Is modulus capable of universal computation?

Question

I heard once that you could translate any digital circuit into a modulus operation, perhaps modulusing against different numbers?

It was a long while ago though and don't remember where I heard it.

Is this true or does it ring any bells?

Not quite it, but... https://en.wikipedia.org/wiki/FRACTRAN – Pseudonym Oct 22 '15 at 03:37 — Pseudonym, Oct 22 '15 at 03:37
very strange and interesting – Alan Wolfe Oct 22 '15 at 03:43 — Alan Wolfe, Oct 22 '15 at 03:43
@Pseudonym Make a full-fledged answer? – Yuval Filmus Oct 22 '15 at 07:47 — Yuval Filmus, Oct 22 '15 at 07:47

score 6 · Accepted Answer · answered Oct 22 '15 at 07:51

You have to distinguish between two meanings of computation:

Computation of functions on a finite domain (e.g. circuits).
Computation of functions on an infinite domain (e.g. Turing machines).

A single division operation cannot be universal in terms of the second meaning, since division is total: it always halts (even division by zero halts). However, it could possibly be universal in terms of the first meaning. For example, if we encode the input using different primes, then the Chinese Remainder Theorem shows that $x \mapsto n \pmod{x}$ is universal, that is, for every Boolean function $f$ on some set $S$ of finite primes there is a number $n$ such that for every $x \in S$, $n \mod{x} = f(x)$.

Is modulus capable of universal computation?

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