Definition 3.2 (Generalized tex2html_wrap_inline$K_T$)
Given any TM T, define
Compare Schnorr's ``process complexity'' for MTMs [#!Schnorr:73!#,#!Vyugin:98!#].
Definition 3.3 (tex2html_wrap_inline$K^M, K^E, K^G$ based on Invariance Theorem)
Consider Def. 2.4.
Let C denote a set of TMs with universal TM UC ().
We drop the index T, writing
This is justified by an appropriate Invariance Theorem
there is a positive constant c such
for all x,
since the size of the compiler that translates arbitrary
programs for T into equivalent programs for UC does not depend on x.
Definition 3.4 (tex2html_wrap_inline$Km_T,Km^M,Km^E,Km^G$)
Given TM T and ,
Consider Def. 2.4.
If C denotes a set of TMs with universal TM UC, then define
KmC(x) = KmUC(x).
KmC is a generalization of Schnorr's [#!Schnorr:73!#]
and Levin's [#!Levin:73a!#] complexity measure KmM for MTMs.
Describability issues.K(x) is not computable by a halting program
[#!Kolmogorov:65!#,#!Solomonoff:64!#,#!Chaitin:69!#], but obviously
G-computable or describable; the z with 0.z =
is even enumerable. Even KE(x) is describable, using the following algorithm:
Run all EOM programs in ``dovetail style'' such
that the n-th step of the i-th program is executed in the n+i-th
); whenever a program outputs x, place it
(or its prefix read so far) in a tentative list L of x-computing
programs or program prefixes; whenever an element of L produces output
delete it from L; whenever an element of L requests an
additional input bit, update L accordingly. After every change of
L replace the current estimate of KE(x) by the length of the
shortest element of L. This estimate will eventually stabilize forever.
Theorem 3.1KG(x) is not describable.
Proof. Identify finite bitstrings with the integers they represent.
If KG(x) were describable then also
where g is any fixed recursive function, and also
Since the number of descriptions p with
l(p) < n-O(1)
2n-O(1), but the number of strings x with l(x) = n equals 2n,
most x cannot be compressed by more than O(1) bits; that is,
for most x.
From (9) we
KG(f(x)) > logg(x) - O(1) for large enough x, because f(x)
picks out one of the incompressible
However, obviously we also would have
using the encoding of Def. 2.8.
Contradiction for quickly growing g with low complexity, such as