<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
mathematical	O
logic	O
,	O
satisfiability	B-Application
modulo	I-Application
theories	I-Application
(	O
SMT	O
)	O
is	O
the	O
problem	O
of	O
determining	O
whether	O
a	O
mathematical	O
formula	O
is	O
satisfiable	O
.	O
</s>
<s>
It	O
generalizes	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
SAT	O
)	O
to	O
more	O
complex	O
formulas	O
involving	O
real	O
numbers	O
,	O
integers	O
,	O
and/or	O
various	O
data	B-General_Concept
structures	I-General_Concept
such	O
as	O
lists	O
,	O
arrays	B-Data_Structure
,	O
bit	B-Data_Structure
vectors	I-Data_Structure
,	O
and	O
strings	O
.	O
</s>
<s>
The	O
name	O
is	O
derived	O
from	O
the	O
fact	O
that	O
these	O
expressions	O
are	O
interpreted	O
within	O
(	O
"	O
modulo	O
"	O
)	O
a	O
certain	O
formal	O
theory	O
in	O
first-order	O
logic	O
with	O
equality	O
(	O
often	O
disallowing	O
quantifiers	B-Language
)	O
.	O
</s>
<s>
SMT	B-Application
solvers	I-Application
are	O
tools	O
which	O
aim	O
to	O
solve	O
the	O
SMT	O
problem	O
for	O
a	O
practical	O
subset	O
of	O
inputs	O
.	O
</s>
<s>
SMT	B-Application
solvers	I-Application
such	O
as	O
Z3	B-Language
and	O
cvc5	O
have	O
been	O
used	O
as	O
a	O
building	O
block	O
for	O
a	O
wide	O
range	O
of	O
applications	O
across	O
computer	B-General_Concept
science	I-General_Concept
,	O
including	O
in	O
automated	B-Application
theorem	I-Application
proving	I-Application
,	O
program	O
analysis	O
,	O
program	O
verification	O
,	O
and	O
software	O
testing	O
.	O
</s>
<s>
Since	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
is	O
already	O
NP-complete	O
,	O
the	O
SMT	O
problem	O
is	O
typically	O
NP-hard	O
,	O
and	O
for	O
many	O
theories	O
it	O
is	O
undecidable	O
.	O
</s>
<s>
The	O
resulting	O
decision	O
procedures	O
are	O
often	O
implemented	O
directly	O
in	O
SMT	B-Application
solvers	I-Application
;	O
see	O
,	O
for	O
instance	O
,	O
the	O
decidability	O
of	O
Presburger	O
arithmetic	O
.	O
</s>
<s>
SMT	O
can	O
be	O
thought	O
of	O
as	O
a	O
constraint	B-Application
satisfaction	I-Application
problem	I-Application
and	O
thus	O
a	O
certain	O
formalized	O
approach	O
to	O
constraint	B-Application
programming	I-Application
.	O
</s>
<s>
Formally	O
speaking	O
,	O
an	O
SMT	O
instance	O
is	O
a	O
formula	O
in	O
first-order	O
logic	O
,	O
where	O
some	O
function	O
and	O
predicate	B-Algorithm
symbols	O
have	O
additional	O
interpretations	O
,	O
and	O
SMT	O
is	O
the	O
problem	O
of	O
determining	O
whether	O
such	O
a	O
formula	O
is	O
satisfiable	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
imagine	O
an	O
instance	O
of	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
SAT	O
)	O
in	O
which	O
some	O
of	O
the	O
binary	O
variables	O
are	O
replaced	O
by	O
predicates	B-Algorithm
over	O
a	O
suitable	O
set	O
of	O
non-binary	O
variables	O
.	O
</s>
<s>
A	O
predicate	B-Algorithm
is	O
a	O
binary-valued	O
function	O
of	O
non-binary	O
variables	O
.	O
</s>
<s>
Example	O
predicates	B-Algorithm
include	O
linear	O
inequalities	O
(	O
e.g.	O
,	O
)	O
or	O
equalities	O
involving	O
uninterpreted	B-Language
terms	I-Language
and	O
function	B-Language
symbols	I-Language
(	O
e.g.	O
,	O
where	O
is	O
some	O
unspecified	O
function	O
of	O
two	O
arguments	O
)	O
.	O
</s>
<s>
These	O
predicates	B-Algorithm
are	O
classified	O
according	O
to	O
each	O
respective	O
theory	O
assigned	O
.	O
