This is an assignment of an introductory course of complexity theory and I need to find a way to do the following:
Given $n,m \in \Bbb N$, is $n \le m!$ ?
The idea is to provide a Post Machine that can decide this in an efficient way, using $n,m$ in a binary codification.
We know that the factorial isn't efficient, so the problem actually is just to find a way to decide this, if it's possible.
I know how to compare if $n\le m$, but the factorial is my problem. I don't how how to compute $m!$ with a Post Machine, if possible, in polynomial-time.
I guess that the most simple way to do this is comparing $n$ with factorials of numbers that are lower than $m$, but the factorial it's still my problem.
My question, is there an algorithm that can help me?