# The Ackermann function

The Ackermann function is notable for being the one of the simplest examples of a total, computable function that isn't primitive recursive.

We will use the definition of $$\A(m,n)\$$ taking in two nonnegative integers where

\begin{align} A(0,n) & = n+1 \\ A(m,0) & = A(m-1,1) \\ A(m,n) & = A(m-1,A(m,n-1)) \end{align}

You may implement

• a named or anonymous function taking two integers as input, returning an integer, or
• a program taking two space- or newline-separated integers on STDIN, printing a result to STDOUT.

You may not use an Ackermann function or hyperexponentiation function from a library, if one exists, but you may use any other function from any other library. Regular exponentiation is allowed.

Your function must be able to find the value of $$\A(m,n)\$$ for $$\m \le 3\$$ and $$\n \le 10\$$ in less than a minute. It must at least theoretically terminate on any other inputs: given infinite stack space, a native Bigint type, and an arbitrarily long period of time, it would return the answer. Edit: If your language has a default recursion depth that is too restrictive, you may reconfigure that at no character cost.

The submission with the shortest number of characters wins.

Here are some values, to check your answer:

$$\begin{array}{c|cccccccccc} A & n = 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline m=0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 2 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21 & 23 \\ 3 & 5 & 13 & 29 & 61 & 125 & 253 & 509 & 1021 & 2045 & 4093 & 8189 \\ 4 & 13 & 65533 & \text{big} \end{array}$$

• How has this not been asked before?? Oct 22, 2014 at 4:56
• I think it'd be more fun to make this fastest-code Oct 22, 2014 at 7:43
• Downvoting because there's no challenge here. The obvious answer -- just naively implementing the function exactly according to its definition -- is always going to be the best answer. So the question is just "Which language has the least number of characters in the obvious expression of Ackermann's function?" The true winner is the programming language, not the person who wrote the obvious program in it. Oct 22, 2014 at 7:52
• I agree that this challenge would be more interesting as a fastest code, currently this challenge seems to be mostly, how can I implement the ackerman function in as little characters without much consideration to speed. Instead of how can I make use of any patterns in the ackerman function to speed it up Oct 22, 2014 at 8:55
• @DavidRicherby "The obvious answer [...] is always going to be the best answer." This is not true of all languages. I feel a little dirty for only having an example in my home language, but there are multiple ways to express Ackermann and in some languages you can get savings by using that fact. This was my intention for the challenge. Oct 22, 2014 at 13:15

0%n=1+n
m%n=iterate((m-1)%)1!!(n+1)


this defines the operator function %.

this works by noticing that m%n (where a is the ackerman function) for nonzero m is (m-1)% applied n+1 times to 1. for example, 3%2 is defined as 2%(3%1) which is 2%(2%(3%0)), and this is 2%(2%(2%1))

• too bad I can't use 0%n instead of n+1 because of precedence Dec 4, 2014 at 21:19
• What am I doing wrong? Mar 12, 2018 at 0:59
• wow, this has been wrong for so long, and nobody, including myself, noticed? good work. so it seems I mistakenly copied the first version of the answer, which was faulty, maybe by mistake or because I thought it worked. Mar 21, 2018 at 7:22

# Binary lambda calculus, 54 bits = 6.75 bytes

Hexdump:

00000000: 1607 2d88 072f 68                        ..-../h


Binary:

000101100000011100101101100010000000011100101111011010


This is λm. mg. λn. g (n g 1)) (λn. λf. λx. f (n f x)), where all numbers are represented as Church numerals.

## GolfScript (30)

{1$1>{1\){1$(\A}*\;}{+)}if}:A;


Online demo

Without the 1> (which special-cases A(1, n)) it takes 9 minutes to compute A(3, 10) on the computer I've tested it on. With that special case it's fast enough that the online demo takes less than 10 seconds.

Note that this isn't a naïve translation of the definition. The recursion depth is bounded by m.

