2015 년 봄학기 강원대학교 컴퓨터과학전공 문양세 이산수학 (Discrete Mathematics) 귀납과 재귀 (Induction and Recursion)

Discrete Mathematics by Yang-Sae Moon Page 2 강의 내용 수학적 귀납법 (Mathematical Inductions) 재귀 (Recursion) 재귀 알고리즘 (Recursive Algorithms) Induction and Recursion

Discrete Mathematics by Yang-Sae Moon Page 3 Introduction Mathematical Induction A powerful technique for proving that a predicate P(n) is true for every natural number n, no matter how large. ( 모든 자연수 n 에 대해서 P(n) 이 true 임을 보일 수 있는 매우 유용한 방법임 ) Essentially a “domino effect” principle. ( 앞에 것이 성립하면 ( 넘어지면 ), 바로 다음 것도 성립한다 ( 넘어진다.) Based on a predicate-logic inference rule: P(0)  n  0 (P(n)  P(n+1))  n  0 P(n) P(0) 가 true 이고, P(n) 이 true 라 가정했을 때 P(n+1) 이 true 이면, 모든 n 에 대해 P(n) 이 true 이다.

Discrete Mathematics by Yang-Sae Moon Page 4 Validity of Induction Intuitively, we can prove that induction is correct. P(1) = T since (P(0) = T)  (P(0)  P(1) = T) P(2) = T since (P(1) = T)  (P(1)  P(2) = T) P(3) = T since (P(2) = T)  (P(2)  P(3) = T) … P(n) = T since (P(n-1) = T)  (P(n-1)  P(n) = T) Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 5 Outline of an Inductive Proof Want to prove  n P(n)… Induction basis (or base case): Prove P(0). ( 기본 단계 ) Induction step: Prove  n P(n)  P(n+1). ( 귀납적 단계 ) -E.g. use a direct proof: -Let n  N, assume P(n). (induction hypothesis) -Under this assumption, prove P(n+1). Inductive inference rule then gives  n P(n). Can also be used to prove  n  c P(n) for a given constant c  Z, where maybe c  0. (Base 로 0 이 아닌 상수 c 를 사용할 수 있다.) In this circumstance, the base case is to prove P(c) rather than P(0), and the inductive step is to prove  n  c (P(n)  P(n+1)). Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 6 Induction Examples (1/4) Prove that the sum of the first n odd positive integers is n 2. That is, prove: ( 처음 n 개 홀수의 합은 n 2 와 동일하다.) Proof by Induction Induction basis: Let n=1. Since 1 = 1 2, P(1) is true. Induction step: Let n  1, assume P(n), and prove P(n+1). P(n)P(n) By induction hypothesis P(n) Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 7 Induction Examples (2/4) Prove that  n>0, n<2 n. Let P(n)=(n<2 n ) : Induction basis: P(1) = (1 < 2 1 ) = (1 < 2) = T. Induction step: For n>0, prove P(n)  P(n+1). -Assuming n<2 n, prove n+1 < 2 n+1. -Note n + 1 < 2 n + 1 (by inductive hypothesis) < 2 n + 2 n (because 1 < 2=2∙2 0  2∙2 n-1 = 2 n ) = 2 n+1 -So n + 1 < 2 n+1, and we’re done. Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 8 Induction Examples (3/4) Prove that the sum of the first n positive integers is n(n+1)/2. Let P(n) = Induction basis: Let P(1) = Induction step: Let n  1, assume P(n), and prove P(n+1). Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 9 Induction Examples (4/4) Prove when n  2. Induction basis: n = 2, Induction step: Assume P(n), and prove P(n+1). Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 10 Second Principle of Induction ( 강 귀납법 ) Characterized by another inference rule: P(0)  n  0: (  0  k  n P(k))  P(n+1)  n  0: P(n) Difference with 1 st principle is that the inductive step uses the fact that P(k) is true for all smaller k<n+1, not just for k=n. (Induction step 에서 k=n 인 경우 대신에, k<n+1 인 모든 k 에 대해 P(k) 가 true 라 가정한다.) P(0) 가 true 이고, P(0),…,P(n) 이 모두 true 라 가정했을 때 P(n+1) 이 true 이면, 모든 n 에 대해 P(n) 이 true 이다. Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 11 Example of Second Principle Show that every n>1 can be written as a product p 1 p 2 …p s of some series of s prime numbers. Let P(n)=“n has that property” ( 모든 양의 정수는 소수의 곱으로 나타낼 수 있다.) Induction basis: n=2, let s=1, p 1 =2. Thus, P(2) = T. Induction step: Let n  2. Assume 2  k  n: P(k). -Consider n+1. If prime, let s=1, p 1 =n+1. Done. -Else n+1=ab, where 1<a  n and 1<b  n. -Then a=p 1 p 2 …p t and b=q 1 q 2 …q u. -Then n+1= p 1 p 2 …p t q 1 q 2 …q u, a product of s=t+u primes. Mathematical Induction

