C2, if:

There is a transition: (q1, h1) = (q2,h2,)

  • The other case is analogous for (q1, h1) = (q2,h2,)
  • Example: eq0aaa yieldseq0
two forms of computability
Two Forms of Computability
  • Definition. A word w is accepted by a Turing machine M if M yields an accepting configuration starting from the configuration eq0w
  • Definition. The language L(M) recognized by a Turing machine M:
  • L(M) = {w | w accepted by M}
  • Definition. A language L is Turing-recognizable if there is a Turing machine that recognizes L
  • Definition. A language L is decidable if there is a Turing machine M such that:
    • L(M) = L
    • For any word w L, M yields a rejecting configuration starting from the configuration eq0w
constructing complex turing machines

Example (programming):

Given two nn matrixes A and B to compute ABT we:

  • Construct a procedure to compute BT, transverse(B,C)
  • Construct a procedure to multiply two matrices, multiply(B,C,D)
  • Call transverse(B,C) and then multiply(A,C,D). D contains the answer
Constructing Complex Turing Machines

Idea:

Use a modular approach similar to the one use to construct complex programs

graphical conventions

Convention. Let A and B be 2 TMs, and a be a character in  then:

a

A

B

Means that:

  • If A halts in a configuration of the form w1qacceptaw2 wherew1or w2 are any words
  • Then B starts with the configuration w1q0aw2
Graphical Conventions

We will use a convention similar to a flow chart to indicate how to put together Turing machines (TMs).

A;

if a then B

graphical conventions 2

A

B

M

Means that:

  • If M halts in a configuration of the form w1qacceptaw2, then A starts with the configuration w1q0aw2
  • If M halts in a configuration of the form w1 qacceptw2 with   a, then B starts with the configuration w1q0w2
Graphical Conventions (2)

a

a

M;

If a then A

else B

graphical conventions 3

A

Means that:

  • If the initial configuration has the form w1q0w2, then A starts with this configuration if   a, where s is the initial state in A
  • If A halts in a configuration of the form w1qacceptw2 with   a, then A starts with the configuration w1q0w2
Graphical Conventions (3)

a

while (not a) do

A

a

basic turing machines

Denotes “write a on the current cell” (a is a character in )

Denotes “move to the right of the current cell” (the book uses “R” instead of “”)

Denotes “move to the left of the current cell” (the book uses “L” instead of “”)

a

Basic Turing Machines
other simple turing machines

b

a

Other Simple Turing Machines

What it does

Machine

a

“if current cell has an a,then replace it with a b”

“if current cell has a character a different than , then move once to the right and write that a”

the turing machine

The  TuringMachine

  • What it does?

Move to the right until if finds 

the turing machine30

The  TuringMachine

Draw diagram for 

slide31
 and 

Move to the left until the first non-blank space cell is found

Move to the right until the first non-blank space cell is found

the conventions

R

R

“move to the right until you find “

“move to the right until you find a symbol other than “

The Conventions
  • 2 simple rules:
  • Rule # 1:
  • Rule # 2:

Note: first check. Then move (think of a “while”)

“Never forget Rule # 1”

(Mr Miogi, The Karate Kid)

example deciding the language ab
Example: Deciding the language (ab)*

“yes”

a

b

a

b

a

b

a

a

“no”