# [FOM] Universality for unbounded computations

Todd Rowland rowland at wolfram.com
Tue Nov 13 21:29:13 EST 2007

```A definition for universality needs to take into account how the term will
be used, and what the examples are. Because of Wolfram's Principle of
Computational Equivalence (PCE), we can expect many new examples besides
those already known. There are ECA 110, the Wolfram 2,3 machine which are
known to emulate any other unbounded computation. Moreover, part of the
PCE says that there will be many systems, in the form of scientific and
practical applications, that will be shown to be universal.  It is
important to know whether something is universal, and so we need a
definition which can be used in practice.

In the traditional framework, when the computations of a deterministic
system X are interpreted by the halting of X, the computations of X can be
emulated by a Turing machine with halting states. In contrast, for
unbounded computations there is nothing analogous to halting.
Intuitively, computations are being performed, and any attempt to augment
the system by adding halting conditions is unnatural (even if it were
meaningful).

The traditional definition of universality is useful for its textbook
uses, e.g., a concrete defintion is needed to establish the existence of
universal machines.  It captures well the intuition of universal
computation for finite computations.  But it does not match the intuition
behind emulating unbounded machines.

The main point which has been under discussion recently is what initial
conditions should be allowed in an emulation. It is fitting to mention the
Thue-Morse sequence as an example of an initial condition background that
is simple enough to be allowed, while being non-repetitive.

One way to construct this sequence is recursively.

B={0};
B[n_] = B[n-1] + F[B[n-1]]
where F[s] == 1-s == BitNot[s] == ReplaceAll[s, {0->1, 1->0}] (replace all
0's with 1's and vice versa)

Not only does this have very simple behavior, but it is also reducible.
If one wants to know the value at position x, there is a finite state
machine that can be run on the binary digits of x.  It is also worth
mentioning that the simplicity of the program is not a very good condition
for a definition. As Wolfram has shown, very simple rules can be
universal, but the simplicity of the behavior of a rule is a very good
indication of its universality status.  It should not be a question of how
tedious the encoding is or how much storage is needed, especially if the
encoding is reducible.

As for the formalization of the PCE, as Kovas Boguta says, it is most
useful as a scientific principle.  That does not rule out that there is a
formal theorem that can show its limitations and applicability.

On a somewhat different topic, concerning the status of Alex Smith's
Complex Systems paper, he is still writing it up.

Anyone can let me know if they are interested in being a referee and
giving a timely report.

Todd Rowland

Managing Editor
Complex Systems
rowland at wolfram.com
```