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Traditionally, decidability of some problem class implies there is a
halting algorithm that prints out the answer, given a problem from
the class. We now relax the notion of decidability by allowing for
infinite computations on EOMs or GTMs whose answers
converge after finite yet possibly unpredictable time.
Essentially, an answer needs to be correct for almost all the time,
and may be incorrect for at most finitely many initial time steps
(compare the concept of *computability in the limit*
[20,19,33,15]
and super-recursive algorithms [6,8]).

**Definition 2.11** (Weak decidability)
Consider a

*characteristic function*
:

if

satisfies a certain
property, and

otherwise. The problem of deciding whether or not
some

satisfies that property is

*weakly decidable* if

is describable (compare Def.

2.10).

**Example 2.1**
*Is a given string a halting program
for a given MTM? The problem is not decidable in the traditional sense
(no halting algorithm solves the general halting problem [43]),
but weakly
decidable and even E-decidable, by a trivial algorithm: print ``0'' on
first output square; simulate the MTM on work tapes and apply it to ,
once it halts after having read no more than bits
print ``1'' on first output square.
*

**Example 2.2**
*It is weakly decidable whether a finite bitstring
is a program for a given TM. Algorithm: print ``0''; feed bitwise into
the internally simulated TM whenever it requests a new input bit; once the
TM has requested bits, print ``1''; if it requests an additional
bit, print ``0''. After finite time the output will stabilize forever.
*

**Example 2.3**
*It is weakly decidable whether number theory is consistent
(contrast this with Gödel's famous negative result [18]).
*

**Theorem 2.1** (Convergence Problem)
Given a GTM, it is not weakly decidable whether a
finite bitstring is a converging program, or whether
some of the output bits will fluctuate forever.

**Proof.**
(Based on the standard diagonalization trick [11,18,43];
compare a related result for analytic TMs [12,21]).
Let us write
if there is a
such that
.
Let us write
if 's output fluctuates
forever in response to (e.g., by flipping from 1 to zero
and back forever).
Let
be an effective
enumeration of all GTMs. Uniquely encode all pairs of finite strings
in
as finite strings
.
Suppose there were
a GTM U such that **(*)**: for all :
if
,
and
otherwise.
Then one could construct a GTM with
if
, and
otherwise.
Let be the index of , then
if
,
otherwise
.
By **(*)**, however,
if
, and
if
.
Contradiction.

** Next:** Complexity of Constructive Descriptions
** Up:** Preliminaries
** Previous:** Formally Describable Functions
Juergen Schmidhuber
2003-02-13