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Introduction to CS Theory Lecture 2 – Discrete Math Revision Piotr Faliszewski pf@cs.rit.edu

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Previous Class The course Logistics Problems we study What is computation? What problems do we solve? Languages Alphabet String Operations on strings Operations on languages Mathematical background Discrete mathematics Inductive proofs

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Inductive Proofs Theorem statement P(k) – some predicate about an integer k Theorem: P(k) holds for every integer k Induction base P(0) holds Inductive step Hypothesis: There is a value k such that P(k) holds Step: P(k+1) holds. Example: Prove that 1 + 2 + 4 + … + 2 n-1 = 2 n -1 Proof P(k) = [1 + 2 + … +2 k-1 = 2 k -1] Induction base 1 = 2 1 -1 Inductive step Hypothesis: There is a value k such that 1 + 2 + … +2 k-1 = 2 k -1 Step: Show that if the hypothesis holds then 1 + 2 + … +2 k = 2 k+1 -1

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Strong Mathematical Induciton Inductive step: We were showing that P[k] => P[k+1] Often this is limiting Strong mathematical induction We can use a stronger inductive step (P[0] and P[1] and … and P[k]) => P[k+1] This way we can rely on all “previous” results

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Strong Mathematical Induciton Strong mathematical induction (P[0] and P[1] and … and P[k]) => P[k+1] Theorem. Every positive natural number n ≥ 2 is either prime or is a multiplication of several primes Proof. By induction on n. Base: n = 2. 2 is a prime, so theorem holds. Inductive step: Hypothesis: There is number k such that each number k’ ≤ k is either prime or can be factored into several primes Step: k+1 is either prime or a multiplication of several primes Consider cases a) k + 1 is prime then we are done b) k + 1 is not a prime, thus k+1 = ab, where a and b are two positive integers. Our hypothesis applies to both a and b. Thus, the theorem holds for k+1.

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Is Induction Really Correct? Exercise: Prove that 2 n ≤ n! Exercise: Prove that all horses are af the same color! A proof by induction. Base step holds Inductive step holds Eh…?! Math is always right, so if you ever saw two horses of different colors then you are inconsistent with mathematics!

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Is Induction Really Correct? Exercise: Prove that 2 n ≤ n! Exercise: Prove that all horses are af the same color! A proof by induction. Base step holds Inductive step holds NOT Oops! Be Very Careful with Induction!!!

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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.

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Recursive Definitions – Fibonacci Sequence Fibonacci sequence f(0) = 0 f(1) = 1 For every n > 1: f(n+1) = f(n) + f(n-1) Computing Fibonacci numbers Bottom-up Top-down Exercise Show that for each natural n it holds that f(n) ≤ (5/3) n Show that 1 + ∑ 0 ≤ i ≤ n f i = f n+2

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Recursive Definitions – Examples Example Definition of |x| 1.|ε| = 0 2.|xa| = |x| + 1, for each x Σ * and a Σ. What is |aab|? Prove that for each two strings it holds that |xy| = |x| + |y| An exercise Give a recursive definition of a reverse of a string. (x r ) Compute a reverse of a string based on this definition Prove that for every string x, |x| = |x r |

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Language of Palindromes 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!

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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?

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Bribery Problem Consider the following setting C – set of candidates V – set of voters Each voter votes for one candidate Candidates with most points win Bribery For a unit price you can buy a voter We want to make our preferred candidate p win How to do this most cheaply? Example c1c1 c2c2 c3c3 p

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Bribery Problem Consider the following setting C – set of candidates V – set of voters Each voter votes for one candidate Candidates with most points win Bribery For a unit price you can buy a voter We want to make our preferred candidate p win How to do this most cheaply? Example c1c1 c2c2 c3c3 p

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Bribery Problem Consider the following setting C – set of candidates V – set of voters Each voter votes for one candidate Candidates with most points win Bribery For a unit price you can buy a voter We want to make our preferred candidate p win How to do this most cheaply? Example c1c1 c2c2 c3c3 p

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Bribery Problem Consider the following setting C – set of candidates V – set of voters Each voter votes for one candidate Candidates with most points win Bribery For a unit price you can buy a voter We want to make our preferred candidate p win How to do this most cheaply? Example c1c1 c2c2 c3c3 p

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Bribery Problem Consider the following setting C – set of candidates V – set of voters Each voter votes for one candidate Candidates with most points win Bribery For a unit price you can buy a voter We want to make our preferred candidate p win How to do this most cheaply? Example c1c1 c2c2 c3c3 p

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Bribery Problem Consider the following setting C – set of candidates V – set of voters Each voter votes for one candidate Candidates with most points win Bribery For a unit price you can buy a voter We want to make our preferred candidate p win How to do this most cheaply? Example Proof of correctness? Induction! On what? On the cost of the optimal solution! c1c1 c2c2 c3c3 p

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Tournament Problem Tournament A directed graph There is a single edge between any two vertices Vertices – players Edges – results of games (arrow points to the loser) Example

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Tournament Problem Tournament Vertices – players Edges – results of games (arrow points to the loser) Problem: We want to pick a team of players, who jointly defeat everyone How can we pick such a set? How large a set do we need? Example

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Tournament Problem Algorithm Input: Tournament T Pick player u who defeats most players Form tournament T’ via removing T and all the players he/she defeats. If T’ is an empty graph the finish. Else, loop back using T’ instead of T Questions Does this algorithm work? How large a set does it find? How to prove this?

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Recursive Definitions and Structural Induction

Recursive Definitions and Structural Induction

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