<s>
A	O
computer-assisted	B-Algorithm
proof	I-Algorithm
is	O
a	O
mathematical	O
proof	O
that	O
has	O
been	O
at	O
least	O
partially	O
generated	O
by	O
computer	O
.	O
</s>
<s>
Most	O
computer-aided	B-Algorithm
proofs	I-Algorithm
to	O
date	O
have	O
been	O
implementations	O
of	O
large	O
proofs-by-exhaustion	O
of	O
a	O
mathematical	O
theorem	O
.	O
</s>
<s>
The	O
idea	O
is	O
to	O
use	O
a	O
computer	B-Application
program	I-Application
to	O
perform	O
lengthy	O
computations	O
,	O
and	O
to	O
provide	O
a	O
proof	O
that	O
the	O
result	O
of	O
these	O
computations	O
implies	O
the	O
given	O
theorem	O
.	O
</s>
<s>
In	O
1976	O
,	O
the	O
four	O
color	O
theorem	O
was	O
the	O
first	O
major	O
theorem	O
to	O
be	O
verified	O
using	O
a	O
computer	B-Application
program	I-Application
.	O
</s>
<s>
Attempts	O
have	O
also	O
been	O
made	O
in	O
the	O
area	O
of	O
artificial	B-Application
intelligence	I-Application
research	I-Application
to	O
create	O
smaller	O
,	O
explicit	O
,	O
new	O
proofs	O
of	O
mathematical	O
theorems	O
from	O
the	O
bottom	O
up	O
using	O
automated	O
reasoning	O
techniques	O
such	O
as	O
heuristic	B-Algorithm
search	O
.	O
</s>
<s>
Such	O
automated	B-Application
theorem	I-Application
provers	I-Application
have	O
proved	O
a	O
number	O
of	O
new	O
results	O
and	O
found	O
new	O
proofs	O
for	O
known	O
theorems	O
.	O
</s>
<s>
One	O
method	O
for	O
using	O
computers	O
in	O
mathematical	O
proofs	O
is	O
by	O
means	O
of	O
so-called	O
validated	B-Algorithm
numerics	I-Algorithm
or	O
rigorous	O
numerics	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
controlling	O
,	O
enclosing	O
and	O
propagating	O
round-off	O
and	O
truncation	O
errors	O
using	O
for	O
example	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Computer-assisted	B-Algorithm
proofs	I-Algorithm
are	O
the	O
subject	O
of	O
some	O
controversy	O
in	O
the	O
mathematical	O
world	O
,	O
with	O
Thomas	O
Tymoczko	O
first	O
to	O
articulate	O
objections	O
.	O
</s>
<s>
Those	O
who	O
adhere	O
to	O
Tymoczko	O
's	O
arguments	O
believe	O
that	O
lengthy	O
computer-assisted	B-Algorithm
proofs	I-Algorithm
are	O
not	O
,	O
in	O
some	O
sense	O
,	O
'	O
real	O
 '	O
mathematical	O
proofs	O
because	O
they	O
involve	O
so	O
many	O
logical	O
steps	O
that	O
they	O
are	O
not	O
practically	O
verifiable	B-Application
by	O
human	O
beings	O
,	O
and	O
that	O
mathematicians	O
are	O
effectively	O
being	O
asked	O
to	O
replace	O
logical	O
deduction	O
from	O
assumed	O
axioms	O
with	O
trust	O
in	O
an	O
empirical	O
computational	O
process	O
,	O
which	O
is	O
potentially	O
affected	O
by	O
errors	O
in	O
the	O
computer	B-Application
program	I-Application
,	O
as	O
well	O
as	O
defects	O
in	O
the	O
runtime	O
environment	O
and	O
hardware	O
.	O
</s>
<s>
Other	O
mathematicians	O
believe	O
that	O
lengthy	O
computer-assisted	B-Algorithm
proofs	I-Algorithm
should	O
be	O
regarded	O
as	O
calculations	O
,	O
rather	O
than	O
proofs	O
:	O
the	O
proof	O
algorithm	O
itself	O
should	O
be	O
proved	O
valid	O
,	O
so	O
that	O
its	O
use	O
can	O
then	O
be	O
regarded	O
as	O
a	O
mere	O
"	O
verification	O
"	O
.	O
</s>
<s>
Arguments	O
that	O
computer-assisted	B-Algorithm
proofs	I-Algorithm
are	O
subject	O
to	O
errors	O
in	O
their	O
source	O
programs	O
,	O
compilers	O
,	O
and	O
hardware	O
can	O
be	O
resolved	O
by	O
providing	O
a	O
formal	O
proof	O
of	O
correctness	O
for	O
the	O
computer	B-Application
program	I-Application
(	O
an	O
approach	O
which	O
was	O
successfully	O
applied	O
to	O
the	O
four-color	O
theorem	O
in	O
2005	O
)	O
as	O
well	O
as	O
replicating	O
the	O
result	O
using	O
different	O
programming	O
languages	O
,	O
different	O
compilers	O
,	O
and	O
different	O
computer	O
hardware	O
.	O
</s>
<s>
Another	O
possible	O
way	O
of	O
verifying	O
computer-aided	B-Algorithm
proofs	I-Algorithm
is	O
to	O
generate	O
their	O
reasoning	O
steps	O
in	O
a	O
machine-readable	O
form	O
,	O
and	O
then	O
use	O
a	O
proof	O
checker	O
program	O
to	O
demonstrate	O
their	O
correctness	O
.	O
</s>
<s>
However	O
,	O
this	O
approach	O
of	O
using	O
a	O
computer	B-Application
program	I-Application
to	O
prove	O
the	O
output	O
of	O
another	O
program	O
correct	O
does	O
not	O
appeal	O
to	O
computer	B-Algorithm
proof	I-Algorithm
skeptics	O
,	O
who	O
see	O
it	O
as	O
adding	O
another	O
layer	O
of	O
complexity	O
without	O
addressing	O
the	O
perceived	O
need	O
for	O
human	O
understanding	O
.	O
</s>
<s>
Another	O
argument	O
against	O
computer-aided	B-Algorithm
proofs	I-Algorithm
is	O
that	O
they	O
lack	O
mathematical	O
elegance	O
—	O
that	O
they	O
provide	O
no	O
insights	O
or	O
new	O
and	O
useful	O
concepts	O
.	O
</s>
<s>
An	O
additional	O
philosophical	O
issue	O
raised	O
by	O
computer-aided	B-Algorithm
proofs	I-Algorithm
is	O
whether	O
they	O
make	O
mathematics	O
into	O
a	O
quasi-empirical	O
science	O
,	O
where	O
the	O
scientific	O
method	O
becomes	O
more	O
important	O
than	O
the	O
application	O
of	O
pure	O
reason	O
in	O
the	O
area	O
of	O
abstract	O
mathematical	O
concepts	O
.	O
</s>
<s>
Inclusion	O
in	O
this	O
list	O
does	O
not	O
imply	O
that	O
a	O
formal	O
computer-checked	O
proof	O
exists	O
,	O
but	O
rather	O
,	O
that	O
a	O
computer	B-Application
program	I-Application
has	O
been	O
involved	O
in	O
some	O
way	O
.	O
</s>
<s>
In	O
2010	O
,	O
academics	O
at	O
The	O
University	O
of	O
Edinburgh	O
offered	O
people	O
the	O
chance	O
to	O
"	O
buy	O
their	O
own	O
theorem	O
"	O
created	O
through	O
a	O
computer-assisted	B-Algorithm
proof	I-Algorithm
.	O
</s>