### Dissection

{             # Function boilerplate
1$# Get a copy of m: stack holds m n m 1>{ # (Optimisation): if m is greater than 1 1 # Take this as the value of A(m, -1) \){ # Repeat n+1 times: # Stack: m A(m, i-1) 1$(\    #     Stack: m m-1 A(m, i-1)
A       #     Stack: m A(m, i)
}*
\;        #   Lose that unwanted copy of m
}{          # Else
+)        #   A(m in {0, 1}, n) = m + n + 1
}if
}:A;          # Function boilerplate

• In CJam, you wouldn't need 1>. After removal (and changing if to ?), computing 3 10 A takes 110 seconds with the online interpreter and six seconds with the Java interpreter. Oct 22, 2014 at 14:03

# Pyth, 19

DaGHR?atG?aGtHH1GhH


Defines a, which works as the Ackermann function. Note that this requires a higher recursion depth than the official pyth compiler allowed up until today to compute a 3 10, so I increased the recursion depth. This is not a change to the language, just to the compiler.

$time pyth -c "DaGHR?atG?aGtHH1GhH ;a 3 10" 8189 real 0m0.092s user 0m0.088s sys 0m0.000s  ### Explanation: DaGH def a(G,H): R return ? G (if G: atG (a(G-1, ? H (if H: aGtH a(G,H-1) 1 else:1) hH else:H+1)  Essentially, it first conditions on the truth value of G whether to recurse or return H+1. If it is recursing, the first argument is always G-1, and it conditions on the truth value of H whether to use a(G,H-1) as the second argument, or to use 1 as the second argument. • In modern Pyth (I assume this was added after this challenge) I think you can more or less change DaGHR to M and a to g. (Did the order of arguments for ? change?) – lynn Dec 16, 2015 at 13:39 • @Mauris Yes, you can use M instead, and yes, ? argument order changed. It's now condition, true, false. It was true, condition, false. Dec 16, 2015 at 23:25 • @Lynn Fun Pyth history facts: This question actually influenced isaacg to change (at least) two things about Pyth: the recursion limit and the change to M! Mar 16, 2016 at 17:39 # J - 26 char ($:^:(<:@[]1:)^:(0<[)>:)


There is an alternate, more functional definition of Ackermann:

Ack 0 n = n+1
Ack m n = Iter (Ack (m-1)) n
Iter f 0 = f 1
Iter f n = f (Iter f (n-1))


It so happens that Iter is very easy to write in J, because J has a way of passing in the m-1 to Ack and also to define the initial value of Iter to be 1. Explained by explosion:

(                      >:)  NB. increment n
^:(0<[)     NB. if m=0, do nothing to n+1; else:
^:                       NB. iterate...
($: ) NB. self ($: is recursion)
(<:@[     )            NB.   with left arg m-1
]                NB.   n+1 times
1:             NB.   starting on 1


This relies on what J calls the gerund form of ^:—basically a way to have more control over all the bounds in a tacit (point-free) fashion.

At the REPL:

   3 ($:^:(<:@[]1:)^:(0<[)>:) 3 61 ack =: ($:^:(<:@[]1:)^:(0<[)>:)
(i.4) ack"0 table (i.11)
+-----+------------------------------------------+
|ack"0|0  1  2  3   4   5   6    7    8    9   10|
+-----+------------------------------------------+
|0    |1  2  3  4   5   6   7    8    9   10   11|
|1    |2  3  4  5   6   7   8    9   10   11   12|
|2    |3  5  7  9  11  13  15   17   19   21   23|
|3    |5 13 29 61 125 253 509 1021 2045 4093 8189|
+-----+------------------------------------------+
6!:2 '3 ($:^:(<:@[]1:)^:(0<[)>:) 10' NB. snugly fits in a minute 58.5831  We need to define ack by name to be able to put it in a table, because $: is a horrible, ugly beast and lashes out at anyone who attempts to understand it. It is self-reference, where self is defined as the largest verb phrase containing it. table is an adverb and so would love to become part of the verb phrase if you give it the chance, so you have to trap $: in a named definition to use it. ### Edit: 24 char? Years later, I found a solution which is two characters shorter. (0&<~(<:@#~$:/@,1:^:)>:)


It's a lot slower, though: 3 ack 8 takes over a minute on my machine. This is because (1) I use a fold / instead of iteration, so J probably has to remember more things than usual, and (2) while 0&<~ performs the same calculation as (0<[), it actually gets executed n+1 times before taking the recursive step when invoking m ack n0&< happens to be idempotent, so it doesn't ruin the calculation, but n gets big fast and ack is highly recursive.

I am doubtful that a more powerful machine could push the new code under a minute, because this is a computer where the old code can find 3 ack 10 in less than 15 seconds.

