2 - Recursive Definition

Automata and Language Theory

A New Method for Defining LanguageIs characteristically a three step process

We specify some basic objects in the setWe give rules for constructing more objects in

the set from the ones we already knowWe declare that no objects except those

constructed in this way are allowed

A New Method for Defining LanguageExample

Suppose that we are trying to define the set of positive even integers for someone who knows about arithmetic, but has never heard of the even numbers.

EVEN is the set of all positive who numbers divisible by 2

A New Method for Defining Language EVEN is the set of all 2n where n = 1 2 3 3…

by recursive definition: The set EVEN is defined by these three rules:

Rule 1 2 is EVENRule 2 If x is EVEN, then so is x+2Rule 3 The only elements in the set EVEN

are those that can be produced from the two rules above

A New Method for Defining LanguageTo prove that 14 is even

By Rule 1, we know that 2 is in EVEN Then by Rule 2, we know that 2+2 is also in EVEN Again by Rule 2, we know that since 4 has just been

shown to e in EVEN, 4+2=6 is also in EVEN The fact that 6 is in EVEN means that when we

apply Rule 2, we deduce that 6+2=8 is in EVEN too……..

And, at last, by applying Rule 2, yet again, to the number 12, we conclude that 12+2=14 is, indeed in EVEN

A New Method for Defining LanguageThe set EVEN is defined by these two rules:

Rule 1 2 is in EVENRule 2 If x and y are both in EVEN, then

so is x + y

A New Method for Defining LanguageProve that 14 is in EVEN in fewer steps:

By rule 1, 2 is EVEN By rule 2, x=2, y=2, x+y=4 is in EVEN By rule 2, x=2, y=4, x+y=6 is in EVEN By rule 2, x=4, y=4, x+y=8 is in EVEN By rule 2, x=6, y=8, x+y=14 is in EVEN

A New Method for Defining LanguageExample

The following is a recursive definition of the positive integers

Rule 1 1 is in INTEGERSRule 2 If x is in INTEGERS, the so is x+1

(not including negatives)

A New Method for Defining LanguageRule 1 1 is in INTEGERSRule 2 If both x and y are in INTEGERS,

then so are x+y and x-y

Since 1-1=0 and for all positivex, o-x=-x, we see that negative integers an zero are all included in this definition

A New Method for Defining LanguageThe set POLYNOMIAL is defined by these three

rulesRule 1 Any number is in POLYNOMIAL.Rule 2 The variable x is in POLYNOMIAL.Rule 3 If p and q are POLYNOMIAL,

then so are p+q, p-q, (p), & pq

Try to prove that 3x2+7x-9 is in POLYNOMIAL

A New Method for Defining LanguageObserve how natural the following definition

are

Rule 1 x is in L1.Rule 2 If w is any word in L1, then xw is

also in L1L1=x+={x xx xxx ….}

A New Method for Defining LanguageRule 1 ^ is in L4Rule 2 if w is any word in L4, then xw is

also in l4L4=x*

….

A New Method for Defining LanguageThe definition of Kleene closure might have

benefited from a recursive definition:Rule 1 If S is a language, then all the

words of S are in S*Rule 2 ^ is in S*Rule 3 If x and y are in S*, then so is their

concatenation xy.

An important Language: Arithmetic Expression

- {0 1 2 3 4 5 6 7 8 9 + - * / ( )}

What can you say about the following- (3+5)+6)- 2(/8+9)- (3+(4-)8)- 2)-(4


Recursive Definition of AERule 1 Any Number(+,-,0) is in AERule 2 If x is in AE, then so are

(i) (x)

(ii) –x (provided x does not already start with a - sign)

Rule 3 If x and y are in AE, then so are:(i) x + y (if the first symbol in y is not + or –

(ii) x - y (if the first symbol in y is not + or –(iii) x*y(iv) x/y(v) x**y (our notation for exponentiation)


Try to prove

(2+4) * (7*(9-3)/4) / 4 * (2+8) - 1

is in AE Prove that

An arithmetic expression cannot contain the character $

No AE can begin or end with the symbol /. No AE can contain the substring //.

Quiz Prove that 78 is in set EVEN using the following recursive definition:

Rule 1 2 and 4 are in EVEN Rule 2 If x is in EVEN, then so x+4

Give the recursive definition for the set POWER-OF-TWO = {1 2 4 8 16……}

2 - Recursive Definition

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