Chapter 11: The Notion of Computation

Section 12: Universality in Turing Machines and Other Systems

[Universal] recursive functions

The general recursive functions from page 907 provided an early example of universality (see page 907). That such functions are universal can be demonstrated by showing for example that they can emulate any tag system. With the state of a 2-color tag system encoded as an integer according to FromDigits[Reverse[list] + 1, 3] the following takes the rule for any such tag system (in the first form from page 894) and yields a primitive recursive function that emulates a single step in its evolution:

TSToPR[{n_, rule_}] := Fold[c @@ Flatten[{#1, Array[p, #2], c[r[z, c[r[p[1], s], c[r[z, p[2]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[2]]], p[1]]], p[#2]]}] & , c[c[r[p[1], s], p[1], c[r[p[1], r[z, c[s, c[s, s]]]], c[c[r[z, c[r[p[1], s], c[r[z, c[s, z]], c[r[p[1], r[z, c[r[p[1], s], c[r[z, p[2]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[2]]], p[1]]]], p[2], p[3]]], p[1]]], p[1], p[1]], p[1]], p[2]]], p[n + 1], MapIndexed[c[r[z, c[r[p[1], p[4]], p[2], p[3], p[4]]], c[r[z, r[c[s, z], c[r[c[s, c[s, z]], z], p[2]]]], p[Length[#2] + 1]], #1[[1]], #1[[2]]] & , Nest[Partition[#1, 2] & , Table[Nest[c[s, #1] & , z, FromDigits[Reverse[IntegerDigits[i, 2, n] /. rule] + 1, 3]], {i, 0, 2^n - 1}], n - 1], {0, n - 1}]], Range[n, 1, -1]]

(For tag system (a) from page 94 this yields a primitive recursive function of size 325.) The result of t steps of evolution is in general given in terms of this function f by Nest[f, init, t], or equivalently r[p[1], f][t, init]. Any fixed number of steps of evolution can thus be emulated by applying a primitive recursive function. But if one wants to find out what happens after an arbitrarily large number of steps, one needs to use the μ operator, yielding a general recursive function. (So for example μ[r[p[1], f]][init] returns the smallest t for which the tag system reaches state {}—and never returns if the tag system does not halt.) Note that the same basic approach can be used to emulate Turing machines with recursive functions; the Turing machine configuration {s, list, n} can be encoded by a integer such as

2^FromDigits[Reverse[Take[list, n-1]]] 3^FromDigits[Take[list, {n+1, -1}]] 5^list[[n]] 7^s

From Stephen Wolfram: A New Kind of Science [citation]