Most traditional computability theory focuses on
properties of halting programs.
Given an MTM or EOM or GTM T with halt instruction
for ``p computes x on T and halts''.
Much of this paper, however, deals with programs
that never halt, and with TMs that do not need halt instructions.
Definition 2.1 (Convergence)
the input string or program read by TM T. Let Tt(p) denote T's finite output
string after t instructions. We say that p and p's output
stabilize and converge towards
iff for each
there exists a postive integer
tn such that for all
Tt(p)n = xn and
Then we write
Although each beginning or prefix of x eventually becomes stable during the possibly
infinite computation, there need not be a halting program that computes
an upper bound of stabilization time, given any p and prefix size.
Compare the concept of computability in the limit
[#!Gold:65!#,#!Putnam:65!#,#!Freyvald:74!#] and [#!Greg:57!#,#!Mostowski:57!#].
Definition 2.2 (TM-Specific Individual Describability)
Given a TM T,
is T-describable or T-computable iff
there is a finite
Objects with infinite shortest descriptions on T are not T-describable.
Definition 2.3 (Universal TMs)
Let C denote a set of TMs. C has a
universal element if there is a TM
such that for each
there exists a constant string
(the compiler) such
that for all possible programs p, if
Definition 2.4 (M, E, G)
Let M denote the set of MTMs, E denote the set of EOMs, G denote the
set of GTMs.
M,E,G all have universal elements, according to
the fundamental compiler theorem (for instance, a fixed compiler
can translate arbitrary LISP programs into equivalent FORTRAN programs).
Definition 2.5 (Individual Describability)
Let C denote a set
of TMs with universal element UC. Some
C-describable or C-computable if it is UC-describable.
E-describable strings are called enumerable. G-describable strings
are called formally describable or simply describable.
Example 2.1 (Pseudorandom universe based on halting problem)
Let x be a universe history in the style of Example 1.1.
Suppose its pseudorandom generator's n-th output bit PRG(n) is 1 if
the n-th program of an ordered list of all possible programs halts, and
0 otherwise. Since PRG(n) is describable, x is too. But there is no
halting algorithm computing PRG(n) for all n, otherwise the
halting problem would be solvable, which it is not [#!Turing:36!#].
Hence in general there is no computer that outputs xand onlyx
without ever editing some previously computed history.
Definition 2.6 (Always converging TMs)
always converges if for all of its possible
there is an
For example, MTMs and EOMs converge always. GTMs do not.
Definition 2.7 (Approximability)
Let 0.x denote a real
0.x is approximable by TM T if
there is a
such that for each real
exists a t0 such that
0.x is approximable if there is at least
one GTM T as above -- compare (2).
If 0.x is approximable, then x is describable, and vice versa.