<s>
Automated	B-Application
theorem	I-Application
proving	I-Application
(	O
also	O
known	O
as	O
ATP	O
or	O
automated	B-Application
deduction	I-Application
)	O
is	O
a	O
subfield	O
of	O
automated	O
reasoning	O
and	O
mathematical	O
logic	O
dealing	O
with	O
proving	O
mathematical	O
theorems	O
by	O
computer	B-Application
programs	I-Application
.	O
</s>
<s>
Automated	O
reasoning	O
over	O
mathematical	O
proof	O
was	O
a	O
major	O
impetus	O
for	O
the	O
development	O
of	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
Frege	O
's	O
Begriffsschrift	B-Application
(	O
1879	O
)	O
introduced	O
both	O
a	O
complete	O
propositional	O
calculus	O
and	O
what	O
is	O
essentially	O
modern	O
predicate	O
logic	O
.	O
</s>
<s>
This	O
approach	O
was	O
continued	O
by	O
Russell	O
and	O
Whitehead	O
in	O
their	O
influential	O
Principia	O
Mathematica	B-Application
,	O
first	O
published	O
1910	O
–	O
1913	O
,	O
and	O
with	O
a	O
revised	O
second	O
edition	O
in	O
1927	O
.	O
</s>
<s>
However	O
,	O
shortly	O
after	O
this	O
positive	O
result	O
,	O
Kurt	O
Gödel	O
published	O
On	O
Formally	O
Undecidable	O
Propositions	O
of	O
Principia	O
Mathematica	B-Application
and	O
Related	O
Systems	O
(	O
1931	O
)	O
,	O
showing	O
that	O
in	O
any	O
sufficiently	O
strong	O
axiomatic	O
system	O
there	O
are	O
true	O
statements	O
which	O
cannot	O
be	O
proved	O
in	O
the	O
system	O
.	O
</s>
<s>
In	O
1954	O
,	O
Martin	O
Davis	O
programmed	O
Presburger	O
's	O
algorithm	O
for	O
a	O
JOHNNIAC	B-Device
vacuum	O
tube	O
computer	O
at	O
the	O
Institute	O
for	O
Advanced	O
Study	O
in	O
Princeton	O
,	O
New	O
Jersey	O
.	O
</s>
<s>
More	O
ambitious	O
was	O
the	O
Logic	B-Application
Theory	I-Application
Machine	I-Application
in	O
1956	O
,	O
a	O
deduction	O
system	O
for	O
the	O
propositional	O
logic	O
of	O
the	O
Principia	O
Mathematica	B-Application
,	O
developed	O
by	O
Allen	O
Newell	O
,	O
Herbert	O
A	O
.	O
Simon	O
and	O
J	O
.	O
C	O
.	O
Shaw	O
.	O
</s>
<s>
Also	O
running	O
on	O
a	O
JOHNNIAC	B-Device
,	O
the	O
Logic	B-Application
Theory	I-Application
Machine	I-Application
constructed	O
proofs	O
from	O
a	O
small	O
set	O
of	O
propositional	O
axioms	O
and	O
three	O
deduction	O
rules	O
:	O
modus	O
ponens	O
,	O
(	O
propositional	O
)	O
variable	O
substitution	O
,	O
and	O
the	O
replacement	O
of	O
formulas	O
by	O
their	O
definition	O
.	O
</s>
<s>
The	O
"	O
heuristic	O
"	O
approach	O
of	O
the	O
Logic	B-Application
Theory	I-Application
Machine	I-Application
tried	O
to	O
emulate	O
human	O
mathematicians	O
,	O
and	O
could	O
not	O
guarantee	O
that	O
a	O
proof	O
could	O
be	O
found	O
for	O
every	O
valid	O
theorem	O
even	O
in	O
principle	O
.	O
</s>
<s>
Gilmore	O
's	O
program	O
used	O
conversion	O
to	O
disjunctive	B-Application
normal	I-Application
form	I-Application
,	O
a	O
form	O
in	O
which	O
the	O
satisfiability	O
of	O
a	O
formula	O
is	O
obvious	O
.	O
</s>
<s>
It	O
follows	O
that	O
an	O
automated	B-Application
theorem	I-Application
prover	I-Application
will	O
fail	O
to	O
terminate	O
while	O
