BuK/IIm11/BuK-KA-Goedelisierung und mu-rekursive Funktionen

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<language>Scheme</language>
'''Kreativaufgabe: Gödelisierung und $\mu$-rekursive Funktionen'''
<code id="508fe8d1b6671" hidden>
(define get-elem
  (lambda (set index)
    (if (= index 1)
      (car set)
      (get-elem (cdr set) (- index 1)))))

;Berechnet die Anzahl der Wörter mit Länge <= letterCount
(define word-number
  (lambda (cardinal-number letter-count)
    (if (= letter-count 0)
      0
      (+
        (pow cardinal-number letter-count)
        (word-number cardinal-number (- letter-count 1))))))
    
(define fCreator
  (lambda (set)
    (lambda (index)
      (if (= index 0)
        ""
        (let((cardinalNumber (cardinalNumber S))
            (lettersRemaining (calculateWordSize S index)))
          (createWord S (- index (numberOfWords cardinalNumber (- lettersRemaining 1))) cardinalNumber lettersRemaining))))))

(define monus-minus
  (lambda (y)
    (if (<= 1 y) 0 (- 1 y))))

(define S '(#\a #\b))

(define cardinalNumber
  (lambda (set)
    (if (null? set)
      0
      (+ 1 (cardinalNumber (cdr set))))))

(define calculateWordSize
  (lambda (set index)
    (letrec ((calculateWordSizeReal
      (lambda (cardinalNumber index sum exp)
        (if (>= sum index)
          0
          (+ 1 
            (calculateWordSizeReal
              cardinalNumber
              index
              (+ sum (potenz cardinalNumber exp))
              (+ 1 exp)))))))
  (calculateWordSizeReal (cardinalNumber set) index 0 1))))

(define createWord
  (lambda (set wordnumber cardinalNumber lettersRemaining)
    (if (= 0 lettersRemaining)
      ""
      (let ((sectionSize (potenz cardinalNumber (- lettersRemaining 1))))
        (let ((rest (modulo wordnumber sectionSize))
            (result (quotient wordnumber sectionSize)))
          (if (= 0 rest)
            (string-append
              (string (getElement set result))
              (createWord set sectionSize cardinalNumber (- lettersRemaining 1)))
            (string-append
              (string (getElement set (+ result 1)))
              (createWord set rest cardinalNumber (- lettersRemaining 1)))))))))

(define numberOfWords
  (lambda (cardinalNumber letterCount)
    (if (= letterCount 0)
      0
      (+
        (potenz cardinalNumber letterCount)
        (numberOfWords cardinalNumber (- letterCount 1))))))

(define getElement
  (lambda (set index)
    (if (= index 1)
      (car set)
      (getElement (cdr set) (- index 1)))))

(define potenz
  (lambda (zahl exponent)
    (if (= exponent 0)
        1
        (* zahl (potenz zahl (- exponent 1))))))

(define wort-laenge-eins
  (lambda (wort)
    (- (length wort) 1)))

(define p
  (lambda (r y)
    (if (and (<= r y) (goedel-nummer? y)) 1 0)))

(define f (fCreator S))

(define is-goedel?
  (lambda (y index)
    (cond
      [(eqv? (my-goedel(string->list(f index))) y) #t]
      [(>(my-goedel(string->list(f index))) y) #f ]  
      [else (is-goedel? y (+ index 1))])))

(define goedel-nummer?
  (lambda (nr)
    (is-goedel? nr 1)))

</code>

<code id="508fe835901e4">

(define S '(#\a #\b))



(define my-goedel
 (lambda (wort)
   (cond
     [(and (null? (cdr wort))(eqv? (car wort) #\a)) 1]
     [(and (null? (cdr wort))(eqv? (car wort)#\b)) 2]
     [(eqv? #\a (car wort)) (+ (my-goedel (cdr wort)) (* 1 (potenz 3 (wort-laenge-eins wort)))) ]
     [(eqv? #\b (car wort)) (+ (my-goedel (cdr wort)) (* 2 (potenz 3 (wort-laenge-eins wort)))) ]
     [else #f])))


(define h
  (lambda (wort)
    (my-goedel (string->list wort))))

</code>

Gödelisierung h (Verschlüsselung im 3-er System):

<run id="508d8bc6ef98b">
(h "aba")
</run>

<code id="508fe8359026c">
(define g
  (lambda (i m z)
    (cond
      [(equal? 0 z) (display "...")  ]
      [(= (monus-minus (p i m)) 0)  (display m)(display ", ") (g (+ m 1) 0 (- z 1)) ]
      [else (g i (+ m 1) z)])))

(define monus-minus
  (lambda (y)
    (if (<= 1 y) 0 (- 1 y))))

(define g-n-mal
  (lambda (n)
    (g 0 0 n)))
</code>
 
Die ersten n Gödelnummern:

<run id="508d8bc6efa00">
(g-n-mal 15)
</run>

<code id="508fe835902d8">

(define is-goedel?
  (lambda (y index)
    (cond
      [(eqv? (my-goedel(string->list(f index))) y) #t]
      [(>(my-goedel(string->list(f index))) y) #f ]  
      [else (is-goedel? y (+ index 1))])))

(define goedel-nummer?
  (lambda (nr)
    (is-goedel? nr 1)))

</code>

Prozedur goedel-nummer? prüft, ob es sich um eine gültige Gödelnummer handelt:

<run id="508d8bc6efa69">
(goedel-nummer? (h "abba"))
</run>
<run id="508d8bc6efadb">
(goedel-nummer? 10)
</run>

<code id="508fe83590345">
(define p
  (lambda (r y)
    (if (and (<= r y) (goedel-nummer? y)) 1 0)))
</code>

Scheme-Prozedur für ein total berechenbares Prädikat p:

<run id="508d8bc6efb4c">
(p 3 3)
</run>
<run id="508d8bc6efbbf">
(p 3 4)
</run>

<code id="508fe835903b3">

(define f (fCreator S))
  
(define k
  (lambda (n)
    (f n)))


</code>

Prozedur k liefert das n-te Wort in der gödelgeordneten Folge der Wörter aus S*:

<run id="508d8bc6efc33">
(h (k 10))
</run>
<run id="508d8bc6efca8">
(k 10)
</run>