Computability

Communication 1

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Communication 2

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Any question about the previous lecture?

Historic hero: Alan Turing

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)

Turing machine

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

Table of instructions

Cost of computation

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

How to do it

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?

Lessons to learn

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

Important questions

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

Reductio ad absurdum

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

Library paradox

Setting

Problem

Resolution

The halting 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?

Resolution (part 1)

Suppose we have a def does_it_halt(an_algorithm) algorithm, which returns True if the execution of an_algorithm() terminates, False otherwise

def a_simple_algorithm(another_algorithm):
    if does_it_halt(another_algorithm):
        run_forever() 
    else:
        return

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

Resolution (part 2)

What happens if we try to execute a_simple_algorithm(a_simple_algorithm)?

Two situations:

1. if does_it_halt says that a_simple_algorithm stops (i.e. does_it_halt(a_simple_algorithm) returns True), then a_simple_algorithm run forever

2. if does_it_halt says that a_simple_algorithm runs forever (i.e. does_it_halt(a_simple_algorithm) returns False), then a_simple_algorithm stops

Contradiction

Whatever is the behaviour of a_simple_algorithm, it always generates a contradiction

While a_simple_algorithm can be developed (we provided the algorithm!), 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