## JavaScript, ES6, 41 34 bytes

f=(m,n)=>m?f(m-1,!n||f(m,n-1)):n+1


Run this in a latest Firefox Console and it will create a function called f which you can call with different values of m and n like

f(3,2) // returns 29


OR

try the code below in a latest Firefox

f=(m,n)=>m?f(m-1,!n||f(m,n-1)):n+1

B.onclick=_=>alert(f(+M.value, +N.value))
#M,#N{max-width:15px;border: 1px solid;border-width:0 0 1px 0}
<div>f(<input id=M />,<input id=N />)</div><br><button id=B>Evaluate</button>

• Testing this with 0,1 in Chrome gives no result. Oct 22, 2014 at 8:36
• Pl read, this only works in latest Firefox due to ES6 Oct 22, 2014 at 8:42
• Wow... we have 4 practically identical JS solutions, all at 34 bytes. I've never seen that before. Dec 15, 2015 at 20:24

# Python 2.7.8 - 80, 54, 48, 46 45

A=lambda m,n:m and A(m-1,n<1or A(m,n-1))or-~n


(Credits to xnor!)

More readable, but with 1 more character:

A=lambda m,n:n+(m<1or A(m-1,n<1or A(m,n-1))-n)


Not that I had to set sys.setrecursionlimit(10000) in order to get a result for A(3,10). Further golfing using logical indexing did not work due to the dramatically growing recursion depth.

• I get a syntax error on the 1else. The start letter e causes trouble for the parser because numbers can be written like 1e3.
– xnor
Oct 22, 2014 at 8:09
• Saved a few chars switching to and/or: A=lambda m,n:m and A(m-1,n<1or A(m,n-1))or-~n
– xnor
Oct 22, 2014 at 8:17
• @xnor: Thanks for the tip! See this discussion for the parsing issue issue. Python 2.7.8 accepts 1else, most other versions don't. Oct 22, 2014 at 8:29
• Thanks for pointer about 1else; it lets me squeeze out a char here and probably other places. But darn is it version-specific! Python 2.7.4 doesn't allow it. Are you using an online version with 2.7.8 or would I have to download it?
– xnor
Oct 22, 2014 at 9:02
• @xnor: It's an offline installation. ideone.com, e.g., fails to parse 1else as well. Oct 22, 2014 at 9:09

# C - 41 bytes

Nothing to it--the small limits mean that all the required values can be calculated in less than 1 second by naively following the function definition.

A(m,n){return!m?n+1:A(m-1,n?A(m,n-1):1);}

int main()
{
int m,n;
for(m = 0; m <= 3; m++)
for(n = 0; n <= 10; n++)
printf("%d %d %d\n", m,n,A(m,n));
return 0;
}


# Javascript ES6 (34)

a=(m,n)=>m?a(m-1,n?a(m,n-1):1):n+1


### Implementation:

a=(m,n)=>m?a(m-1,n?a(m,n-1):1):n+1
td[colspan="2"] input{width: 100%;}
<table><tbody><tr><td>m=</td><td><input id="m" type="number" value="0" /></td></tr><tr><td>n=</td><td><input id="n" type="number" value="0" /></td></tr><tr><td colspan="2"><input type="button" value="Calculate!" onclick="document.getElementById('out').value=a(document.getElementById('m').value, document.getElementById('n').value)" /></td></tr><tr><td colspan="2"><input id="out" disabled="disabled" type="text" /></td></tr></tbody></table>

# Coq, 40

nat_rec _ S(fun _ b n=>nat_iter(S n)b 1)


This is a function of type nat -> nat -> nat. Since Coq only allows the construction of total functions, it also serves as a formal proof that the Ackermann recurrence is well-founded.

Demo:

Welcome to Coq 8.4pl6 (November 2015)

Coq < Compute nat_rec _ S(fun _ b n=>nat_iter(S n)b 1) 3 10.
= 8189
: nat


Note: Coq 8.5, released after this challenge, renamed nat_iter to Nat.iter.

# JavaScript (ES6) - 34

A=(m,n)=>m?A(m-1,!n||A(m,n-1)):n+1


And a test:

> A=(m,n)=>m?A(m-1,!n||A(m,n-1)):n+1;s=new Date().getTime();console.log(A(3,10),(new Date().getTime() - s)/1000)
8189 16.441


# Jelly, 13 bytes

+‘ð;x’ß@ƒ1ðỊ?


