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•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

•Automata accepts regular languages only

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

•Pushdown Automata accepts context-free languages only

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

•We can easily write an algorithm (in C) recognizing if a sequence of characters have the form anbncn or not

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

Turing decomposed operations in the device as follows:

•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:

•A pencil will be “observing” symbols from a paper

1. The current symbol being observed2. The “state of mind” of the device

This device is called a Turing Machine

•The pencil can move left and right

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

... 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)

•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

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

…a1 a2

Control

Tape

cell

head

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)

•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

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

• 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 (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

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)

is a collection of transitions defined by the function:

: (Q {qaccept, qreject }) Q {, }

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

Discussion: Are you sure that such a simple model can simulate

my C++ program for computing the Minimum Spanning Tree?

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

Configuration:

•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

(q,h,LS,RS)

Configuration for Turing Machines (2)

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

Examples:

- Initial configuration with 3 a’s for the eraser TM: eq0aaa

- Accepting configuration starting with eq0aaa: qaccept

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

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

Computation in Turing Machines

•A configuration C1 = LS1q1h1RS1 yields a configuration C2

= LS2h2q2RS2 in one step, written C1 C2, if:

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

C yields C’, written C * C’:

There is a sequence of configurations in one step:

C C0 C1 … C’

• The other case is analogous for (q1, h1) = (q2,h2,)

• Example: eq0aaa yields eq0

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) = LFor any word w L, M yields a rejecting configuration starting from the configuration eq0w

Constructing Complex Turing Machines

Idea: Use a modular approach similar to the one use to

construct complex programs

Example (programming): Given two nn matrixes A and B to compute ABT we:

1. Construct a procedure to compute BT, transverse(B,C)

2. Construct a procedure to multiply two matrices, multiply(B,C,D)

3. Call transverse(B,C) and then multiply(A,C,D). D contains the answer

Graphical Conventions

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

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

Aa

B

Means that:

1. If A halts in a configuration of the form w1qacceptaw2

where w1or w2 are any words

2. Then B starts with the configuration w1q0aw2

A;if a then B

Graphical Conventions (2)

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 qacceptw2

with a, then B starts with the configuration w1q0w2

A

a

B

Ma M;

If a then Aelse B

Graphical Conventions (3)

Means that:

• If the initial configuration has the form w1q0w2,

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 w1qacceptw2

with a, then A starts with the configuration w1q0w2

Aa while (not a) do

Aa

Basic Turing Machines

a 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 “”)

Other Simple Turing Machines

ba

Machine What it does

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

a “if current cell has a

character a different than , then move once to the right and write that a”

The Turing Machine

•What it does?

––

Move to the right until if finds

The �Turing Machine

–

–

Draw diagram for �

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• 2 simple rules:

• Rule # 1:

• Rule # 2:

R“move to the right until you find “

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

“Never forget Rule # 1”

(Mr Miogi, The Karate Kid)

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

Example: Deciding the language (ab)*

a b a b

a b a a

“yes”

“no”