</s>
<s>
For	O
instance	O
,	O
linear	O
inequalities	O
over	O
real	O
variables	O
are	O
evaluated	O
using	O
the	O
rules	O
of	O
the	O
theory	O
of	O
linear	O
real	O
arithmetic	O
,	O
whereas	O
predicates	B-Algorithm
involving	O
uninterpreted	B-Language
terms	I-Language
and	O
function	B-Language
symbols	I-Language
are	O
evaluated	O
using	O
the	O
rules	O
of	O
the	O
theory	O
of	O
uninterpreted	B-Language
functions	I-Language
with	O
equality	O
(	O
sometimes	O
referred	O
to	O
as	O
the	O
empty	B-Language
theory	I-Language
)	O
.	O
</s>
<s>
Other	O
theories	O
include	O
the	O
theories	O
of	O
arrays	B-Data_Structure
and	O
list	O
structures	O
(	O
useful	O
for	O
modeling	O
and	O
verifying	O
computer	B-Application
programs	I-Application
)	O
,	O
and	O
the	O
theory	O
of	O
bit	B-Data_Structure
vectors	I-Data_Structure
(	O
useful	O
in	O
modeling	O
and	O
verifying	O
hardware	B-General_Concept
designs	I-General_Concept
)	O
.	O
</s>
<s>
Most	O
SMT	B-Application
solvers	I-Application
support	O
only	O
quantifier-free	O
fragments	O
of	O
their	O
logics	O
.	O
</s>
<s>
An	O
SMT	O
instance	O
is	O
a	O
generalization	O
of	O
a	O
Boolean	B-Algorithm
SAT	I-Algorithm
instance	O
in	O
which	O
various	O
sets	O
of	O
variables	O
are	O
replaced	O
by	O
predicates	B-Algorithm
from	O
a	O
variety	O
of	O
underlying	O
theories	O
.	O
</s>
<s>
SMT	O
formulas	O
provide	O
a	O
much	O
richer	O
modeling	B-Language
language	I-Language
than	O
is	O
possible	O
with	O
Boolean	B-Algorithm
SAT	I-Algorithm
formulas	O
.	O
</s>
<s>
For	O
example	O
,	O
an	O
SMT	O
formula	O
allows	O
one	O
to	O
model	O
the	O
datapath	B-General_Concept
operations	O
of	O
a	O
microprocessor	B-Architecture
at	O
the	O
word	O
rather	O
than	O
the	O
bit	O
level	O
.	O
</s>
<s>
By	O
comparison	O
,	O
answer	B-Application
set	I-Application
programming	I-Application
is	O
also	O
based	O
on	O
predicates	B-Algorithm
(	O
more	O
precisely	O
,	O
on	O
atomic	O
sentences	O
created	O
from	O
atomic	B-Algorithm
formula	I-Algorithm
)	O
.	O
</s>
<s>
Unlike	O
SMT	O
,	O
answer-set	O
programs	O
do	O
not	O
have	O
quantifiers	B-Language
,	O
and	O
cannot	O
easily	O
express	O
constraints	O
such	O
as	O
linear	O
arithmetic	O
or	O
difference	O
logic	O
—	O
ASP	O
is	O
at	O
best	O
suitable	O
for	O
Boolean	O
problems	O
that	O
reduce	O
to	O
the	O
free	B-Language
theory	I-Language
of	O
uninterpreted	B-Language
functions	I-Language
.	O
</s>
<s>
Implementing	O
32-bit	O
integers	O
as	O
bitvectors	B-Data_Structure
in	O
ASP	O
suffers	O
from	O
most	O
of	O
the	O
same	O
problems	O
that	O
early	O
SMT	B-Application
solvers	I-Application
faced	O
:	O
"	O
obvious	O
"	O
identities	O
such	O
as	O
x+y	O
=	O
y+x	O
are	O
difficult	O
to	O
deduce	O
.	O
</s>
<s>
Constraint	B-Application
logic	I-Application
programming	I-Application
does	O
provide	O
support	O
for	O
linear	O
arithmetic	O
constraints	O
,	O
but	O
within	O
a	O
completely	O
different	O
theoretical	O
framework	O
.	O
</s>
<s>
SMT	B-Application
solvers	I-Application
have	O
also	O
been	O
extended	O
to	O
solve	O
formulas	O
in	O
higher-order	B-Algorithm
logic	I-Algorithm
.	O
</s>
<s>
Early	O
attempts	O
for	O
solving	O
SMT	O
instances	O
involved	O
translating	O
them	O
to	O
Boolean	B-Algorithm
SAT	I-Algorithm
instances	O
(	O
e.g.	O
,	O
a	O
32-bit	O
integer	O
variable	O
would	O
be	O
encoded	O
by	O
32	O
single-bit	O
variables	O
with	O
appropriate	O
