Introduction to CS Theory Lecture 1 – Introduction Piotr Faliszewski

Introduction  Instructor Piotr Faliszewski Office:  Course website:  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-free languages Recursive languages All languages

Effectively Decidable Languages P NP Regular languages Context-free languages 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, Examples of nonstrings (for alphabet {0,1})   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 L 1 = { x  {0,1} * | x contains at least as many 0s as 1s} L 2 = { x  {0,1} * | x contains at least as many 1s as 0s} L 3 = {ε} L 4 = { x  {0,1} * | x viewed as an integer is a prime} L 5 = L 1  L 2 L 6 = L 1

Operations on Strings Let x, y  Σ * Some operations on strings xy – concatenation x i – 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 x 3 = xxx = ababab x 0 = ??? 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, L 1, L 2 be languages over the same alphabet Σ L 1 L 2 = {xy | x in L 1 and y in L 2 } – the concatenation of two languages L i – concatenation of i L’s  What is L 0 ? L * = L 0  L 1  L 2  L 3  …  Kleene’s star L + = LL * = L * L  Exercises Is there a language L such that L * is finite? Let L 1, L 2 be two finite languages. What is the cardinality of L 1 L 2 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 2 n 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 2 k subsets. Thus A has 2 k+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! > 2 n 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 Factorial 0! = 1 For 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  L 2.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  L 2.For any x in L it holds that each of: 0x1 1x0 01x 10x x01 x10 belongs to L. Is this definition correct?  Exercise Prove that for each string x over alphabet Σ it holds that |x| = |x r | x r is the reverse of x Structural induction!