intro to computer science module 2

Foundations of Computer Science and Software Engineering

module 2/30

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 2 (1 hour)

MATHEMATICAL PRILIMENARIES – LOGIC


Introduction to Computer Science – Preliminaries


Huh?

Propositional Logic


Logic is a system of reasoning to systematically determine the soundness (or validity) of an argument

Propositional Logic


The simplest logical system is that of “propositional logic” – which is roughly equivalent to what is also known in computer science and computer engineering as “Boolean algebra”


Propositional Logic



The simplest logical system is that of “propositional logic” – which is roughly equivalent to what is also known in computer science and computer engineering as “Boolean algebra”

Sentences (or declarative statements) that express or declare some proposition, are assigned the value True or False

Propositional Logic



Propositional Logic



Examples of sentences that can be assigned True or False:

Propositional Logic


Simple sentences (that are assigned a truth value such as True or False) can be combined into compound statements (that are also be assigned a truth value)

Propositional Logic



Everyone in France Speaks Chinese

The moon is made of cheese


Propositional Logic



Everyone in France Speaks ChineseANDThe moon is made of cheese


Propositional Logic



Everyone in France Speaks Chinese

The moon is made of cheese


Propositional Logic



Everyone in France Speaks ChineseORThe moon is made of cheese


Propositional Logic



If Everyone in France Speaks ChineseThen The moon is made of cheese


Propositional Logic



It is NOT True thatEveryone in France Speaks Chinese


Propositional Logic


If we have two propositions (sentences, or declarative statements), p and q, then we can say:


Propositional Logic


According to the above, we can make an infinite number of sentences, such as


Propositional Logic


But if we know the truth values of two simple sentences, how do we determine the truth value of a compound sentence?

TTFF

TFTF

TFFF

TTFF

TFTF

TTTF

TTFF

TFTF

TF

FT

TFTT

p and q is true whenboth are true, andfalse otherwise

p or q is true wheneither one is true, otherwise it is false

if p then q is false only when p is true and q is false

not p is false when p is true & true when p is false


Propositional Logic


iff (if and only if), or as it is also called bi-directional if or equivalence


Equivalence


Two formulas P and Q are equivalent iff (if and only if) they have the same truth value for every truth value assignment of their symbols. We write


Equivalence



EXAMPLE


Equivalence


All possible truth value assignments of P and Q


EXAMPLE


Equivalence


All possible truth value assignments of P and Q

These two statements have the same truth value for every truth value assignment of P and Q, and thus they are equivalent


EXAMPLE



ValidityA formula is valid if it is true for every possible truth assignment of its symbols. Or, a formula is valid if it cannot be made false for any truth assignment




AS A TRIVIAL EXAMPLE is a formula that is always true – cannot be false!





Let us try something more interesting. Is the following a valid formula?






It seems like this is a sound (valid) statement, since the formula states the following:







saying it is not true that both p and q are true







saying it is not true that both p and q are true is exactly like saying






saying it is not true that both p and q are true is exactly like saying either p is false or q is false








saying it is not true that both p and q are true is exactly like saying either p is false or q is false

Which seems reasonable



ValidityWhile it sounds reasonable to claim that is a valid statement, we need to systematically check if this statement is true for every truth assignment of p and q



ValidityWhile it sounds reasonable to claim that is a valid statement, we need to systematically check if this statement is true for every truth assignment of p and q

This statement is valid since it cannot be made false, or since it is true for every possible truth assignment of p and q



SatisfiabilityA formula is satisfiable if there is at least one possible truth assignment that will make it true – otherwise, we say the formula is unsatisfiable

AS A TRIVIAL EXAMPLE is a formula that is satisfiable since it can be made true when p is true or when q is false





Now, is the formula satisfiable?






Are there truth values for r, p and q that will make it true?






Are there truth values for r, p and q that will make it true?If it can be made true, in how many ways?



Satisfiability



Satisfiability

This statement can be made true ...



Satisfiability

If we have the following:



Satisfiability

Which means we have the following:



Satisfiability

The OR statement can be made true in three ways



Satisfiability

So we have 3 possible ways to make our formula true (so it satisfiable in 3 ways)



Satisfiability


An alternative to this process is to make the entire truth table and select at the end the rows for which the last formula comes out to be true.

For a formula with 3 symbols this might not be too much, but what if we had 6 or 10 symbols?



Satisfiability


Try it out! Plug the different truth values of the three combinations we arrived at to see if the original formula comes out to be true!