weights	O
and	O
word-level	O
operations	O
such	O
as	O
'	O
plus	O
 '	O
would	O
be	O
replaced	O
by	O
lower-level	O
logic	O
operations	O
on	O
the	O
bits	O
)	O
and	O
passing	O
this	O
formula	O
to	O
a	O
Boolean	B-Algorithm
SAT	I-Algorithm
solver	O
.	O
</s>
<s>
This	O
approach	O
,	O
which	O
is	O
referred	O
to	O
as	O
the	O
eager	O
approach	O
,	O
has	O
its	O
merits	O
:	O
by	O
pre-processing	O
the	O
SMT	O
formula	O
into	O
an	O
equivalent	O
Boolean	B-Algorithm
SAT	I-Algorithm
formula	O
existing	O
Boolean	B-Algorithm
SAT	I-Algorithm
solvers	O
can	O
be	O
used	O
"	O
as-is	O
"	O
and	O
their	O
performance	O
and	O
capacity	O
improvements	O
leveraged	O
over	O
time	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
loss	O
of	O
the	O
high-level	O
semantics	O
of	O
the	O
underlying	O
theories	O
means	O
that	O
the	O
Boolean	B-Algorithm
SAT	I-Algorithm
solver	O
has	O
to	O
work	O
a	O
lot	O
harder	O
than	O
necessary	O
to	O
discover	O
"	O
obvious	O
"	O
facts	O
(	O
such	O
as	O
for	O
integer	O
addition	O
.	O
)	O
</s>
<s>
This	O
observation	O
led	O
to	O
the	O
development	O
of	O
a	O
number	O
of	O
SMT	B-Application
solvers	I-Application
that	O
tightly	O
integrate	O
the	O
Boolean	O
reasoning	O
of	O
a	O
DPLL-style	O
search	O
with	O
theory-specific	O
solvers	O
(	O
T-solvers	O
)	O
that	O
handle	O
conjunctions	O
(	O
ANDs	O
)	O
of	O
predicates	B-Algorithm
from	O
a	O
given	O
theory	O
.	O
</s>
<s>
Dubbed	O
DPLL(T )	O
,	O
this	O
architecture	O
gives	O
the	O
responsibility	O
of	O
Boolean	O
reasoning	O
to	O
the	O
DPLL-based	O
SAT	O
solver	O
which	O
,	O
in	O
turn	O
,	O
interacts	O
with	O
a	O
solver	O
for	O
theory	O
T	O
through	O
a	O
well-defined	O
interface	O
.	O
</s>
<s>
The	O
theory	O
solver	O
only	O
needs	O
to	O
worry	O
about	O
checking	O
the	O
feasibility	O
of	O
conjunctions	O
of	O
theory	O
predicates	B-Algorithm
passed	O
on	O
to	O
it	O
from	O
the	O
SAT	O
solver	O
as	O
it	O
explores	O
the	O
Boolean	O
search	O
space	O
of	O
the	O
formula	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
theory	O
solver	O
must	O
be	O
incremental	O
and	O
backtrackable	B-Algorithm
.	O
</s>
<s>
(	O
On	O
the	O
other	O
hand	O
,	O
the	O
theory	O
of	O
real	O
closed	O
fields	O
,	O
and	O
thus	O
the	O
full	O
first	O
order	O
theory	O
of	O
the	O
real	O
numbers	O
,	O
are	O
decidable	O
using	O
quantifier	B-Language
elimination	O
.	O
</s>
<s>
Since	O
multiplication	O
by	O
constants	O
can	O
be	O
implemented	O
as	O
nested	O
additions	O
,	O
the	O
arithmetic	O
in	O
many	O
computer	B-Application
programs	I-Application
can	O
be	O
expressed	O
using	O
Presburger	O
arithmetic	O
,	O
resulting	O
in	O
decidable	O
formulas	O
.	O
</s>
<s>
Examples	O
of	O
SMT	B-Application
solvers	I-Application
addressing	O
Boolean	O
combinations	O
of	O
theory	O
atoms	O
from	O
undecidable	O
arithmetic	O
theories	O
over	O
the	O
reals	O
are	O
ABsolver	O
,	O
which	O
employs	O
a	O
classical	O
DPLL(T )	O
architecture	O
with	O
a	O
non-linear	O
optimization	O
packet	O
as	O
(	O
necessarily	O
incomplete	O
)	O
subordinate	O
theory	O
solver	O
,	O
and	O
,	O
building	O
on	O
a	O
unification	O
of	O
DPLL	B-Application
SAT-solving	O
and	O
interval	O
constraint	O
propagation	O
called	O
the	O
iSAT	O
algorithm	O
.	O
</s>
<s>
The	O
table	O
below	O
summarizes	O
some	O
of	O
the	O
features	O
of	O
the	O
many	O
available	O
SMT	B-Application
solvers	I-Application
.	O
</s>