searching	O
for	O
a	O
proof	O
precisely	O
when	O
the	O
statement	O
being	O
investigated	O
is	O
undecidable	O
in	O
the	O
theory	O
being	O
used	O
,	O
even	O
if	O
it	O
is	O
true	O
in	O
the	O
model	O
of	O
interest	O
.	O
</s>
<s>
For	O
this	O
,	O
it	O
is	O
generally	O
required	O
that	O
each	O
individual	O
proof	O
step	O
can	O
be	O
verified	O
by	O
a	O
primitive	B-Architecture
recursive	I-Architecture
function	I-Architecture
or	O
program	O
,	O
and	O
hence	O
the	O
problem	O
is	O
always	O
decidable	O
.	O
</s>
<s>
Since	O
the	O
proofs	O
generated	O
by	O
automated	B-Application
theorem	I-Application
provers	I-Application
are	O
typically	O
very	O
large	O
,	O
the	O
problem	O
of	O
proof	O
compression	O
is	O
crucial	O
and	O
various	O
techniques	O
aiming	O
at	O
making	O
the	O
prover	O
's	O
output	O
smaller	O
,	O
and	O
consequently	O
more	O
easily	O
understandable	O
and	O
checkable	O
,	O
have	O
been	O
developed	O
.	O
</s>
<s>
Another	O
distinction	O
is	O
sometimes	O
drawn	O
between	O
theorem	B-Application
proving	I-Application
and	O
other	O
techniques	O
,	O
where	O
a	O
process	O
is	O
considered	O
to	O
be	O
theorem	B-Application
proving	I-Application
if	O
it	O
consists	O
of	O
a	O
traditional	O
proof	O
,	O
starting	O
with	O
axioms	O
and	O
producing	O
new	O
inference	O
steps	O
using	O
rules	O
of	O
inference	O
.	O
</s>
<s>
Other	O
techniques	O
would	O
include	O
model	B-Application
checking	I-Application
,	O
which	O
,	O
in	O
the	O
simplest	O
case	O
,	O
involves	O
brute-force	O
enumeration	O
of	O
many	O
possible	O
states	O
(	O
although	O
the	O
actual	O
implementation	O
of	O
model	B-Application
checkers	I-Application
requires	O
much	O
cleverness	O
,	O
and	O
does	O
not	O
simply	O
reduce	O
to	O
brute	O
force	O
)	O
.	O
</s>
<s>
There	O
are	O
hybrid	O
theorem	B-Language
proving	I-Language
systems	I-Language
which	O
use	O
model	B-Application
checking	I-Application
as	O
an	O
inference	O
rule	O
.	O
</s>
<s>
Commercial	O
use	O
of	O
automated	B-Application
theorem	I-Application
proving	I-Application
is	O
mostly	O
concentrated	O
in	O
integrated	O
circuit	O
design	O
and	O
verification	O
.	O
</s>
<s>
Since	O
the	O
Pentium	B-Device
FDIV	I-Device
bug	I-Device
,	O
the	O
complicated	O
floating	B-General_Concept
point	I-General_Concept
units	I-General_Concept
of	O
modern	O
microprocessors	O
have	O
been	O
designed	O
with	O
extra	O
scrutiny	O
.	O
</s>
<s>
AMD	O
,	O
Intel	O
and	O
others	O
use	O
automated	B-Application
theorem	I-Application
proving	I-Application
to	O
verify	O
that	O
division	O
and	O
other	O
operations	O
are	O
correctly	O
implemented	O
in	O
their	O
processors	O
.	O
</s>
<s>
In	O
the	O
late	O
1960s	O
agencies	O
funding	O
research	O
in	O
automated	B-Application
deduction	I-Application
began	O
to	O
emphasize	O
the	O
need	O
for	O
practical	O
applications	O
.	O
</s>
<s>
One	O
of	O
the	O
first	O
fruitful	O
areas	O
was	O
that	O
of	O
program	O
verification	O
whereby	O
first-order	B-Application
theorem	I-Application
provers	I-Application
were	O
