Discrete Structure

1. Discrete Structure Li Tak Sing(李德成) Lecture 13

2. More examples on inductively defined sets • Find an inductive definition for each set S of strings. • Even palindromes over the set {a,b} • Odd palindromes over the set {a,b} • All palindromes over the set {a,b} • The binary numerals.

3. Solution • Basis:induction: • Basis:induction: • Basis:induction: • Basis:induction:

4. More examples on inductively defined sets • Find an inductive definition for each set S of lists. • {<a>, <a,a>, <a,a,a>,..} • {<1>, <2,1>, <3,2,1>,..} • {<a,b>, <b,a>, <a,a,b>, <b,b,a>, <a,a,a,b>, <b,b,b,a>,...} • {L| L is a list with even length over {0,1,2}}

5. Solution • Basis:induction: • Basis:induction: • Basis:induction: • Basis:induction:

6. 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 -.

7. Solution • Basis:induction:,

8. More examples on inductively defined sets • Find an inductive definition for each subset S of NN. • S={(x,y)| y=x or y=x+1} • S={(x,y) | x is even and yx/2

9. Solution • Basis:induction: • Basis:induction:

10. 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.

11. 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 xS, specify a value for f(x). • Give rules that, for any inductively defined element xS, will define f(x) in terms of previously defined value of f.

12. 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 xS, specify a set of actions for P(x). • Give rules that, for any inductively defined element xS, will define the actions of P(x) in terms of previously defined actions of P.

13. 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

14. Numbers • Adding odd numbers • f(n)=1+3+...+(2n+1) • Definition: • f(0)=1 • f(n)=f(n-1)+(2n+1)

15. The rabbit program • The Fibonacci numbers are the numbers in the sequence 0,1,1,2,3,5,8,13where 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.

16. Sum and product notation • Sum of sequence a1,a2,....,an • Product of a sequence a1,a2,....,an

17. Factorial • n!=12.....n • 0!=1 • n!=(n-1)!n

18. Examples • Construct a recursive definition for each of the following functions, where all variables are natural numbers. • 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.

20. 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))

21. 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

22. 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)

23. 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)

24. 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)

25. 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))

26. Example • Write recursive definition for the following list functions. • The function "last" that returns the last elemnt of a nonempty list. For example last(<a,b,c>)=c • 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>.

27. Solution