Computer scientist (+ works in mathematics, logic, philosophy, and biology)

Father of Theoretical Computer Science (see Turing machine) and of Artificial Intelligence (see Turing test)

Crucial contribution in deciphering the German Nazy Enigma machine during the Second World War - he was appointed an Officer of the Order of the British Empire (OBE)

It is a theoretical machine

Turing machine can be used to simulate any algorithm

If you can define an algorithm for solving a particular computational problem, then there exists a Turing machine that can solve the same problem as well, and vice versa

It is composed by:

an infinite memory tape containing cells - each cell can contain 0 (blank symbol) or 1

an head for reading/writing/moving on the tape

a storage for recording the current state

a finite table of instructions + start and end states

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

Start state: A

No end states

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

Current state | Tape symbol | Write symbol | Move head | Next state |
---|---|---|---|---|

A |
0 or 1 |
1 |
right |
B |

B |
0 or 1 |
0 |
right |
A |

The previous simple machine can compute indefinitely (there are no ending states)

Other machines could compute a result in a reasonable finite time

There can be also algorithms that must spend an exaggerated time (even if still finite) to return a result

Execution cost of an algorithm: how much time, indicatively, an algorithm needs for returning a result

Computational complexity theory: classifying computational problems

Computational problem: a problem that be solved algorithmically by a computer

Classification to an algorithm is provided according to a specific hierarchy of classes that express the difficulty in solving such problems

Analysis of algorithms: understanding the amount of time, storage and other resources that are needed to execute such algorithm, defined in terms of a mathematical function

Typical question: how many instructions an algorithm has to execute to return the final result from that input?

The efficiency of an algorithm is directly derived and guided by the way such algorithm has been developed

It is possible to develop two different algorithms addressing the same computational problem that take two drastic different times for returning the result

Can we use algorithms for computing whatever we want?

Alternatively, there exists a limitation on what we can compute?

Alternatively, is it possible to define a computational problem that cannot be solved by any algorithm?

Turing created his machine for answering this question

One of the most used approaches to demonstrate that something cannot exist

Idea: to come to a paradoxical and self-contradictory situation in which, for instance, the existence of an algorithm contradicts its existence itself

Setting

Problem

Proposed by David Hilbert in 1900

Description: is it possible to develop an algorithm that takes another algorithm as input and determines if such input algorithm will terminate its execution at some point or it will run forever?

From Turing point of view: can we develop a Turing machine which is able to certainly decide if another machine will terminate its execution or will not?

Suppose we have a does it halt

algorithm which takes in input another algorithm and returns *yes* if the input algorithm terminates, and *no* otherwise

Reuse the does it halt

algorithm to develop a new algorithm

Run forever

exists since we can develop a Turing machine that does not terminate

What happens if we try to execute the last algorithm passing itself as input?

Two situations:

if

does it halt

says that the last algorithm stops, then such last algorithm runs foreverif

does it halt

says that the last algorithm runs forever, then such last algorithm stops

Whatever is the behaviour of the last algorithm, it always generates a contradiction

While the last algorithm can be developed (all the widgets in the diagram can be actually developed), then it is does it halt

that cannot be developed

Turing's machines and their analyses posed clear limits to what we can compute, since there are specific computational problems that cannot be solved