<s>
Solomonoff	B-General_Concept
's	I-General_Concept
theory	I-General_Concept
of	I-General_Concept
inductive	I-General_Concept
inference	I-General_Concept
is	O
a	O
mathematical	O
proof	O
that	O
if	O
a	O
universe	O
is	O
generated	O
by	O
an	O
algorithm	O
,	O
then	O
observations	O
of	O
that	O
universe	O
,	O
encoded	O
as	O
a	O
dataset	O
,	O
are	O
best	O
predicted	O
by	O
the	O
smallest	O
executable	O
archive	O
of	O
that	O
dataset	O
.	O
</s>
<s>
This	O
formalization	O
of	O
Occam	O
's	O
razor	O
for	O
induction	O
was	O
introduced	O
by	O
Ray	O
Solomonoff	O
,	O
based	O
on	O
probability	O
theory	O
and	O
theoretical	B-General_Concept
computer	I-General_Concept
science	I-General_Concept
.	O
</s>
<s>
Without	O
loss	O
of	O
generality	O
,	O
we	O
can	O
thus	O
consider	O
that	O
any	O
observable	O
data	O
is	O
a	O
finite	O
bit	B-Data_Structure
string	I-Data_Structure
.	O
</s>
<s>
The	O
universal	O
prior	O
probability	O
of	O
any	O
prefix	O
p	O
of	O
a	O
computable	O
sequence	O
x	O
is	O
the	O
sum	O
of	O
the	O
probabilities	O
of	O
all	O
programs	O
(	O
for	O
a	O
universal	B-Architecture
computer	I-Architecture
)	O
that	O
compute	O
something	O
starting	O
with	O
p	O
.	O
Given	O
some	O
p	O
and	O
any	O
computable	O
but	O
unknown	O
probability	O
distribution	O
from	O
which	O
x	O
is	O
sampled	O
,	O
the	O
universal	O
prior	O
and	O
Bayes	O
 '	O
theorem	O
can	O
be	O
used	O
to	O
predict	O
the	O
yet	O
unseen	O
parts	O
of	O
x	O
in	O
optimal	O
fashion	O
.	O
</s>
<s>
The	O
more	O
computing	O
power	O
they	O
are	O
given	O
,	O
the	O
closer	O
their	O
predictions	O
are	O
to	O
the	O
predictions	O
of	O
inductive	O
inference	O
(	O
their	O
mathematical	B-Algorithm
limit	I-Algorithm
is	O
Solomonoff	O
's	O
inductive	O
inference	O
)	O
.	O
</s>
<s>
Another	O
direction	O
of	O
inductive	O
inference	O
is	O
based	O
on	O
E	O
.	O
Mark	O
Gold	O
's	O
model	O
of	O
learning	O
in	O
the	O
limit	B-Algorithm
from	O
1967	O
and	O
has	O
developed	O
since	O
then	O
more	O
and	O
more	O
models	O
of	O
learning	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
process	O
of	O
inductive	O
inference	O
is	O
performed	O
by	O
an	O
abstract	O
automaton	O
called	O
an	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
(	O
Burgin	O
,	O
2005	O
)	O
.	O
</s>
<s>
Inductive	O
Turing	B-Architecture
machines	I-Architecture
represent	O
the	O
next	O
step	O
in	O
the	O
development	O
of	O
computer	B-General_Concept
science	I-General_Concept
providing	O
better	O
models	O
for	O
contemporary	O
computers	O
and	O
computer	O
networks	O
(	O
Burgin	O
,	O
2001	O
)	O
and	O
forming	O
an	O
important	O
class	O
of	O
super-recursive	O
algorithms	O
as	O
they	O
satisfy	O
all	O
conditions	O
in	O
the	O
definition	O
of	O
algorithm	O
.	O
</s>
<s>
Namely	O
,	O
each	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
is	O
a	O
type	O
of	O
effective	O
method	O
in	O
which	O
a	O
definite	O
list	O
of	O
well-defined	O
instructions	O
for	O
completing	O
a	O
task	O
,	O
when	O
given	O
an	O
initial	O
state	O
,	O
will	O
proceed	O
through	O
a	O
well-defined	O
series	O
of	O
successive	O
states	O
,	O
eventually	O
terminating	O
in	O
an	O
end-state	O
.	O
</s>
<s>
The	O
difference	O
between	O
an	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
and	O
a	O
Turing	B-Architecture
machine	I-Architecture
is	O
that	O
to	O
produce	O
the	O
result	O
a	O
Turing	B-Architecture
machine	I-Architecture
has	O
to	O
stop	O
,	O
while	O
in	O
some	O
cases	O
an	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
can	O
do	O
this	O
without	O
stopping	O
.	O
</s>
<s>
This	O
condition	O
is	O
satisfied	O
by	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
,	O
as	O
their	O
results	O
are	O
exhibited	O
after	O
a	O
finite	O
number	O
of	O
steps	O
,	O
but	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
do	O
not	O
always	O
tell	O
at	O
which	O
step	O
the	O
result	O
has	O
been	O
obtained	O
.	O
</s>
<s>
Simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
are	O
equivalent	O
to	O
other	O
models	O
of	O
computation	O
.	O
</s>
<s>
More	O
advanced	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
are	O
much	O
more	O
powerful	O
.	O
</s>
<s>
It	O
is	O
proved	O
(	O
Burgin	O
,	O
2005	O
)	O
that	O
limiting	O
partial	O
recursive	O
functions	O
,	O
trial	O
and	O
error	O
predicates	O
,	O
general	O
Turing	B-Architecture
machines	I-Architecture
,	O
and	O
simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
are	O
equivalent	O
models	O
of	O
computation	O
.	O
</s>
<s>
However	O
,	O
simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
and	O
general	O
Turing	B-Architecture
machines	I-Architecture
give	O
direct	O
constructions	O
of	O
computing	O
automata	O
,	O
which	O
are	O
thoroughly	O
grounded	O
in	O
physical	O
machines	O
.	O
</s>
<s>
Simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
and	O
general	O
Turing	B-Architecture
machines	I-Architecture
are	O
related	O
to	O
limiting	O
partial	O
recursive	O
functions	O
and	O
trial	O
and	O
error	O
predicates	O
as	O
Turing	B-Architecture
machines	I-Architecture
are	O
related	O
to	O
partial	O
recursive	O
functions	O
and	O
lambda-calculus	O
.	O
</s>
<s>
Note	O
that	O
only	O
simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
have	O
the	O
same	O
structure	O
(	O
but	O
different	O
functioning	O
semantics	O
of	O
the	O
output	O
mode	O
)	O
as	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
Other	O
types	O
of	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
have	O
an	O
essentially	O
more	O
advanced	O
structure	O
due	O
to	O
the	O
structured	O
memory	O
and	O
more	O
powerful	O
instructions	O
.	O
</s>
<s>
Some	O
researchers	O
confuse	O
computations	O
of	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
with	O
non-stopping	O
computations	O
or	O
with	O
infinite	O
time	O
computations	O
.	O
</s>
<s>
First	O
,	O
some	O
of	O
computations	O
of	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
halt	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
of	O
conventional	O
Turing	B-Architecture
machines	I-Architecture
,	O
some	O
halting	O
computations	O
give	O
the	O
result	O
,	O
while	O
others	O
do	O
not	O
give	O
.	O
</s>
<s>
Second	O
,	O
some	O
non-stopping	O
computations	O
of	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
give	O
results	O
,	O
while	O
others	O
do	O
not	O
give	O
.	O
</s>
<s>
Rules	O
of	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
determine	O
when	O
a	O
computation	O
(	O
stopping	O
or	O
non-stopping	O
)	O
gives	O
a	O
result	O
.	O
</s>
<s>
Namely	O
,	O
an	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
produces	O
output	O
from	O
time	O
to	O
time	O
and	O
once	O
this	O
output	O
stops	O
changing	O
,	O
it	O
is	O
considered	O
the	O
result	O
of	O
the	O
computation	O
.	O
</s>
<s>
Third	O
,	O
in	O
contrast	O
to	O
the	O
widespread	O
misconception	O
,	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
give	O
results	O
(	O
when	O
it	O
happens	O
)	O
always	O
after	O
a	O
finite	O
number	O
of	O
steps	O
(	O
in	O
finite	O
time	O
)	O
in	O
contrast	O
to	O
infinite	O
and	O
infinite-time	O
computations	O
.	O
</s>
<s>
There	O
are	O
two	O
main	O
distinctions	O
between	O
conventional	O
Turing	B-Architecture
machines	I-Architecture
and	O
simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
The	O
first	O
distinction	O
is	O
that	O
even	O
simple	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
can	O
do	O
much	O
more	O
than	O
conventional	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
The	O
second	O
distinction	O
is	O
that	O
a	O
conventional	O
Turing	B-Architecture
machine	I-Architecture
always	O
informs	O
(	O
by	O
halting	O
or	O
by	O
coming	O
to	O
a	O
final	O
state	O
)	O
when	O
the	O
result	O
is	O
obtained	O
,	O
while	O
a	O
simple	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
in	O
some	O
cases	O
does	O
inform	O
about	O
reaching	O
the	O
result	O
,	O
while	O
in	O
other	O
cases	O
(	O
where	O
the	O
conventional	O
Turing	B-Architecture
machine	I-Architecture
is	O
helpless	O
)	O
,	O
it	O
does	O
not	O
inform	O
.	O
</s>
<s>
Indeed	O
,	O
everyday	O
desktop	O
computer	O
applications	O
like	O
word	O
processors	O
and	O
spreadsheets	O
spend	O
most	O
of	O
their	O
time	O
waiting	O
in	O
event	B-General_Concept
loops	I-General_Concept
,	O
and	O
do	O
not	O
terminate	O
until	O
directed	O
to	O
do	O
so	O
by	O
users	O
.	O
</s>
<s>
Evolutionary	O
approach	O
to	O
inductive	O
inference	O
is	O
accomplished	O
by	O
another	O
class	O
of	O
automata	O
called	O
evolutionary	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
(	O
Burgin	O
and	O
Eberbach	O
,	O
2009	O
;	O
2012	O
)	O
.	O
</s>
<s>
An	O
evolutionary	O
inductive	O
Turing	B-Architecture
machine	I-Architecture
is	O
a	O
(	O
possibly	O
infinite	O
)	O
sequence	O
E	O
=	O
{	O
A[t]	O
;	O
t	O
=	O
1	O
,	O
2	O
,	O
3	O
,	O
...	O
}	O
of	O
inductive	O
Turing	B-Architecture
machines	I-Architecture
A[t]	O
each	O
working	O
on	O
generations	O
X[t]	O
which	O
are	O
coded	O
as	O
words	O
in	O
the	O
alphabet	O
of	O
the	O
machines	O
A[t]	O
.	O
</s>