applied	O
to	O
the	O
problem	O
of	O
verifying	O
the	O
correctness	O
of	O
computer	B-Application
programs	I-Application
in	O
languages	O
such	O
as	O
Pascal	O
,	O
Ada	O
,	O
etc	O
.	O
</s>
<s>
This	O
was	O
the	O
first	O
automated	B-Application
deduction	I-Application
system	O
to	O
demonstrate	O
an	O
ability	O
to	O
solve	O
mathematical	O
problems	O
that	O
were	O
announced	O
in	O
the	O
Notices	O
of	O
the	O
American	O
Mathematical	O
Society	O
before	O
solutions	O
were	O
formally	O
published	O
.	O
</s>
<s>
First-order	O
theorem	B-Application
proving	I-Application
is	O
one	O
of	O
the	O
most	O
mature	O
subfields	O
of	O
automated	B-Application
theorem	I-Application
proving	I-Application
.	O
</s>
<s>
More	O
expressive	O
logics	O
,	O
such	O
as	O
Higher-order	B-Algorithm
logics	I-Algorithm
,	O
allow	O
the	O
convenient	O
expression	O
of	O
a	O
wider	O
range	O
of	O
problems	O
than	O
first	O
order	O
logic	O
,	O
but	O
theorem	B-Application
proving	I-Application
for	O
these	O
logics	O
is	O
less	O
well	O
developed	O
.	O
</s>
<s>
The	O
quality	O
of	O
implemented	O
systems	O
has	O
benefited	O
from	O
the	O
existence	O
of	O
a	O
large	O
library	O
of	O
standard	O
benchmark	O
examples	O
—	O
the	O
Thousands	O
of	O
Problems	O
for	O
Theorem	O
Provers	O
(	O
TPTP	O
)	O
Problem	O
Library	O
—	O
as	O
well	O
as	O
from	O
the	O
CADE	B-General_Concept
ATP	I-General_Concept
System	I-General_Concept
Competition	I-General_Concept
(	O
CASC	O
)	O
,	O
a	O
yearly	O
competition	O
of	O
first-order	O
systems	O
for	O
many	O
important	O
classes	O
of	O
first-order	O
problems	O
.	O
</s>
<s>
E	B-Language
is	O
a	O
high-performance	O
prover	O
for	O
full	O
first-order	O
logic	O
,	O
but	O
built	O
on	O
a	O
purely	O
equational	O
calculus	O
,	O
originally	O
developed	O
in	O
the	O
automated	O
reasoning	O
group	O
of	O
Technical	O
University	O
of	O
Munich	O
under	O
the	O
direction	O
of	O
Wolfgang	O
Bibel	O
,	O
and	O
now	O
at	O
Baden-Württemberg	O
Cooperative	O
State	O
University	O
in	O
Stuttgart	O
.	O
</s>
<s>
Otter	B-Application
,	O
developed	O
at	O
the	O
Argonne	O
National	O
Laboratory	O
,	O
is	O
based	O
on	O
first-order	O
resolution	O
and	O
paramodulation	O
.	O
</s>
<s>
Otter	B-Application
has	O
since	O
been	O
replaced	O
by	O
Prover9	B-Application
,	O
which	O
is	O
paired	O
with	O
Mace4	B-Application
.	O
</s>
<s>
E	B-Language
and	O
SETHEO	O
have	O
been	O
combined	O
(	O
with	O
other	O
systems	O
)	O
in	O
the	O
composite	O
theorem	O
prover	O
E-SETHEO	O
.	O
</s>
<s>
It	O
has	O
won	O
the	O
FOF	O
division	O
(	O
among	O
other	O
divisions	O
)	O
at	O
the	O
CADE	B-General_Concept
ATP	I-General_Concept
System	I-General_Concept
Competition	I-General_Concept
regularly	O
since	O
2001	O
.	O
</s>
<s>
SPASS	B-Application
is	O
a	O
first	O
order	O
logic	O
theorem	O
prover	O
with	O
equality	O
.	O
</s>
<s>
This	O
is	O
developed	O
by	O
the	O
research	O
group	O
Automation	O
of	O
Logic	O
,	O
Max	O
Planck	O
Institute	O
for	O
Computer	B-General_Concept
Science	I-General_Concept
.	O
</s>