Try it online!

Again uses proud haskeller's approach: A(m+1,n) is A(m,?) applied n+1 times to 1.

The base case should've been simply A(0,n) = n+1, but unfortunately it (the code at the bottom) times out badly, so I had to use A(m,n) = m+n+1 if m <= 1 instead.

### How it works

+‘ð;x’ß@ƒ1ðỊ?    Dyadic link. Left = m, Right = n
AAðBBBBBBBðC?    If C, do A; otherwise, do B
Ị     If m <= 1...
+‘               Return m+n+1
;x’ß@ƒ1       Otherwise do this:
;x            Concatenate m to n copies of m (which gives n+1 copies of m)
’           Decrement each number (n+1 copies of m-1)
ƒ1       Reduce with seed 1...
ß@         This function, args swapped (since new m is on the right side)


# Jelly, 12 bytes, times out on A(3,9)

;x’ß@ƒ1ð‘}ḷ?


Try it online!

## Mathematica, 46 bytes

0~a~n_:=n+1
m_~a~n_:=a[m-1,If[n<1,1,a[m,n-1]]]


Takes pretty much exactly a minute for a[3,10]. Note that Mathematica's default recursion limit is too small for a[3,8] and beyond (at least on my machine), but that can be fixed by configuring

let a2 = tail $iterate (a1 !!) 1 -- the function 0 times -- etc  Then we simply add another level of iteration based on that pattern, and do some pointless juggling. • You could alias iterate to a single letter name i.e. i=iterate;ack=i ...  Oct 23, 2014 at 11:55 • @proudhaskeller oh yeah, didn't think about that. Thanks! Borrowing your operator-name use as well. Oct 23, 2014 at 12:11 # Julia, 3431 28 bytes m\n=m>0?~-m\(n<1||m\~-n):n+1  This is a named anonymous function. It is a straightforward implementation of the recursive definition, abusing Julia's ability to redefine operators. Try it online! # Python 3, 58 49 bytes a=lambda m,n:m and a(m-1,n and a(m,n-1)or 1)or-~n  Try it online! I didn't see a Python 3 answer (there was a Python 2), so I decided to fix that and put my own answer in here. (-(43+21) thanks to caird, -9 thanks to HN) • 78 bytes Apr 28, 2021 at 9:29 • 57 bytes Apr 28, 2021 at 9:33 • 49 bytes Apr 28, 2021 at 9:37 • I was going to mention that n<1or would save even more bytes but then it becomes identical to the aforementioned Python 2 answer... – Neil Apr 28, 2021 at 10:16 # Racket 67 (define(a m n)(if(= m 0)(+ n 1)(a(- m 1)(if(= n 0)1(a m(- n 1))))))  ## Javascript with lambdas, 34 A=(m,n)=>m?A(m-1,n?A(m,n-1):1):n+1  A tipical answer, can't make anything shorter. # Tiny Lisp, 70 (out of competition) This runs out of competition, as the language is newer than the question, and it also doesn't succeed to run the (A 3 10) as required in the question, due to a stack overflow. (d A(q((m n)(i m(i n(A(s m 1)(A m(s n 1)))(A(s m 1)1))(s n(s 0 1)))))) This defines a function A which calculates the Ackermann function. Formatted: (d A (q( (m n) (i m (i n (A (s m 1) (A m (s n 1) ) ) (A (s m 1) 1 ) ) (s n (s 0 1) ) ) ) ) )  We are using all builtin macros (d (define) and q (quote) and i (if)) and one builtin function (s – subtract) here. i executes its true part when the condition is a number > 0 (and otherwise the false part), so we don't have to do an explicit comparison here. s is the only arithmetic operation available, we use it for the n-1/m-1, as well as as (s n (s 0 1)) for n+1. Tiny lisp is using tail recursion optimization, but this only helps for the outer A call in the result, not for the A(m, n-1) call which is used for the parameters. With my tiny lisp implementation in Ceylon on the JVM, it works up to (A 3 5) = 253, but it seems to break down when trying to calculate (A 2 125) directly (which should give the same result). If calculating that after (A 3 4) = 125, the JVM seems to got to optimize the functions enough to inline some intermediate function calls in my interpreter, allowing more recursion depth. Strange. The reference implementation gets up to (A 3 5) = 253 and also (A 2 163) = 329, but doesn't succeed (A 2 164), and therefore even less (A 3 6) = (A 2 253). • this might be competitive save for the whitespace and parenthesis ;) – cat Dec 26, 2015 at 20:14 # brainfuck, 90 bytes >>>>+>,>,<<[>[>[-[->>>+<<<]<[->+>>+<<<]>-[-<+>]>+>>>>>]<[->+>>]]<[>>+[-<<<+>>>]<<-]<<<]>>.  