Question

The Ackermann function is usually defined as follows: In+1 A(m, n) = {Am - 1,1) ( Alm – 1, A(m, n - 1)) if m =0 if m >0 and n=0 if m >0 and n > 0. Use the definition of the Ackermann function to find Ack(3,2). Please show your work step by step.

Answer 1

A ck (3,2)= A(3,2) Here m= 370 and n=270 By Given definition -A13721)-=-A(2,-A-13,7)) -- Now we find. Here mso and nal > A13111) = A 2 , A (-3,0)) Now we find - A(3,0). .. - Here m= 330 and n=o . definition (above) A(3,0) = A(2,1) Now we find A(2,1) Here m=2> 0 and n=1 to: A 12,1) - A C1_A(2, 0)) - -Now we find A12,0) A(2,0) = A(,1) G. How we find All, ALTH - Ato A-1-1,-0.)) How we find a(,0) A(1,0) = Alo, 1) - -Now we find a font- Alo, 1) = 2

Date put A(0, 1) = 2 in ③ we get - A.4, o = 2 --put_LA_C1,0) = 2 in © we get - A(1, 1) = A 10, 2) A(0, 2) = 3 > All, 17 = 3 Put Ali, 1) = 3 in 6 we get - A(2,0) = 3 . . pur A c2, o) in 6 we get A (2, 1) - AM, 3) * @ since A (1,3) = A (O, A(1,2)) A(1,2)= A TO, ACT,117) S = A(0, 3) (All, 1) = 3 ) put in ③ we get Al1,3) = A 10, 4) A(1,3)=.5 from Ⓡ A (2,11 – 5 -put A12,1)=5 in ) we get - A 13,01 = S put in ③ we get LA(3,12 = A(2, 5). - A 12,5) = All, A(2,4)) - @ A(2,4) - All, Al2,377 LA(2,3) = A (1,A(2, 2))

Date__ FA-12,1)=5) A(2,-2-) = ACI, A(2,1)-).- FAC, 5) - Alo, Ati, 5-) t 14 A(1,5) = 10, 11776)) {A (13):5) A11,4) - A lO, A (1, 3)) LEA 10, s) -A 11,4)=6 put in ③ we get A(1,5) = A(0,6). LA4,5) =__ put in 16 we get A(2,2) = A(0,7). = 8 - put in ③ we get A(273) = A(1,8) A 11,8) - ALO, A11,7)) A11,79 = Alo, A(1,6) _A (1,6)= ACO, A (1,5)) - = A10, 7). -Alli71 = A(0,87 - 9 LA 4,8) = A 0,9). = 10 PNS

Dale اما امي ما لي LA1213) = 10 put in ③ we get - A_(2,4) = AC10) -=A4,A(1,9)) All, 9) = ALU, A(1,8)) = A(0, 10) A-12, 4) = A 10, 11) -A-62,4) = 12 : From A (2,5) - Al!,12) AMT, 12) = A TO, Ati, 1)) LA ( 11) = A CO, A(1,10)) | = (0 , 11) = 12 A(1,12) = A (0,12) = 13 A(2,5) = 13. from 6 A(3, 1) = 13 put in 0 we get. A (3,2) = A (2,13) - - A(2,13) = A (1, A(2,121) A(2,12) - All, A(2,1)). LA 12,11) 2_ A4-y–A_12,10)) At 2,701 = All, A(2,3))

A(2,5) = All, Al2,8)) LA(218) - All, A(247)) _A_1277)_-_A41,-A-12, 6.)). -A-(2,6) - AH A1215- = All, 13) ( A12,5)= 13) All, 13) . = A(0,A(1,,12)) = A(0,13)_ JA(1,12 ) = 13) *A42767447 A (2,7) = A(1, 14) - Alo, All! = A(0,14) =IS _A (2,8) - ACT A(2,7)) D_ALL, 15) =__A (On A (1;14)). 2A (0,15 ) = 16 LA_(2,-9.)__ALH1_161 _ A1.077A-G415)) AL0Y-16) 17 A12, 10) = ACTIFT =: Alo, A(1,16) ) = A (og 17 - 18 1

A (2,11) - All 185 = A (0, A11,17)) = A co, 181 = 19 A(2,12 ) = ACT, 19)_ = Alo All, 18)) = A (0,19) = 20 A (2,13) = A ( ?, 20) = A CO, A(1,19)) SA(0,20). =_2 from (16) A(3,2) = 21 Answer Lack (3,2) = 21