Intro to computer science module 4a

Foundations of Computer Science and Software Engineering

module 4a/30

WALID S. SABA


ALBERT EINSTIEN

TO THE TEACHER


TO THE STUDENT

SOCRATES


About this CourseThis is an introductory course that will introduce computer science and software engineering to the novice, and it is also a course that will teach even veteran computer scientists some new concepts, or explain concepts they thought they understood well. Concepts are introduced with hands on experience using object-oriented pseudo code (which can very easily be translated to C#, Java, C++, etc.) and functional programming pseudo code (which can easily be translated into F#, Erlang, Python, Haskell, etc.)

About the AuthorWalid Saba has 20 years of experience in information technology, where he worked at such places as the American Institutes for Research, AT&T Bell Labs, Metlife, Nortel Networks, IBM and Cognos. He has also spent 7 years in academia where he has taught computer science at Carelton University, the New Jersey Institute of Technology, the University of Windsor, and the American University of Beirut. He has published over 30 technical articles, including an award wining paper that he recieved at KI-2008 in Germany. Walid holds a PhD in Computer Science which he obtained from Carleton University in1999

MODULE 4 (1 hour)


MATHEMATICAL PRILIMENARIES – RECURSION AND INDUCTIVE STRUCTURES (part 1)

Introduction to Computer Science – Preliminaries

Infinite Objects

In most programming languages you can define an integer variable and assign it an initial value in a statement like this:

int n = 31789;


Infinite Objects


int n = 31789;

In other programming languages we might define a function that expects an integer value as input:

square n = n * n


Infinite Objects


int n = 31789;

square n = n * n

How does the compiler of the programming language check if the assigned value or the value passed as input is a valid integer?

In other programming languages we might define a function that expects an integer value as input:


Infinite Objects

IsAValidInteger

Input string

true/false

Somewhere, there must be some truth-valued (or Boolean) function, say IsAValidInteger, that takes a string and tests whether that string is a valid integer


Infinite Objects

IsAValidInteger

Input string

true/false

bool IsAValidInteger(string input){ if (input == "0") return true; if (input == "1") return true; if (input == "2") return true; ... if (input == "1732") return true; ...}

// this function might look like this

But what would the logic in the function IsAValidInteger look like?


Infinite Objects

IsAValidInteger

Input string

true/false

bool IsAValidInteger(string input){ if (input == "0") return true; if (input == "1") return true; if (input == "2") return true; ... if (input == "1732") return true; ...}

// this function might look like this

But what would the logic in the function IsAValidInteger look like?

But how do we check all possible cases? Where do we stop?

Recursion and Infinite Objects


It turns out that the only way to have a finite definition of an infinite object is by recursion, the easiest and one of the most important concepts in computer science

Let us define the function that will take any string whatsoever and check whether it is a valid number or not ...


First, we need to define a function that checks if the input is a valid digit:

Recursion


RecursionFirst, we need to define a function that checks if the input is a valid digit:


Recursion

A single character is a valid digit if it is 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9



Recursion

Example of how this predicate is applied




Another example of how this predicate is applied



What if the input is not a single character?

Modern programming languages are what we call strongly-typed – meaning the input to the function must be of a specific type. Thus, this function will not even accept another type of input, and thus those situations will not even be considered, and thus they can never return true!

We will discuss this more later in the course.

Recursion


Now we can define the recursive function number as follows:


Recursion


An empty sequence of characters is not a valid number



Recursion


A sequence of a single character is a valid number if that character is a valid digit



Recursion


A sequence of n characters is a valid number if the first character is a valid digit and the rest of the sequence is in turn a valid number



Recursion


How is the function number recursive?

Recursion


How is the function number recursive?

number is a function that is defined in terms of itself


Here’s how number(“3047”) comes out to be true





















Recursion


What would you have to change to make this function accept any sequence of binary digits (that is, any sequence of 1’s and 0’s)?

Recursion


What would you have to change to make this function accept any sequence of binary digits (that is, any sequence of 1’s and 0’s)?