Try it online! Assumes an implementation with arbitrary sized cell size, with IO as numbers. -6 bytes if you don't mind using negative cells. Finishes in about 30 seconds for 3,8 in the linked interpreter, provided you tick the correct settings. Type inputted numbers prepended with \s, e.g. 3,9 is \3\9. # Ral, 81 61 bytes ,,/:11+-111:++:+:++:+:+?/0=:1/-/1/10*-0*1:+?+0=11:+?1++/1:+?.  Try it online! Commented Code Stack after Description ,, 0, m, n Input m and n Loop: /:11+-111:++:+:++:+:+? r, n, m Jump to SmallM if m < 2 /0=:1/-/1/10*- r, m-1, 1, m, n-1 Perform [m, n] => [m-1, 1, m, n-1] 0*1:+? r, m-1, 1, m, n-1 Jump to Loop if n>0 +0= r, m-1, 1 Remove the last two elements 11:+? r, m-1, 1 Jump to Loop SmallM: 1++ n+m+1, r Perform [m, n] => [n+m+1] /1:+? n+m+1 Jump to Loop if r>0 . Print n+m+1  Explanation Because there are no built in function calls (or recursion) in Ral, we need to simulate the stack frame. One way to do that is as follows: 1. Start with the stack [m, n] 2. Repeatedly apply the following rules, until a single element remains: • [..., 0, n] => [..., n+1] • [..., m, 0] => [..., m-1, 1] • [..., m, n] => [..., m-1, m, n-1] 3. Print the remaining element However, since there is no way to know the current length of the stack in Ral, an additional flag is stored to test if the recursion should continue, resulting in the following rules: • [..., 0, 0, n] => Print n+1 and exit • [..., 1, 0, n] => [..., n+1] • [..., m, 0] => [..., m-1, 1] • [..., m, n] => [..., m-1, 1, m, n-1] # K (ngn/k), 16 bytes {x/[;1]1+}/[;1+]  Try it online! Same structure as my K answer for fast growing hierarchy, and happens to be the same length too. Can compute f(0..3)(0..10) in a couple seconds. Again uses "A(m+1,n) is A(m,?) applied n+1 times to 1." {x/[;1]1+}/[;1+] input: m, returns function that takes n { }/[;1+] iterate m times on the function 1+ (increment): 1+ create a function that increments n and then x/[;1] run x/[n+1;1] == (n+1)x/1 == iterate function x n+1 times to 1  ## APL, 31 {⍺=0:⍵+1⋄⍵=0:1∇⍨⍺-1⋄(⍺-1)∇⍺∇⍵-1}  Pretty straightforward. Uses the ⍨ character once to save one byte by reversing arguments. Takes m as the left argument and n as the right argument. TryAPL.org # Go, 260243240 122 bytes I didn't see that the question allowed anon funcs. far from competitive but i'm learning this language and i wanted to test it out. func (m,n int)int{r:=0 switch{case m==0&&n!=0:r=n+1 case m!=0&&n==0:r=a(m-1,1) case m!=0&&n!=0:r=a(m-1,a(m,n-1))} return r} use it like go run ack.go and then supply two numbers, m and n. if m>4 or n>30, execution time may be in excess of half a minute. for m=3 n=11: $ time go run ack
16381
real    0m1.434s
user    0m1.432s
sys     0m0.004s


edit: saved total 17 bytes by switching to switch over if/else and dot-imports

• You can do even better! switch 0 {case m:r=n+1 case n:r=a(m-1,1) default:r=a(m-1,a(m,n-1))} Go's switch statement is so beautifully flexible! May 14, 2016 at 2:11
• @EMBLEM Thanks, it's been so long since I've written a line of Go, but I'm glad to see there are other Go-golfers about :D
– cat
May 14, 2016 at 3:02

## Ceylon, 8887 85

alias I=>Integer;I a(I m,I n)=>m<1then n+1else(n<1then a(m-1,1)else a(m-1,a(m,n-1)));


This is a straightforward implementation. Formatted:

alias I => Integer;
I a(I m, I n) =>
m < 1
then n + 1
else (n < 1
then a(m - 1, 1)
else a(m - 1, a(m, n - 1)));


The alias saves just one byte, without it (with writing Integer instead of I) we would get to 86 bytes. Another two bytes can be saved by replacing == 0 by < 1 twice.

