Discrete Structure

Li Tak Sing(李德成 )

Lecture 13

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More examples on inductively defined setsFind an inductive definition for each set S

of strings.1. Even palindromes over the set {a,b}

2. Odd palindromes over the set {a,b}

3. All palindromes over the set {a,b}

4. The binary numerals.

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Solution

1. Basis:induction:

2. Basis:induction:

3. Basis:induction:

4. Basis:induction:

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More examples on inductively defined setsFind 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}}

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Solution

1. Basis:induction:

2. Basis:induction:

3. Basis:induction:

4. Basis:induction:

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More examples on inductively defined setsFind 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 -.

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Solution

Basis:induction:,

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More examples on inductively defined setsFind 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

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Solution

1. Basis:induction:

2. Basis:induction:

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Recursive Functions and Procedures

ProcedureA 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.

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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.

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Constructing a recursively defined procedureIf 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.

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Numbers

Sum of integers.f(n)=0+1+2+...+nDefinition:

f(n)= if n=0 then 0 else f(n-1)+nAlternatively, it can be written as

f(0)=0f(n)=f(n-1)+nThis is known as the pattern matching method

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Numbers

Adding odd numbersf(n)=1+3+...+(2n+1)Definition:

f(0)=1f(n)=f(n-1)+(2n+1)

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The rabbit program

The Fibonacci numbers are the numbers in the sequence0,1,1,2,3,5,8,13

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.

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Sum and product notation

Sum of sequence a1,a2,....,an

Product of a sequence a1,a2,....,an

n

ini aaaa

121 .....

ni

n

i

aaaa .....211

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Factorial

n!=12.....n0!=1n!=(n-1)!n

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Examples

Construct a recursive definition for each of the following functions, where all variables are natural numbers.1. f(n)=0+2+4+...+2n.

2. f(n)=floor(0/2)+floor(1/2)+....+floor(n/2).

3. f(n,k)=k+(k+1)+(k+2)+...+(k+n).

4. f(n,k)=0+k+2k+...+nk.

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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))

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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

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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)

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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)

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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)=Acat(x::L,A)=x::cat(L,A)

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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))

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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>.

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Solution

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