Applications of Propositional Logic


DIGITAL REPRESENTATION OF DECIMAL DIGITS

Every digit is displayed by turning some variables ON (true) and some variables OFF (false)

Applications of Propositional Logic


DIGITAL REPRESENTATION OF DECIMAL DIGITS

?

Every digit is displayed by turning some variables ON (true) and some variables OFF (false)

Limitations of Propositional Logic


Consider the following declarative sentences:




Assuming that p and q are true (that is, assuming we agreed that p and q are true), then r should follow from p and q. In other words this statement should be a valid (always true)





But this statement is not valid – it is not true for all possible truth values of p, q and r





But this statement is not valid – it is not true for all possible truth values of p, q and rWHAT WENT WRONG?

First-Order Logic (FOL)


It turns out that certain valid inferences cannot be made in propositional logic, and a more powerful logic is needed



It turns out that certain valid inferences cannot be made in propositional logic, and a more powerful logic is needed

First-Order Logic (FOL), or the Predicate Calculus, as it is also called, introduces predicates, quantifiers and variables



Variables introduced by quantifiers have scope




This is the scope of the variable x





This is the scope of the variable y






Note that the scope of y is within the scope of x






Note that the scope of y is within the scope of x

Later in the course we will see that the notion of introducing and defining variables in a certain scope in programming languages is exactly the same as that in first-order logic



Example predicates

Is true if the square of x is y



Example predicates


Is true if x is positive



Example predicates


Is true if x is positive

For any x and for any y, if y is the square of x, then y is positive

Introduction to Computer Science – PreliminariesIntroduction to Computer Science – Preliminaries

Inference Rules in Logic: Modus Ponens (MP)An important rule of inference in propositional logic is called modus ponens, which can be described as follows:


An important rule of inference in propositional logic is called modus ponens, which can be described as follows:

The rule says: if p implies q, and we can establish p, therefore we can conclude q.

Inference Rules in Logic: Modus Ponens (MP)


The modus ponens inference rule states the following: if p implies q, and p, then q



The modus ponens inference rule states the following: if p implies q, and p, then q

The modus ponens inference rule is a valid inference rule, since it is true for all possible truth value assignments of p and q



Other Inference RulesOther basic but important inference rules include the following:



If p is true, and q is true, then we can infer (p and q)



If (p and q) is true, the we can infer p and we can infer q




If (p and q) is true, the we can infer p and we can infer q

(elimination)if (not p or q) is true, and we know p is true, then q must be true



Inferences in FOLLet us now re-consider this argument and use first order logic to make this inference:


Inferences in FOLLet us now re-consider this argument and use first order logic to make this inference:

Let us introduce these predicates:

True if x is a mathematician

True if x is clever


Inferences in FOL


Inferences in FOL

1.


Inferences in FOL

1.

2.


Inferences in FOL

1.

2.

3.


Inferences in FOL

1.

2.

3.

4.


Using FOL in Representation and Reasoning Let us see how we can use FOL in data and information representation and reasoning

Suppose we want to represent the blocks world depicted graphically below, where the goal is stack all the blocks from A (at the top) to E (at the bottom) and where the rule that we should follow is that a block cannot be placed on top of another block unless it precedes it alphabetically


Using FOL in Representation and Reasoning

Representing the domain – predicates to describe the domain



If block x is not stacked, and block x precedes y, and block y is free, then block x can be placed on top of block y

Representing the domain – rules of movement



Representing the domain – domain knowledge

If block x precedes y, and block y precedes z, then block x precedes block y



Representing the domain – describing the state of affairs


FOL in Computing Science

FOL can be used to define sets and relations

As an example, let us define the predicate even

0 is even (or, that 0 is even is true!)

Any number is even if the one preceding it is not



Let us test this definition




Check using the above definition of even if even(5) comes out to be true or false




Later in the course you will see how we can write a simple function/program to test whether some input number is an even number of not



Sets, which are unordered collections of objects, can be defined using predicates in FOL

The sets A and B have the same objects. A is defined intensionally, i.e., using logical formulas, and B is defined extensionally, i.e., by actually enumerating the actual objects





Define the set that would contain the elements {3, 7, 9, 11} without listing the elements but using the predicate even





Sets, relations (between sets) and functions will be the subject of the next module!

Introduction to Computer Science

END OF MODULE 2

MATHEMATICAL PRILIMENARIES – SETS, RELATIONS AND FUNCTIONSNext module

intro to computer science module 2

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