Introduction to CS Theory

Lecture 1 – IntroductionPiotr Faliszewskipf@cs.rit.edu

Introduction

Instructor Piotr Faliszewski Office: 70-3575 Email: pf@cs.rit.edu

Course website: http://www.cs.rit.edu/~pf/theory/

Office hours Monday through Thursday 2pm—4pm

Logistics

Look at the courses website!

Important highlights Cooperation on homeworks

Teams of two allowed Both people get the same grade—you cannot

complain about your teammate not working Quizzes! Disputing your grade

At most a week after you received the grade

Discrete math quiz Next Monday

Introduction

This is a course in mathematics! Basic discrete math

Dealing with sets, functions, sequences etc. Logic Inductive proofs (!!!)

Goals of the course Understand the nature of computation Develop problem solving skills Increase mathematical sophistication

You should already be familiar with those, but we will review a bit as well

Computer Science Needs Precision!

From a letter of Charles Babbage to Alfred Tennyson in response to his poem "The Vision of Sin“:

"In your otherwise beautiful poem, one verse reads,

Every moment dies a man,Every moment one is born.

... If this were true, the population of the world would be at a standstill. In truth, the rate of birth is slightly in excess of that of death. I would suggest:

Every moment dies a man,Every moment 1 1/16 is born.

Strictly speaking, the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 will be sufficiently accurate for poetry."

What is this course about?

Two fundamental questions about computation What can and cannot be

computed? What can and cannot be

computed efficiently?

But, before we can answer these… What IS computation? What PROBLEMS do

we want to solve?

What is computation? Many different models

of a computer Finite automata Push-down automata Turing machines

Decision problems and languages

Languages and Decision Problems

Decision problem Name Input

the mathematical entities about whose properties we ask

Question A well-defined yes/no

question

Language A set of strings Languages encode

decision problems

Example Name: Connectivity Input: Graph G, vertices

s and t Question: Are vertices s

and t connected in G

Languages and Decision Problems

Our two fundamental questions What decision problems

can and cannot be solved?

What decision problem can and cannot be solved efficiently

We focus on languages

Chomsky Hierarchy

Models of computation Regular languages /

finite automata Context-free

languages / push-down automata

Turing machines

Hierarchy of Languages

Finite languages

Context-freelanguages

Recursive languages

All languages

Effectively Decidable Languages

P

NP

Regularlanguages

Context-freelanguages

Recursive languages

And now…

Let’s get some real work done!

Your responsibility: Part I of the book But, I will cover:

Languages Mathematical induction

1.5 Languages

Language A set of strings over

some alphabet

Alphabet A finite set of indivisible

objects, usually denoted as Σ

Examples {0, 1, 2, …, 9} {0, 1} {a, b, c, …, z} {1}

String A string over Σ is a finite,

possibly empty, sequence of elements of Σ

Some strings over {0,1}: ε

empty string 0, 1, 001, 01010101

Examples of nonstrings (for alphabet {0,1}) 012011 11111…

Length of a string x |x| Examples?

Defining Languages

Set of all strings Σ* -- set of all strings

over Σ {0,1}*

{a}*

We can define languages via set operations Intersection, union Difference Complement operation

L = Σ* - L

Examples of languages

L1 = { x {0,1}* | x contains at least as many 0s as 1s}

L2 = { x {0,1}* | x contains at least as many 1s as 0s}

L3 = {ε}

L4 = { x {0,1}* | x viewed as an integer is a prime}

L5 = L1 L2

L6 = L1

Operations on Strings

Let x, y Σ*

Some operations on strings xy – concatenation

xi – concatenation of x i times with itself

x is a substring of y if there are strings w and z such that y=wxz

x is a prefix of y if there is a string z such that y=xz

x is a suffix of y if there is a string z such that y=zx

Examples x = ab y = aabb

xy = abaabb xε = x = εx

x3 = xxx = ababab x0 = ???

x is a substring of y z = bb z is a suffix of y x is a prefix of x

Operations on Languages

Let L, L1, L2 be languages over the same alphabet Σ

L1L2 = {xy | x in L1 and y in L2} – the concatenation of two languages

Li – concatenation of i L’s What is L0?

L* = L0 L1 L2 L3 … Kleene’s starL+ = LL* = L*L

Exercises Is there a language L such that L* is finite? Let L1, L2 be two finite languages. What is the cardinality of

L1L2

Is there a language such that L = L*?

Mathematical Induction

Mathematical induction Technique to prove facts

about finite entities E.g., properties of integers

Inductive proof: Base case Inductive step

Intuition … a little like a for loop in a

program: we construct the proof in iterations.

Theorem. Let A be a finite set with n elements. There are 2n distinct subsets of A.

Proof.Base: A = Inductive step:

Assume that the theorem holds for all natural numbers up to k.

Let A be a set with k+1 elements and B an arbitrary subset of k elements of A. Let x be such that:

A = B {x}

By the inductive assumption, B has 2k subsets. Thus A has 2k+1 subsets. (Why?)

Mathematical Induction – Examples

Let us prove the following:

Σ = {0,1}, each string x such that x = 0y1, where y Σ*, contains 01 as a substring

For every natural number n, n ≥ 2, n either is a prime or a product of two or more primes

How about the following theorem:

For every natural number n it holds that n! > 2n

And I can prove by induction that all horses are of the

same color!

Recursive Definitions

Recursive definition Define a small set of

basic entities Show how to obtain

larger ones from those already constructed

A bit like induction!

Examples

Factorial0! = 1For each natural n, (n+1)! = (n+1)*n!

Recursive definition of Σ*:1. ε Σ*

2. If x Σ* and a Σ then xa Σ*

3. Nothing is in Σ* unless it is obtained by the two previous rules.

Recursive Definitions – Examples

Two exercises Give a recursive

definition of the length of a string.

Give a recursive definition of a reverse of a string.

Palindromes A string is a palindrome if

it reads the same backward and forward

Let’s define a language of palindromes!

Language L of palindromes, Σ = {a, b}.1. ε L2. If x L then both axa and

bxb belong to L.3. Nothing belongs to L

unless it is obtained by rules 1 and 2.

Hmm… This looks wrong to

me!

Recursive Definitions – Examples

Language L of all strings with as many 0s as 1s. (Σ = {0,1})1. ε L2. For any x in L it holds

that each of: 0x1 1x0 01x 10x x01 x10belongs to L.

Is this definition correct?

Exercise

Prove that for each string x over alphabet Σ it holds that |x| = |xr|

xr is the reverse of x

Structural induction!