Inductive specifications

• Code

    

Inductive specifications and recursive programming

1. 0 E S
2. whenever x E S, x + 3 E S

0 3 6 9 12 ...

Inductive specification of "multiples of 3"

In general:

1. Some specific value is in S
2. If certain values are in S, 
   then certain other values are in S

Inductive specification of data types:

"List of Numbers"

1. ()
2. if L is a "ListofNumbers" and n is a number,
   then the pair (n.L) is a "ListofNumbers"

()
(14)
(3 14)

...


	BNF for inductive specifications

- especially useful to specify programming language
syntax --> compilers.

- theory of computing


 ::== ()
 ::== ( . )

OR 

 ::== () | ( . )

terminals, Syntactic categories

more Shortcuts:

Kleene Star
 ::== ({}*) // 0 or more numbers

Kleene Plus

 ::== ({}+)  // 1 or more numbers




Prove that a given data value is syntactic category X

Derivation:

(14.()) is a listofnumbers


=> (.)
=> (14.)
=> (14.())

Another way equally ok.


=> (.)
=> (.())
=> (14.())


Scheme example (language specification example):
using BNF

datum is an literal data representation

		    ::== ({}*)

	    ::== ({}+ . )

	    ::== #({}*)

		    ::==  |  
			 |  |  
			 |  | 
			 | 

BNF rules are context free 
programming languages are not

Prose accompanying BNF 
provides context in language specification

Compilers use limited context senstive BNF rules



Recursive specification of programs

1. if n is 0, e(n,x) = 1
2. if n > 0, assume e(n-1,x) is known/well defined
   then
   e(n,x) = x * e(n-1,x)

(define e
  (lambda (n x)
    (if (zero? n)
	1
	(* x (e (- n 1) x)))))


When defining a program based on structural
induction, the structure of the program should
be patterned after the structure of the data.

(define listofnumbers?

 ::== () | ( . )

(define list-of-numbers?
  (lambda (lst)
    (if (null? lst)
	#t
	(if (pair? lst)
	    (if (number? (car lst))
		(list-of-numbers? (cdr lst))
		#f)
	    #f))))