An $m$-ary function that represents all $n$-ary functions

Let $m=4$ and $S:=\mathbb{Z_k}$ of residues modulo $k\geq 3$ (all arithmetical operations will be considered modulo $k$).

We will define $f(x,y,a,b)$ as follows: $$ \begin{equation} f(x,y,a,b)= \begin{cases} (x+1) & \text{if } y=x\\ 0 & \text{if } y=(x+1)\\ a & \text{if } y\not\in\{x,x+1\} \wedge x=0\\ b & \text{if } y\not\in\{x,x+1\} \wedge x\not=0\\ \end{cases} \end{equation} $$

As we will see, this essentially smuggles two unrelated functions under one umbrella and uses the equality of first two arguments as the distinguishing factor.

The first two lines of the definition of $f$ allow us to use any single variable to produce explicit constants: $$\begin{array}{lll} 0 & = & f(x,f(x,x)) \\ 1 & = & f(0, 0) = f(f(x,f(x,x)), f(x,f(x,x))) \\ & \ldots &\\ (k-1) & = & f(k-2, k-2) \\ \end{array}$$

We can also implement addition of a constant to variable: $$\begin{array}{lll} x+1 & = & f(x,x) \\ x+2 & = & f(x+1, x+1) = f(f(x,x),f(x,x))\\ & \ldots &\\ \end{array}$$

Furthermore, since we are working modulo $k$, so we can also subtract constants too (subtracting $1$ is equivalent to adding $(k-1)$).

Since $|S|\geq 3$, the value $(x+2)$ is always distinct both from $x$ and $(x+1)$ so an expression of the form $f(x,(x+2),a,b)$ can serve as the if-then-else primitive: $$ITE(x,a,b) := f(x, x+2, a, b)$$

This, in turn, gives us the array indexing primitive (given an array, it returns $x$-th member of the array $a$ or the last element if $x$ is too big): $$\begin{equation} SELECT(x, [a_0, a_1, \ldots, a_t])= \begin{cases} a_0 & \text{if }t=0\\ ITE(x, a_0, SELECT\left(x-1, [a_1, \ldots, a_t])\right) & \text{otherwise}\\ \end{cases} \end{equation}$$

Note that the constants, addition/subtraction, if-then-else and select primitives are all just syntactic sugar and simply abbreviate a much longer valid expression.

Now, any unary function $g(x_1)$ can be represented as a valid expression; after all, a unary function is fully defined by the array of its values: $$g(x_1)=SELECT(x_1, [g(0),g(1),\ldots,g(k-1)])$$

Functions of higher arity just involve nesting of selects: $$\begin{array}{llll} g(x_1,x_2) & = & SELECT(x_1, [ & \\ & & & SELECT(x_2, [g(0,0), g(0,1), \ldots, g(0,k-1)]),\\ & & & SELECT(x_2, [g(1,0), g(1,1), \ldots, g(1,k-1)]),\\ & & & \ldots \\ & & & SELECT(x_2, [g(k-1,0), g(k-1,1), \ldots, g(k-1,k-1)])\\ & & ])\\ \end{array}$$

Generalization to any higher dimension is straightforward.

Thus, we have shown that $m=4$ suffices for any $|S|\geq 3$ and one $f$ can be used for any $n$ (note that the definition of $f$ depends on $S$, so although our definition of it works "uniformly" for any $|S|$, it is not a single function $f$ for all $S$; just for all $n$).

This leaves the question open for $m=2$ and $m=3$.

There is a universal binary ($m=2$) function for any finite domain.

As in Peter Košinár's answer, take to $S$ be the integers modulo $|S|.$ I assume $|S|\geq 3.$ Define

$$f(x,y)=\begin{cases}x+1&\text{ if $x=y,$}\\ 0&\text{ otherwise.}\end{cases}$$

Some easy gizmos:

• the successor function $s(x)=x+1$ can be defined in terms of $f$ by $s(x)=f(x,x)$
• the constant $0$ can be defined by $f(f(x,x),x)$
• any constant $c$ can be defined by expressions of the form $s(s(\dots s(s(0))\dots)),$ with $c$ applications of $s$ (taking a representative $c\in\{0,1,\dots,|S|-1\}$)

I will give constructions for:

1. the function $e_i:S^1\to S$ that sends $i$ to $1$ and everything else to $0$
2. a function $NOR:S^2\to S$ such that $NOR(0,0)=1$ and $NOR(0,1)=NOR(1,0)=NOR(1,1)=0.$ (It doesn't matter what happens with other inputs.)
3. a function $d:S^{|S|}\to S$ that outputs $c$ if the $c$'th input (counting from $0$) is $1$ and all other inputs are $0.$ For example $d(1,0,0,\dots,0)=0$ and $d(0,0,1,0,0,\dots,0)=2.$ (It doesn't matter what $d$ does to inputs not of this form for any $c.$)

(Mnemonic: "d" and "e" decode and encode, from a one-hot representation.)

The constructions are:

1. Use $e_i(x)=f(s^{|S|-i}(x),0)$ for $i\in\{0,1,\dots,|S|-1\}.$ Here $s^{|S|-i}$ means applying $s$ exactly $|S|-i$ times, effectively adding $|S|-i.$ This works because $s^{|S|-i}(x)=0$ if and only if $e_i(x)=1.$

2. $NOR(x,y)=e_1(f(x,y)).$

3. I will inductively construct functions $d_i:S^i\to S$ such that $d_i(x)=c$ whenever $x$ has a $1$ in position $c$ and zeroes elsewhere. I'll take the base case to be $i=2,$ which is easy: $d_i(x_0,x_1)=1.$ Given a construction for $d_i,$ construct $d_{i+1}$ by $d_{i+1}(x_0,\dots,x_{i+1})=f(d_i(OR(x_0,x_1),x_2,x_3,\dots,x_{i+1}),d_i(x_1,OR(x_0,x_2),x_3,\dots,x_{i+1})),$ where $OR(x,y)=NOR(NOR(x,y),NOR(x,y)).$ Then take $d=d_{|S|}.$

We can now express an arbitrary function $g:S^n\to S$ in three parts: first convert all the inputs to binary using the $n|S|$ expressions of the form $e_i(x_j)$; then use the fact that $NOR$ is universal for binary functions to express each indicator function of whether the output should be $k,$ i.e. the functions $e_k(g(x)),$ in terms of the $e_i(x_j)$; then use $d$ to convert this binary representation of the output of $g$ into the correct output - in notation, $d(g_0(x),\dots,g_{|S|-1}(x))$ where $g_k(x)=e_k(g(x))$ (which can be expressed without $g$ as mentioned previously).