The Turing machine

Olena Lastivka

Definition Turing machine is a theoretical

device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, Turing machine can be adapted to simulate the logic of any computer algorithm.

Alan Turing

The "Turing" machine was described by Alan Turing in 1936, who called it an "automatic-machine". The Turing machine is not intended as a practical computing technology, but rather as a thought experiment representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.

History background In 1936 Alan Turing was

member of the Royal Army and worked in the British codebreaking centre, located at Bletchley Park in South East England. His major task during World War II was breaking the Enigma, a device that was used by the Germans, in order to encode military messages.

German Enigma

History background

A single message could be interpreted in more than 150 quadrillion ways (> 150,000,000,000,000 or 150 * 1012), therefore the British scientists made use of technical support. The “Turing Bombe”1, a cluster of four Turing machines, could successfully decode a secret message within several hours and thus was a huge advantage for the Royal British Army in their battle against Nazi Germany.

“Turing Bombe”

DescriptionThe Turing machine consists of three main

components:tape which is divided into cells, one next to the

other. Each cell contains a symbol from some finite alphabet. The alphabet contains a special blank symbol and one or more other symbols. Tape is endless in both directions.

head that can read data from or write data to the tape

application, which is a table of instructions that tell the machine what to do next.

State

State register stores the state of the Turing machine, one of finitely many. There is one special start state with which the state register is initialized. So Turing machine’s actions depends not only on the symbol, which the head reads from the tape but also on the state it is in.

These states, writes Turing, replace the "state of mind" a person performing computations would ordinarily be in.

Operations

Turing machine can do only 3 operations:

Replace current symbol on the tape with the other one (same symbol is also possible)

Move right or left for one cellChange the state (it is also possible to

keep the same state)

Processing

Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write 1; if the symbol seen is 1, shift to the right, and change into state 17; in state 17, if the symbol seen is 0, write 1 and change to state 6;" etc.

Simple example

It is a definition that the head initially points at the first element on our tape. The input in our example is number five, represented by a sequence of the digit “1”. The state is “state 1”.

Simple example In our example the Turing machine is in

“state 1” and points at an element that has the value “1”, so the value remains “1” and “state 1” is kept as well. The head now moves right and the next element is processed.

Simple example

As long as we are in “state 1” and the value on the tape does not change, the head will keep on moving right. Therefore we will skip the next couple of movements and have a look at the very end of our string.

Simple example As soon as the head reaches the blank

element at the end of our string, the previous rule does not fit anymore. Although we are still in “state 1”, the head now points at a blank element. Consequently this time the Turing machine will execute the second line of our source code:

Simple example The head writes a “1” onto the tape, but

the Turing machine remains in “state 1”. However the device now halts and the calculation is completed. Our final output will look like this:

So these two lines of pseudo-code enable the Turing machine to calculate the successor of a natural number. No matter whether our input is 4 or 4,000,000, our output will always be the next natural number (here: 5 or 4,000,001).

Church-Turing thesis

Any possible calculation can be performed by an algorithm, running on a computer that provides sufficient time and storage space.

Turing-complete

Inventions that are in accordance with the Church-Turing thesis, because they can solve every given mathematical expression, are called “Turing-complete”.

Turing-complete Although the Turing machine is rather

old, it is still present in a wide range of scientific areas, covering the sectors of information technology, mathematics, physics, biology or even chemistry (e.g. quantum computers). In most of these branches the term “Turing-complete” became very important and is for example used to classify programming languages, algorithms or technical devices today.

General idea of computations

Although most people know that “three times three” is nine, nearly none of them will think about what it actually means, namely “3 + 3 + 3”. However the Turing machine makes use of this technique and can even calculate more sophisticated terms by converting them back to our basic mathematical operations (here: addition).

Conclusion

Admittedly it is just pure theory, when we say that a Turing machine can solve every given mathematical problem. No question about that it is possible in theory, as it is a given, defined by the Church-Turing thesis, but in practice we will quickly realize that this idea is very limited.

Sources:http://en.wikipedia.org/wiki/Church–

Turing_thesishttp://en.wikipedia.org/wiki/Turing_machi

nehttp://en.wikipedia.org/wiki/Alan_Turing

by Olena Lastivka