Lecture 1: Introduction
Prof. David L. Dill
Department of Computer Science
1
Outline
•
What is this course about? What is it good for?
•
Course administration
•
Basic concepts: Strings, languages, and problems.
•
Proof expectations.
Reading: Chapter 1 of the textbook
2
Representing Sets (discussion)
Supppose you, as a programmer, need to represent a small, finite, set.
•
What does “represent” mean? (what questions/operations?)
Answer: You can answer questions about it.
Simple common question: Is
x
∈
S
?
Other questions: Is
S
=
∅
? Is
S
∩
T
=
∅
? Etc.
•
What representations would be appropriate?
Ok, suppose you want to represent
infinite
sets. How do you do it?
•
What does “represent” mean? (what questions/operations?)
Answer: You can answer questions about it.
Simple common question: Is
x
∈
S
?
•
What representations would be appropriate?
That’s what the course is about.
3
One view of formal language theory
Automata and complexity theory is concerned with properties of
formal
languages
.
In formal language, automata, and complexity theory, a
language
is just a set of
strings.
(Like many mathematical definitions, this leaves behind most of what we think of
as “languages,” but can be made precise. And it leads to very profound results.)
Basically, any object or value that is of interest to computer science can be
represented as a string.
So a set of anything can be considered a language.
4
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What is a representation?
Suppose you have representation that can be stored in a computer.
Can all sets be represented?
[Poll: how many students know about countably infinite vs. uncountably infinite?]
No: Compare the number of possible strings (which is countable) with the number
of sets of string (uncountable).
A particular set that cannot be represented is the set of all irrational numbers –
there are “too many” irrational numbers.
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 Winter '08
 Motwani,R
 Formal language, Regular expression, Character Set, Automata theory, formal language theory

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