Recursively Specified Data

1
Recursively Specified Data

• Section 1.1

2
1.1.1 Inductive Specification

• Define the set S to be the smallest set of
  natural numbers satisfying the following two
  properties
• 1) 0 ? S, and
• 2) Whenever x ? S, then x 3 ? S.
• 0, 3, 6, 9,

3
Typical Inductive Definition

• 1) Some specific values must be in S.
• 2) If certain values are in S, then certain
  other values are also in S.

4
List-of-numbers

• The set list-of-numbers is the smallest set of
  values satisfying the two properties
• 1) The empty list is a list-of-numbers, and
• 2) If L is a list-of-numbers and n is a
  number, then the pair (n . L) is a
  list-of-numbers
• ( ) is a list-of-numbers
• (3 . ( ) ) is a list-of-numbers
• (9 . ( 3 . ( ) ) ) is a list of numbers
• (9 3)

5
Backus-Naur Form (BNF)

• A language defined to describe the syntactic
  structure of programming languages
• Non-terminal symbols
• Terminal symbols
• Productions
• Also used to define sets of values
• ltlist-of-numbersgt ( )
• ltlist-of-numbersgt (ltnumbergt .
  ltlist-of-numbersgt)

6
Leftmost Derivation with BNF

• ltl-o-ngt ( )
• ltl-o-ngt (ltnumgt . ltl-o-ngt)
• Replacements
• ltl-o-ngt (ltnumgt . ltl-o-ngt)
• ( ltnumgt . (ltnumgt . ltl-o-ngt))
• ( ltnumgt . (ltnumgt . ( ltnumgt .
  ltl-o-ngt)))
• ( ltnumgt . ( ltnumgt . ( ltnumgt
  . ( ))))
• ( ltnumgt . ( ltnumgt . ( 8 . (
  ))))
• ( ltnumgt . ( 4 . ( 8 . ( ))))
• ( 7 . ( 4 . ( 8 . ( ))))

7
BNF Shortcuts

• ltl-o-ngt ( ) ( ltnumgt . ltl-o-ngt )
• Kleen star zero or more items
• ltl-o-ngt ( ltnumbergt )
• Kleen plus one or more items

8
S-Lists

• lts-listgt (ltsym-exprgt)
• ltsym-exprgt ltsymgt lts-listgt
• ltsym-exprgt (ltsym-exprgtltsym-exprgtltsym-exprgt)
• ( (ltsym-exprgt) ltsym-exprgt
  (ltsym-exprgt))
• ((ltsymgt) ltsym-exprgt
  (ltsym-exprgt))
• ((ltsymgt) ltsymgt (ltsym-exprgt))
• ((ltsymgt) ltsymgt (ltsymgt))
• (( x ) ltsymgt (ltsymgt))
• (( x ) ltsymgt ( a ))
• (( x ) p ( a ))

9
Binary Trees

• ltbtreegt ltnumgt (ltsymgt ltbtreegt ltbtreegt )
• ltbtreegt (ltsymgt ltbtreegt ltbtreegt)
• ( x ltbtreegt ltbtreegt)
• (x (ltsymgtltbtreegtltbtreegt)
  ltbtreegt)
• (x ( y ltbtreegtltbtreegt)
  ltbtreegt)
• (x ( y ltnumgtltbtreegt)
  ltbtreegt)
• (x ( y ltnumgtltnumgt) ltbtreegt)
• (x ( y ltnumgtltnumgt) ltnumgt)
• (x ( y 9 ltnumgt) ltnumgt)
• (x ( y 9 7) ltnumgt)
• (x ( y 9 7) 6)

x
y
6
9
7
10
Lambda Calculus Grammar

• ltexpressiongt
• ltidentifiergt
• ( lambda (ltidentifiergt) ltexpressiongt)
• (ltexpressiongtltexpressiongt)

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Scheme Data Categories

• ltlistgt (ltdatumgt)
• ltdotted-datumgt (ltdatumgt . ltdatumgt)
• ltvectorgt (ltdatumgt)
• ltdatumgt ltnumbergt ltsymbolgt
• ltbooleangt
  ltstringgt
• ltlistgt
  ltdotted-datumgt
• ltvectorgt

12
Context Free Productions

• Here is a context-free rule
• ltxgt ltxgt ltsymgt
• A context sensitive rule would look like this
• a ltxgt b a ltygt b
• Context sensitive parts of a language are often
  described by other techniques.

13
Programs Based on Induction

• E( 0, b) 1 (n 0)
• E(n,b) b x E(n-1,b) (n gt 0)
• (define E
• (lambda (n b)
• (if (zero? n)
• 1
• ( b (E (- n 1) b)))))

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Count-nodes

• ltbtreegt ltnumbergt (ltsymgtltbtreegtltbtreegt))
• (define count-nodes
• (lambda (s)
• (if (number? S)
• 1
• ( (count-nodes (cadr s))
• (count-nodes (caddr
  s))
• 1 ))))