Recursion

Communication 1

Monday 25 November 2019 there will be the evaluation of the course

Please do not miss it!

Please, do it carefully and honestly, since it is one of the most important inputs I will have to understand what can be improved in the next year course

Communication 2

Clarification on test-driven development: all the tests must be passed in order to claim that an algorithm returns what it is expected

If a test execution return False, the test is not passed

If you need to check the non-compliancy of the execution of a function on purpose, then you have to create an additional testing function that returns True if the condition of the test is not passed

Remeber: first write the test, then develop the algorithm/function

Any question about the previous lecture?

Historic hero: Douglas Hofstadter

He is a cognitive scientist

Research on self-reference: situations where something refers to itself

Author of Gödel, Escher, Bach: An Eternal Golden Braid where he investigated in depth the concept of self-reference

In 1980, he was awarded of the Pulitzer's for that book

One of the main book themes: concept of intelligence (including artificial intelligence)

Little Harmonic Labyrinth

Achilles and the Tortoise's adventures in entering in the inconsistent composite world depicted by Escher's Convex and Concave

Entering into and exiting from a world via the pushing-potion and popping-tonic respectively

Achilles and the Tortoise narrate (or are part of) a lot of stories, which include citations and references as well as self-citations and self-references

The genies and the lamps

Self-references for paradoxes

Achilles' wish concerns the denial of the wish it self, which results in a paradox by means of a self-reference

Other notable examples of self-references:

• Natural language:
  this sentence is false

• Graphical language:
  Escher's Drawing hands

Recursion

We have a recursion when something is defined in terms of itself or of its type – i.e. when its definition contains a self-reference

Do we use it in our daily life?

<boolean_expression> ::= "(" "not" ​<boolean_expression> ")"
<boolean_expression> ::= "(" <boolean_expression> "or" <boolean_expression> ")"
<boolean_expression> ::= "(" <boolean_expression> "and" <boolean_expression> ")"
<boolean_expression> ::= "True"
<boolean_expression> ::= "False"

Chomsky argued that recusion is essential in human language: Alice thinks Bob said Christine read books

Mirrors and sounds

Recursion in Computer Science

It is an alternative to the iteration (i.e. foreach and while loops)

It is often used when a solution to a particular computational problem depends on the partial solutions of smaller instances of the same problem

Computer Scientists tame recursion to avoid infinite loops:

1. one or more base cases, which describe the terminating scenarios

2. one or more recursion steps, where the algorithm is run gain with a reduced input

What could possibly go wrong?

def run_forever_recursive():
    run_forever_recursive()

The run_forever algorithm implemented by means of a recursion step – with no base cases

Recursive algorithm: an example

Computational problem: multiply two non-negative integers

E.g.: 3 * 4 = 3 + 3 + 3 + 3

n1 * n2 = n1 + (n1 * (n2 - 1))

3 * 4 =
3 + (3 * (4 - 1)) =
3 + (3 + (3 * (4 - 2))) =
3 + (3 + (3 + (3 * (4 - 3)))) =
3 + (3 + (3 + (3 + (3 * (4 - 4))))) =
3 + 3 + 3 + 3 + 0 =
12

Multiplication: the algorithm

def multiplication(int_1, int_2):
    if int_2 == 0:
        return 0
    else:
        return int_1 + multiplication(int_1, int_2 - 1)