Vaja 1.10 g1smo

zapiski/1_g1smo.org

@@ -137,3 +137,134 @@ TODO!
 ** 1.7 TODO
 
 ** 1.8 TODO
+
+** 1.10 Ackermannov postopek
+
+Sledeči postopek je Ackermannov:
+
+#+begin_src scheme
+  (define (A x y)
+    (cond ((= y 0) 0)
+          ((= x 0)
+           (* 2 y))
+          ((= y 1) 2)
+          (else (A (- x 1)
+                   (A x (- y 1))))))
+#+end_src
+
+Kakšni so rezultati sledečih izrazov?
+
+- `(A 1 10)`
+
+#+begin_quote
+(A 1 10)
+(A 0 (A 1 9))
+(A 0 (A 0 (A 1 8)))
+(A 0 (A 0 (A 0 (A 1 7))))
+(A 0 (A 0 (A 0 (A 0 (A 1 6)))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 32)))))
+(A 0 (A 0 (A 0 (A 0 64))))
+(A 0 (A 0 (A 0 128)))
+(A 0 (A 0 256))
+(A 0 512)
+
+1024
+#+end_quote
+
+- `(A 2 4)`
+
+#+begin_quote
+(A 2 4)
+(A 1 (A 2 3))
+(A 1 (A 1 (A 2 2)))
+(A 1 (A 1 (A 1 (A 2 1)))
+(A 1 (A 1 (A 1 2)))
+(A 1 (A 1 (A 1 2)))
+(A 1 (A 1 (A 0 (A 1 1)))
+(A 1 (A 1 (A 0 2)))
+(A 1 (A 1 (A 0 2)))
+(A 1 (A 1 4))
+(A 1 (A 0 (A 1 3)))
+(A 1 (A 0 (A 0 (A 1 2))))
+(A 1 (A 0 (A 0 (A 0 (A 1 1)))))
+(A 1 (A 0 (A 0 (A 0 2))))
+(A 1 (A 0 (A 0 4)))
+(A 1 (A 0 8))
+(A 1 16)
+(A 0 (A 1 15))
+(A 0 (A 0 (A 1 14))
+...
+(A 0 (A 0 (A 1 14))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1))))))))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32)))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64))))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128)))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256))))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512)))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024))))))
+(A 0 (A 0 (A 0 (A 0 (A 0 2048)))))
+(A 0 (A 0 (A 0 (A 0 4096))))
+(A 0 (A 0 (A 0 8192)))
+(A 0 (A 0 16384))
+(A 0 32768)
+
+65536
+#+end_quote
+
+- `(A 3 3)`
+#+begin_quote
+(A 3 3)
+(A 2 (A 3 2))
+(A 2 (A 2 (A 3 1)))
+(A 2 (A 2 2))
+(A 2 (A 1 (A 2 1)))
+(A 2 (A 1 2))
+(A 2 (A 0 (A 1 1)))
+(A 2 (A 0 2))
+(A 2 4)
+(A 1 (A 2 3))
+(A 1 (A 1 (A 2 2)))
+(A 1 (A 1 (A 1 (A 2 1))))
+(A 1 (A 1 (A 1 2)))
+(A 1 (A 1 (A 0 (A 1 1))))
+(A 1 (A 1 (A 0 2)))
+(A 1 (A 1 4))
+(A 1 (A 0 (A 1 3)))
+(A 1 (A 0 (A 0 (A 1 2))))
+(A 1 (A 0 (A 0 (A 0 (A 1 1)))))
+(A 1 (A 0 (A 0 (A 0 2))))
+(A 1 (A 0 (A 0 4)))
+(A 1 (A 0 8))
+(A 1 16)
+
+65536
+#+end_quote
+
+Če imamo še sledeče postopke:
+#+begin_src scheme
+(define (f n) (A 0 n))
+(define (g n) (A 1 n))
+(define (h n ) (A 2 n))
+(define (k n) (* 5 n n))
+#+end_src
+
+Jedrnato opredeli matematične funkcije izračuna postopkov `f`, `g` in `h`. `k`, recimo izračuna `5n^2`.
+
+#+begin_quote
+`f`: 2n
+`g`: 2^n
+`h`: 2^{2^n}
+#+end_quote