Recursion


Let the symbol '::=' stand for “is made up of” and the symbol ':' stand for “attached to”

Recursion


STRINGS


A string such as “page” is constructed as follows

Recursion


STRINGS



Recursion


The string “page”

STRINGS



Recursion


is made up of

STRINGS



Recursion


the character ‘p’

STRINGS



Recursion


attached to

STRINGS



Recursion


the string “age”

STRINGS



Recursion


LISTS


A list such as [3,7,10,22] is constructed as follows

Recursion



The list [3,7,10,22]

LISTS


Recursion



is made up of

LISTS


Recursion



the object 3

LISTS


Recursion



attached to

LISTS


Recursion



the list [7,10,22]

LISTS


Recursion


Recursion must at some point end – there must be a base case

Recursion



Recursion



Recursion



Recursion



Recursion



Recursion



The base case

Recursion



Recursion



Recursion



Recursion



Recursion



Recursion



The base case

Recursion


As we will see later in the course, trees are data structures that have important applications in computer science

Recursion



One special type of a tree is the binary tree, which is either empty (we say it is a null tree), or if not empty it is a tree that has a node with left and right (sub)trees

Recursion



22

= =One special type of a tree is the binary tree, which is either empty (we say it is a null tree), or if not empty it is a tree that has a node with left and right (sub)trees

Recursion



22

31

=

=


=

Recursion



22

31

25

==

=

=One special type of a tree is the binary tree, which is either empty (we say it is a null tree), or if not empty it is a tree that has a node with left and right (sub)trees

Recursion



22

18 31

25

=

==

==


Recursion



22

18 31

16 25

=

= ===

=


Recursion



22

18 31

16 25 43

=

= = ====


Recursion



22

18 31

16 25 43

=

= = ====


This is the root node of the right subtree

Recursion


All recursive objects have a base case and a construction case

Recursion


A string is either empty or it is a character attached to another string


Recursion



A list is either an empty list or it is some element attached to another list


Recursion




A number is either 0 or it is 1 plus some other number


Recursion




A number is either 0 or it is 1 plus some other number

A binary tree is either null or it is a node with left and right trees


The List Data Structure


In a class of programming languages – known as functional programming languages a list matches any of two patterns:



The list could be empty




The list could have at least one object e followed by the rest of the list es




The first pattern matches only the empty list. The second pattern matches actual lists as follows:















One way to visualize how the list data structure is implemented is to see it as a collection of nodes that are pointing to each other





17 0 22 1




17 0 22 1




17 0 22 1




17 0 22 1




17 0 22 1


The list data structure can hold objects of any type, including lists





In most pure functional programming languages the objects in the list must be of the same type





A list of numbers





A list of pairs where each pair is a character and a number





A list of strings





A list of ‘lists of numbers’





A list of triples, where each triple has a number, a list of strings and a character


When writing functions (programs) that take these recursive structures as inputs, it is natural to traverse these structures the way they were created, namely, recursively.

Recursive programs – an introduction







Since there are two distinct possibilities of the input, namely an empty list, or a list that has at least one element, we must consider these two possibilities:



Since there are two distinct possibilities of the input, namely an empty list, or a list that has at least one element, we must consider these two possibilities:



Let us introduce a method by which recursive programs can be defined almost automatically





Let us start with the function length that computes the size of a list





The function length takes a list so we must consider the two possible input cases





What is a reasonable value to return as the size when the list is empty?





0 sounds a reasonable value to return as the size of an empty list





What else should we do if we trust that the function length applied recursively on the rest of the list does work and returns the length of the rest?





If length applied on es returns the size of the rest, then the size of the entire list is just 1 more


How does this function work?




















Done! Start evaluating the final result














Using the same reasoning as above, can you complete the definition of the function sum that takes a list of numbers and returns the sum of all the numbers in the list?



Using the same reasoning as above, can you complete the definition of the function sum that takes a list of numbers and returns the sum of all the numbers in the list?


In the next module we will start writing some interesting functions and programs over a number of data structures

Introduction to Computer Science

END OF MODULE 4a

MATHEMATICAL PRILIMENARIES – RECURSION AND INDUCTIVE STRUCTURES (part b)

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Education

Intro to computer science module 4a