We have two functions: $$ f: \{0,1\}^* \to \{0,1\}^* $$ $$ g: \{0,1\}^* \to \{0,1\}^* $$ that commute: $$ f[g(x)] = g[f(x)] $$
These two functions can be calculated in polynomial time (in the length of the input). Moreover, the outputs have the same length of the inputs: $|f(x)| = |x|$ and $|g(x)| = |x|$ .
A trivial example of functions that commute can be easily constructed by splitting the strings into two parts and defining: $$ f(x,y) = ( a(x), y ) $$ and $$ g(x,y) = ( x, b(y) ) $$ where the functions $a(x)$ and $b(y)$ can be calculated in polynomial time (in their inputs).
I was able to construct slightly more complex examples, but not much more complex. In all the examples, the evolution obtained by repeatedly applying $f$ seems to be independent of the evolution obtained by repeatedly applying $g$. More rigorously, in my examples, the following proposition holds:
Proposition 1
There are two functions $n'$ and $m'$ depending on $n$ and $m$, at most polynomial, such that there is an algorithm that, for any integers $n$ and $m$, calculates the function $f^n[g^m(x)]$, operating in polynomial time (in the length of its input), and taking the following inputs: the numbers $n$ and $m$, $f^{n'}(x)$, and $g^{m'}(x)$.
I remark that this happens even if $n$ and $m$ increase exponentially in $|x|$.
In the trivial example above, setting $n'=n$ and $m'=m$, we see that $f^n(x)= ( a^n(x), y )$ and $g^m(x) = (x, b^m(y) )$, from which it is easy to calculate $f^n[g^m(x,y)] = ( a^n(x), b^m(y) ) $.
Is Prop. 1 a general theorem? Alternatively, is there a counter-example to Prop. 1?
Literature references, improved formalization of the problem, nomenclature, and any other suggestion are welcome.