Discrete Mathematics by Yang-Sae Moon Page 12 강의 내용 수학적 귀납법 (Mathematical Inductions) 재귀 (Recursion) 재귀 알고리즘 (Recursive Algorithms) Induction and Recursion

Discrete Mathematics by Yang-Sae Moon Page 13 Recursive Definitions ( 재귀의 정의 ) Recursion In induction, we prove all members of an infinite set have some property P by proving the truth for larger members in terms of that of smaller members. ( 귀납적 정의에서는, “ 무한 집합의 모든 멤버가 어떠한 성질 P 를 가짐 ” 을 보이기 위하여, “ 작은 멤버들을 사용하여 큰 멤버들이 참 (P 의 성질을 가짐 ) 임 ” 을 증명하 는 방법을 사용하였다.) In recursive definitions, we similarly define a function, a predicate or a set over an infinite number of elements by defining the function or predicate value or set-membership of larger elements in terms of that of smaller ones. ( 재귀적 정의에서는, “ 작은 멤버들에 함수 / 술어 / 집합을 적용 ( 정의 ) 하여 모든 멤 버들을 정의 ” 하는 방법을 사용한다.)

Discrete Mathematics by Yang-Sae Moon Page 14 Recursion ( 재귀 ) Recursion is a general term for the practice of defining an object in terms of itself (or of part of itself). ( 재귀란 객체를 정의하는데 있어서 해당 객체 자신을 사용하는 것을 의미한다.) An inductive proof establishes the truth of P(n+1) recursively in terms of P(n). ( 귀납적 증명은 P(n+1) 이 참임을 증명하기 위하여 재귀적으로 P(n) 을 사용하는 것으로 해석할 수 있다.) There are also recursive algorithms, definitions, functions, sequences, and sets. Recursion

Discrete Mathematics by Yang-Sae Moon Page 15 Recursively Defined Functions Simplest case: One way to define a function f:N  S (for any set S) or series a n =f(n) is to: Define f(0). For n>0, define f(n) in terms of f(0),…,f(n−1). E.g.: Define the series a n : ≡ 2 n recursively: Let a 0 : ≡ 1. For n>0, let a n : ≡ 2a n-1. Recursion

Discrete Mathematics by Yang-Sae Moon Page 16 Another Example Suppose we define f(n) for all n  N recursively by: Let f(0)=3 For all n  N, let f(n+1)=2f(n)+3 What are the values of the following? f(1)= f(2)= f(3)= f(4)= Recursion

Discrete Mathematics by Yang-Sae Moon Page 17 Recursive Definition of Factorial Given an inductive definition of the factorial function F(n) : ≡ n! : ≡ 2  3  …  n. Base case: F(0) : ≡ 1 Recursive part: F(n) : ≡ n  F(n-1). −F(1)=1 −F(2)=2 −F(3)=6 Recursion

Discrete Mathematics by Yang-Sae Moon Page 18 The Fibonacci Series The Fibonacci series f n≥0 is a famous series defined by: f 0 : ≡ 0, f 1 : ≡ 1, f n≥2 : ≡ f n−1 + f n− Recursion

Discrete Mathematics by Yang-Sae Moon Page 19 Inductive Proof about Fibonacci Series Theorem: f n < 2 n. Proof: By induction. Base cases:f 0 = 0 < 2 0 = 1 f 1 = 1 < 2 1 = 2Base cases:f 0 = 0 < 2 0 = 1 f 1 = 1 < 2 1 = 2 Inductive step: Use 2 nd principle of induction (strong induction). Assume  k<n, f k < 2 k. Then f n = f n−1 + f n−2 is < 2 n−1 + 2 n−2 < 2 n−1 + 2 n−1 = 2 n. Inductive step: Use 2 nd principle of induction (strong induction). Assume  k<n, f k < 2 k. Then f n = f n−1 + f n−2 is < 2 n−1 + 2 n−2 < 2 n−1 + 2 n−1 = 2 n.  Recursion

Discrete Mathematics by Yang-Sae Moon Page 20 Recursively Defined Sets An infinite set S may be defined recursively, by giving: A small finite set of base elements of S. ( 유한 개수의 기본 원소를 제시 ) A rule for constructing new elements of S from previously- established elements. ( 새로운 원소를 만드는 방법을 제시 ) Example: Let 3  S, and let x+y  S if x,y  S. What is S? (= {3, 6, 9, 12, 15, …}) Recursion

Discrete Mathematics by Yang-Sae Moon Page 21 강의 내용 수학적 귀납법 (Mathematical Inductions) 재귀 (Recursion) 재귀 알고리즘 (Recursive Algorithms) Induction and Recursion

