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Natural Numbers

Natural Numbers. or, "All you ever need is compounds, variants, and recursion". Today's Lecture. We've seen types like: [any] -> natural [num] -> num [num] -> [num] [X], [X] -> [X] Let's check out this guy: flatten : [[X]] -> [X] And look at "lists without baggage"

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Natural Numbers

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  1. Natural Numbers or, "All you ever need is compounds, variants, and recursion".

  2. Today's Lecture • We've seen types like: • [any] -> natural [num] -> num • [num] -> [num] [X], [X] -> [X] • Let's check out this guy: • flatten : [[X]] -> [X] • And look at "lists without baggage" • Close look at "recursion over what?"

  3. Lists of different colors • We've seen • [any] : the kitchen sink • [num] : sequences, n-D points, graph … • distance-from-0 = (sqrt (sum (map-square l))) • What about • [car], [course-info], [posn] • Could [[course-info]] be useful?

  4. Flatten ; flatten : [[X]] -> [X] ; Example : (flatten [ [1,2,3], [4], [5,6] ]) ;  [1,2,3,4,5,6] (define (flatten x) (cond [(empty? x) empty] [else (append (first x) (flatten (rest x)))])))

  5. Make your own naturals ; A Natural is ; - 'Zero ; - (make-next nat) where nat is a Natural (define-struct next (nat)) ; Template: ; (define (f m) ; (cond [(symbol? m) ...] ; [else (... (f (next-nat m)))]))

  6. Importing them ; natIt : number -> Natural ; purpose : convert numbers into Naturals ; example : 0 -> 'Zero, 1 -> (make-next ‘zero), ; 2 -> (make-next (make-next 1)) ... (define (natIt n) (cond [(= n 0) 'Zero] [else (make-next (natIt (- n 1)))]))

  7. Exporting them ; numIt : Natural -> number ; purpose : convert a Natural to a number ; example : 0 <- 'Zero, 1 <- (make-next 1), ; 2 <- (make-next (make-next 1)) ... (define (numIt n) (cond [(symbol? n) 0] [else (+ 1 (numIt (next-nat n)))]))

  8. Addition ; add : Natural, Natural -> Natural (define (add n m) (cond [(symbol? n) m] [else (make-next (add (next-nat n) m))])) • Induction on n

  9. Multiplication ; mult : Natural, Natural -> Natural (define (mult n m) (cond [(symbol? n) 'Zero] [else (add m (mult (next-nat n) m))])) • Induction on n

  10. Exponentiation ; exp : Natural, Natural -> Natural (define (expo n m) (cond [(symbol? m) (make-next 'Zero)] [else (mult n (expo n (next-nat m)))])) • Induction on m

  11. Greater or equal to ; geq : Natural, Natural -> Natural (define (geq n m) ;Induction on m (cond [(symbol? m) true] [else (cond [(symbol? n) false] [else (geq (next-nat n) (next-nat m))])]))

  12. Subtraction ; minus : Natural, Natural -> Natural (define (minus n m) ;Induction on m & n (cond [(symbol? m) n] [(symbol? n) 'Zero] [else (minus (next-nat n) (next-nat m))]))

  13. Division ; divi : Natural, Natural -> Natural (define (divi n m) ;StrongInduction on n (cond [(symbol? (next-nat m)) n] [(not (geq n m)) 'Zero] [else (make-next (divi (minus n m) m))]))

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