With the default settings of ceylon run, it will work up to A(3,12) = 32765 (and A(4,0) = 13), but A(3,13) (and therefore also A(4,1)) will throw a stack overflow error. (A(3,12) takes about 5 seconds, A(3,11) about 3 on my computer.)

Using ceylon run-js (i.e. running the result of compiling to JavaScript on node.js) is a lot slower (needs 1 min 19 s for A(3,10)), and breaks already for A(3, 11) with a »Maximum call stack size exceeded« (using default settings) after running for 1 min 30 s.

## Ceylon without recursion, 228

As a bonus, here is a non-recursive version (longer, of course, but immune to stack overflows – might get an out-of-memory error at some point).

import ceylon.collection{A=ArrayList}Integer a(Integer[2]r){value s=A{*r};value p=s.addAll;while(true){if(exists m=s.pop()){if(exists n=s.pop()){if(n<1){p([m+1]);}else if(m<1){p([n-1,1]);}else{p([n-1,n,m-1]);}}else{return m;}}}}


Formatted:

import ceylon.collection {
A=ArrayList
}

Integer a(Integer[2] r) {
value s = A { *r };
value p = s.addAll;
while (true) {
if (exists m = s.pop()) {
if (exists n = s.pop()) {
if (n < 1) {
p([m + 1]);
} else if (m < 1) {
p([n - 1, 1]);
} else {
p([n - 1, n, m - 1]);
}
} else {
// stack is empty
return m;
}
}
}
}


It is quite slower on my computer than the recursive version: A(3,11) takes 9.5 seconds, A(3,12) takes 34 seconds, A(3,13) takes 2:08 minutes, A(3,14) takes 8:25 minutes. (I originally had a version using lazy iterables instead of the tuples I now have, which was even much slower, with the same size).

A bit faster (21 seconds for A(3,12)) (but also one byte longer) is a version using s.push instead of s.addAll, but that needed to be called several times to add multiple numbers, as it takes just a single Integer each. Using a LinkedList instead of an ArrayList is a lot slower.

## R - 54 52

I've used this as an excuse to try and get my head around R, so this is probably really badly done:)

a=function(m,n)"if"(m,a(m-1,"if"(n,a(m,n-1),1)),n+1)


Example run

> a(3,8)
[1] 2045


I get a stack overflow for anything beyond that

## T-SQL- 222

I thought I would try to get T-SQL to do it as well. Used a different method because recursion isn't that nice in SQL. Anything over 4,2 bombs it.

DECLARE @m INT=4,@n INT=1;WITH R AS(SELECT 2 C, 1 X UNION ALL   SELECT POWER(2,C),X+1FROM R)SELECT IIF(@m=0,@n+1,IIF(@m=1,@n+2,IIF(@m=2,2*@n+3,IIF(@m=3,POWER(2,@n+3)-3,IIF(@m=4,(SELECT TOP(1)C FROM R WHERE x= @n+3)-3,-1)))))

• for your R submisson, looks like you don't need the {} although there's no helping the stack overflow limit, since R doesn't have TCO... Mar 21, 2018 at 12:45
• @Giuseppe thanks ... in my defense, I was new to it then :) Mar 21, 2018 at 17:25

# Tcl, 67 bytes

proc tcl::mathfunc::A m\ n {expr {$m?A($m-1,$n?A($m,$n-1):1):$n+1}}


Try it online!

# Tcl, 77 bytes

proc A m\ n {expr {$m?[A [expr$m-1] [expr {$n?[A$m [expr $n-1]]:1}]]:$n+1}}


Try it online!