Discrete Mathematics by Yang-Sae Moon Page 22 Introduction Recursive Algorithms Recursive definitions can be used to describe algorithms as well as functions and sets. ( 재귀적 정의를 수행한 경우, 손쉽게 알고리즘의 함수 / 집합으로 기술할 수 있다.) 예제 : A procedure to compute a n. procedure power(a≠0: real, n  N) if n = 0 then return 1 else return a · power(a, n−1)

Discrete Mathematics by Yang-Sae Moon Page 23 Efficiency of Recursive Algorithms The time complexity of a recursive algorithm may depend critically on the number of recursive calls it makes. ( 재귀 호출에서의 시간 복잡도는 재귀 호출 횟수에 크게 의존적이다.) 예제 : Modular exponentiation to a power n can take log(n) time if done right, but linear time if done slightly differently. ( 잘하면 O(log(n)) 이나, 조금만 잘못하면 O(n) 이 된다.) Task: Compute b n mod m, where m≥2, n≥0, and 1≤b<m. Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 24 Modular Exponentiation Algorithm #1 Uses the fact that b n = b·b n−1 and that x·y mod m = x·(y mod m) mod m. Note this algorithm takes  (n) steps! procedure mpower(b≥1,n≥0,m>b  N) {Returns b n mod m.} if n=0 then return 1; else return (b·mpower(b,n−1,m)) mod m; Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 25 Modular Exponentiation Algorithm #2 Uses the fact that b 2k = b k·2 = (b k ) 2, x·y mod m = x·(y mod m) mod m, and x·x mod m = (x mod m) 2 mod m. What is its time complexity? procedure mpower(b,n,m) {same signature} if n=0 then return 1 else if 2|n then return mpower(b,n/2,m) 2 mod m else return (mpower(b,n−1,m)·b) mod m ( 첫 번째 mpower() 는 한번 call 되고, 그 값을 제곱하는 것임 )  (log n) steps Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 26 A Slight Variation of Algorithm #2 Nearly identical but takes  (n) time instead! The number of recursive calls made is critical. procedure mpower(b,n,m) {same signature} if n=0 then return 1 else if 2|n then return (mpower(b,n/2,m)·mpower(b,n/2,m)) mod m else return (mpower(b,n−1,m)·b) mod m Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 27 Recursive Euclid’s Algorithm ( 예제 ) Note recursive algorithms are often simpler to code than iterative ones… (Recursion 이 코드를 보다 간단하게 하지만 …) However, they can consume more stack space, if your compiler is not smart enough. ( 일반적으로, recursion 은 보다 많은 stack space 를 차지한다.) procedure gcd(a,b  N) if a = 0 then return b else return gcd(b mod a, a) Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 28 Recursion vs. Iteration (1/3) Factorial – Recursion procedure factorial(n  N) if n = 1 then return 1 else return n  factorial(n – 1) Factorial – Iteration procedure factorial(n  N) x := 1 for i := 1 to n x := i  x return x Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 29 Recursion vs. Iteration (2/3) Fibonacci – Recursion procedure fibonacci(n: nonnegative integer) if (n = 0 or n = 1) then return n else return (fibonacci(n–1) + fibonacci(n–2)) Fibonacci – Iteration procedure fibonacci(n: nonnegative integer) if n = 0 then return 0 else x := 0, y := 1 for i := 1 to n – 1 z := x + y, x := y, y := z return z Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 30 재귀 (recursion) 는 프로그램을 간단히 하고, 이해하기가 쉬 운 장점이 있는 반면에, 각 호출이 스택을 사용하므로, depth 가 너무 깊어지지 않도 록 조심스럽게 프로그래밍해야 함 컴퓨터에게 보다 적합한 방법은 반복 (iteration) 을 사용한 프 로그래밍이나, 적당한 범위에서 재귀를 사용하는 것이 바람 직함 Recursion vs. Iteration (3/3) Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 31 Merge Sort ( 예제 ) (1/2) The merge takes (n) steps, and merge-sort takes (n log n). The merge takes  (n) steps, and merge-sort takes  (n log n). procedure sort(L = 1,…, n ) if n>1 then m :=  n/2  {this is rough ½-way point} L := merge(sort( 1,…, m ), sort( m+1,…, n )) return L Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 32 The Merge Routine procedure merge(A, B: sorted lists) L = empty list i:=0, j:=0, k:=0 while i<|A|  j<|B| {|A| is length of A} if i=|A| then L k := B j ; j := j + 1 else if j=|B| then L k := A i ; i := i + 1 else if A i < B j then L k := A i ; i := i + 1 else L k := B j ; j := j + 1 k := k+1 return L Merge Sort ( 예제 ) (2/2) – skip Recursive Algorithms

Discrete Mathematics by Yang-Sae Moon Page 33 강의 내용 수학적 귀납법 (Mathematical Inductions) 재귀 (Recursion) 재귀 알고리즘 (Recursive Algorithms) Induction and Recursion

Discrete Mathematics by Yang-Sae Moon Page 34 Homework #5 Recursive Algorithms