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Discrete Structure Li Tak Sing(李德成) Lecture 13 1 More examples on inductively defined sets Find an inductive definition for each set S of strings. 1. 2. 3. 4. Even palindromes over the set {a,b} Odd palindromes over the set {a,b} All palindromes over the set {a,b} The binary numerals. 2 Solution 1. Basis:Λ ∈ induction: ∈ ℎ , ∈ 2. Basis:, ∈ induction: ∈ ℎ , ∈ 3. Basis:Λ, , ∈ induction: ∈ ℎ , ∈ 4. Basis:0,1 ∈ induction: ∈ ≠ 0 ℎ 1, 0 ∈ 3 More examples on inductively defined sets Find an inductive definition for each set S of lists. 1. {<a>, <a,a>, <a,a,a>,..} 2. {<1>, <2,1>, <3,2,1>,..} 3. {<a,b>, <b,a>, <a,a,b>, <b,b,a>, <a,a,a,b>, <b,b,b,a>,...} 4. {L| L is a list with even length over {0,1,2}} 4 Solution 1. Basis:< >∈ induction: ∈ ℎ (, ) ∈ 2. Basis:< 1 >∈ induction: ∈ ℎ (ℎ + 1, ) ∈ 3. Basis:< , >, < , >∈ induction: ∈ ℎ (ℎ , ) ∈ 4. Basis:<>∈ induction: ∈ , ∈ {0,1,2} ℎ ∈ 5 More examples on inductively defined sets Find an inductive definition for the set B of binary trees that represent arithmetic expressions that are either numbers in N or expressions that use operations + or -. 6 Solution Basis:(<>, 0, <>) ∈ induction: , ∈ ℎ , +, , , −, ∈ , <>, , <> ∈ , ℎ (<>, + 1, <>) ∈ 7 More examples on inductively defined sets Find an inductive definition for each subset S of NN. 1. S={(x,y)| y=x or y=x+1} 2. S={(x,y) | x is even and yx/2 8 Solution 1. Basis:(0,0) ∈ induction: , ∈ ℎ + 1, + 1 , , + 1 ∈ 2. Basis:(0,0) ∈ induction: , 0 ∈ ℎ + 2,0 ∈ (, ) ∈ + 1 ≤ 2 ℎ (, + 1) ∈ 9 Recursive Functions and Procedures Procedure A program that performs one or more actions. A procedure may return one or more values through its argument list. For example, a statement like allocate(m,a,s) might perform the action of allocating a block of m memory cells and return the values a and s, where a is the beginning address of the block and the s tells whether the allocation was successful. 10 Definition of recursively defined A function or a procedure is said to be recursively defined if it is defined in terms of itself. If S is an inductively defined set, then we can construct a function f with domain S as follow: For each basis element xS, specify a value for f(x). Give rules that, for any inductively defined element xS, will define f(x) in terms of previously defined value of f. 11 Constructing a recursively defined procedure If S if an inductively defined set, we can construct a procedure P to process the elements of S as follows: For each basis element xS, specify a set of actions for P(x). Give rules that, for any inductively defined element xS, will define the actions of P(x) in terms of previously defined actions of P. 12 Numbers Sum of integers. f(n)=0+1+2+...+n Definition: f(n)= if n=0 then 0 else f(n-1)+n Alternatively, it can be written as f(0)=0 f(n)=f(n-1)+n This is known as the pattern matching method 13 Numbers Adding odd numbers f(n)=1+3+...+(2n+1) Definition: f(0)=1 f(n)=f(n-1)+(2n+1) 14 The rabbit program The Fibonacci numbers are the numbers in the sequence 0,1,1,2,3,5,8,13 where each number after the first two is computed by adding the preceding two numbers. Assume that at the beginning there is one pair of rabbits. They give birth to another pair of rabbit in one month. Let f(n) represents the number of pairs of rabbits at the n-th month. At that time, there were only f(n-2) mature rabbits which give birth to f(n-2) new rabbits. So the total number of rabbits is the total number of rabbits at the (n-1)th month plus these newly born f(n-2) rabbits. So f(n)=f(n-1)+f(n-2) The sequence 0,1,1,2,3,... is called the Fibonacci numbers. 15 Sum and product notation Sum of sequence a1,a2,....,an n a a 1 a 2 ..... a n i i 1 Product of a sequence a1,a2,....,an n a i a 1 a 2 ..... a n i 1 16 Factorial n!=12.....n 0!=1 n!=(n-1)!n 17 Examples Construct a recursive definition for each of the following functions, where all variables are natural numbers. 1. 2. 3. 4. f(n)=0+2+4+...+2n. f(n)=floor(0/2)+floor(1/2)+....+floor(n/2). f(n,k)=k+(k+1)+(k+2)+...+(k+n). f(n,k)=0+k+2k+...+nk. 18 1. 0 = 0 = − 1 + 2 2. 0 = 0 2 = − 1 + 2 3. 0,0 = 0 0, = 0, − 1 + 1 , = − 1, + + 4. 0, = 0 , = − 1, + 19 Lists f(n)=<n,n-1,..,1,0> f(n)= if n=0 then <0> else cons(n,f(n-1)) Using the pattern matching method f(0)=<0> f(n)=cons(n,<n-1,...,1,0>) =cons(n,f(n-1)) 20 Recursive procedures Let P(n) be the procedure that prints out the numbers in the list <n,n-1,...,0>. P(n): if n=0 then print(0) else print(n); P(n-1) fi 21 The distribute function dist(3,<1,2,3>)=<(3,1),(3,2),(3,3)> How to define this function recursively? dist(x,L)= if L=<> then <> else (x,head(L))::dist(x,tail(L)) Pattern matching method: dist(x,<>)=<> dist(x,a::L)=(x,a)::dist(x,L) 22 The pairs function pairs(<a,b,c>,<d,e,f>)=<(a,d),(b,e),(c,f)> pairs(A,B)=if A=B=<> then <> else (head(A),head(B))::pairs(tail(A),tail(B)) pairs(<>,<>)=<>, pairs(x::T, y::U)=(x,y)::pairs(T,U) 23 Concatenation of Lists cat(<a,b>,<c,d,e>)=<a,b,c,d,e> cat(L,M)=if L=<> then M else head(L)::cat(tail(L),M) Pattern matching method: cat(<>,A)=A cat(x::L,A)=x::cat(L,A) 24 Sorting a list by insertion sort(<>)=<> sort(x::L)=insert(x,sort(L)) insert(x,S)=if S=<> then <x> else if x<head(S) then x::s else head(S)::insert(x,tail(S)) 25 Example Write recursive definition for the following list functions. 1. The function "last" that returns the last elemnt of a nonempty list. For example last(<a,b,c>)=c 2. The function "front" that returns the list obtained by removing the last element of a nonempty list. For example front(<a,b,c>)=<a,b>. 26 Solution 1. < > = ∷ = = =<> ℎ ℎ() 2. < > =<> ∷ = ∷ = =<> ℎ <> ℎ ∷ ( ) 27