In the online compiler it fails to run due to time-out, but in a local Tcl interpreter it runs well. I profiled of each root call to A function, to see how much time the calculation took for each pair {m,n} subject to be tested:

m=0, n=0, A=1, time=3.5e-5 seconds
m=0, n=1, A=2, time=2e-6 seconds
m=0, n=2, A=3, time=8e-6 seconds
m=0, n=3, A=4, time=1e-6 seconds
m=0, n=4, A=5, time=2e-6 seconds
m=0, n=5, A=6, time=1e-6 seconds
m=0, n=6, A=7, time=1e-6 seconds
m=0, n=7, A=8, time=1e-6 seconds
m=0, n=8, A=9, time=1e-6 seconds
m=0, n=9, A=10, time=0.0 seconds
m=0, n=10, A=11, time=1e-6 seconds
m=1, n=0, A=2, time=4e-6 seconds
m=1, n=1, A=3, time=6e-6 seconds
m=1, n=2, A=4, time=1e-5 seconds
m=1, n=3, A=5, time=1.2e-5 seconds
m=1, n=4, A=6, time=1.5e-5 seconds
m=1, n=5, A=7, time=2e-5 seconds
m=1, n=6, A=8, time=2e-5 seconds
m=1, n=7, A=9, time=2.6e-5 seconds
m=1, n=8, A=10, time=3e-5 seconds
m=1, n=9, A=11, time=3e-5 seconds
m=1, n=10, A=12, time=3.3e-5 seconds
m=2, n=0, A=3, time=8e-6 seconds
m=2, n=1, A=5, time=2.2e-5 seconds
m=2, n=2, A=7, time=3.9e-5 seconds
m=2, n=3, A=9, time=6.3e-5 seconds
m=2, n=4, A=11, time=9.1e-5 seconds
m=2, n=5, A=13, time=0.000124 seconds
m=2, n=6, A=15, time=0.000163 seconds
m=2, n=7, A=17, time=0.000213 seconds
m=2, n=8, A=19, time=0.000262 seconds
m=2, n=9, A=21, time=0.000316 seconds
m=2, n=10, A=23, time=0.000377 seconds
m=3, n=0, A=5, time=2.2e-5 seconds
m=3, n=1, A=13, time=0.000145 seconds
m=3, n=2, A=29, time=0.000745 seconds
m=3, n=3, A=61, time=0.003345 seconds
m=3, n=4, A=125, time=0.015048 seconds
m=3, n=5, A=253, time=0.059836 seconds
m=3, n=6, A=509, time=0.241431 seconds
m=3, n=7, A=1021, time=0.971836 seconds
m=3, n=8, A=2045, time=3.908884 seconds
m=3, n=9, A=4093, time=15.926341 seconds
m=3, n=10, A=8189, time=63.734713 seconds


It fails for the last pair {m,n}={3,10}, as it takes a very little more than one minute.

For higher values of m, it will be needed to increase the recursionlimit value.

I coult get it shorter to 65 bytes, but it will not meet the question's requirement "Your function must be able to find the value of A(m,n) for m ≤ 3 and n ≤ 10 in less than a minute.". Without the {} it will timeout on TIO and not do the demo of the last two entries.

# Tcl, 65 bytes

proc tcl::mathfunc::A m\ n {expr $m?A($m-1,$n?A($m,$n-1):1):$n+1}


Try it online!

# C (gcc), 36 bytes

A(m,n){m=m?A(m-1,n?A(m,n-1):1):n+1;}


Naive strategy with a little bit of optimization.

Try it online!

# Verilog (Icarus Verilog), 81 bytes

Based on the C answer. Verilog is not really intended for this kind of thing - Yosys crashes on this - but it's possible with Icarus.

function automatic[15:0]a;input[15:0]m,n;a=(m==0)?n+1:(n==0)?a(m-1,1):a(m-1,a(m,n-1));endfunction


Testbench:

module test();

function automatic[31:0]a;input[31:0]m,n;a=m?a(m-1,n?a(m,n-1):1):n+1;endfunction

initial begin
$display("%d", a(3,10)); end endmodule  Icarus is really not intended for this kind of constant abuse, though: $ time ./a.out
8189

real    0m55.422s
user    0m55.375s
sys     0m0.063s


# Haskell, 47 bytes

a 0 n=n+1
a m 0=a(m-1)1
a m n=a(m-1)(a m (n-1))


(Run using ghci <filename>) Almost directly copies from the problem definition (thank you pattern matching!).

• You may want to include a TIO link so everyone can test your submission?
– RGS
Mar 22, 2020 at 18:38
• I think a(m-1)(a m (n-1)) can be a(m-1).a m\$n-1. Mar 23, 2020 at 1:44