<s>
Platform	O
Features	O
Notes	O
Name	O
OS	O
License	O
SMT-LIB	O
CVC	O
DIMACS	O
Built-in	O
theories	O
API	O
SMT-COMP	O
ABsolver	O
Linux	B-Application
CPL	O
linear	O
arithmetic	O
,	O
non-linear	O
arithmetic	O
C++	B-Language
no	O
DPLL-based	O
Alt-Ergo	B-Language
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
CeCILL-C	B-License
(	O
roughly	O
equivalent	O
to	O
LGPL	B-Application
)	O
empty	B-Language
theory	I-Language
,	O
linear	O
integer	O
and	O
rational	O
arithmetic	O
,	O
non-linear	O
arithmetic	O
,	O
polymorphic	O
arrays	B-Data_Structure
,	O
enumerated	O
datatypes	O
,	O
AC	O
symbols	O
,	O
bitvectors	B-Data_Structure
,	O
record	O
datatypes	O
,	O
quantifiers	B-Language
OCaml	B-Language
2008	O
Polymorphic	O
first-order	O
input	O
language	O
à	O
la	O
ML	O
,	O
SAT-solver	O
based	O
,	O
combines	O
Shostak-like	O
and	O
Nelson-Oppen	O
like	O
approaches	O
for	O
reasoning	O
modulo	O
theories	O
Barcelogic	O
Linux	B-Application
Proprietary	O
empty	B-Language
theory	I-Language
,	O
difference	O
logic	O
C++	B-Language
2009	O
DPLL-based	O
,	O
congruence	O
closure	O
Beaver	O
Linux	B-Application
,	O
Windows	B-Application
BSD	B-Operating_System
bitvectors	B-Data_Structure
OCaml	B-Language
2009	O
SAT-solver	O
based	O
Boolector	O
Linux	B-Application
MIT	B-License
bitvectors	B-Data_Structure
,	O
arrays	B-Data_Structure
C	B-Language
2009	O
SAT-solver	O
based	O
CVC3	O
Linux	B-Application
BSD	B-Operating_System
empty	B-Language
theory	I-Language
,	O
linear	O
arithmetic	O
,	O
arrays	B-Data_Structure
,	O
tuples	O
,	O
types	O
,	O
records	O
,	O
bitvectors	B-Data_Structure
,	O
quantifiers	B-Language
C/C	O
++	O
2010	O
proof	O
output	O
to	O
HOL	B-Algorithm
CVC4	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
,	O
FreeBSD	B-Operating_System
BSD	B-Operating_System
rational	O
and	O
integer	O
linear	O
arithmetic	O
,	O
arrays	B-Data_Structure
,	O
tuples	O
,	O
records	O
,	O
inductive	O
data	O
types	O
,	O
bitvectors	B-Data_Structure
,	O
strings	O
,	O
and	O
equality	O
over	O
uninterpreted	B-Language
function	I-Language
symbols	O
C++	B-Language
2021	O
version	O
1.8	O
released	O
May	O
2021	O
cvc5	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
BSD	B-Operating_System
rational	O
and	O
integer	O
linear	O
arithmetic	O
,	O
arrays	B-Data_Structure
,	O
tuples	O
,	O
records	O
,	O
inductive	O
data	O
types	O
,	O
bitvectors	B-Data_Structure
,	O
strings	O
,	O
sequences	O
,	O
bags	O
,	O
and	O
equality	O
over	O
uninterpreted	B-Language
function	I-Language
symbols	O
C++	B-Language
,	O
Python	B-Language
,	O
Java	B-Language
2021	O
version	O
1.0	O
released	O
April	O
2022	O
Decision	O
Procedure	O
Toolkit	O
(	O
DPT	O
)	O
Linux	B-Application
Apache	B-Application
OCaml	B-Language
no	O
DPLL-based	O
iSAT	O
Linux	B-Application
Proprietary	O
non-linear	O
arithmetic	O
no	O
DPLL-based	O
MathSAT	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
Proprietary	O
empty	B-Language
theory	I-Language
,	O
linear	O
arithmetic	O
,	O
nonlinear	O
arithmetic	O
,	O
bitvectors	B-Data_Structure
,	O
arrays	B-Data_Structure
C/C	O
++	O
,	O
Python	B-Language
,	O
Java	B-Language
2010	O
DPLL-based	O
MiniSmt	O
Linux	B-Application
LGPL	B-Application
non-linear	O
arithmeticOCaml	O
2010	O
SAT-solver	O
based	O
,	O
Yices-based	O
Norn	O
SMT	B-Application
solver	I-Application
for	O
string	O
constraints	O
OpenCog	B-Application
Linux	B-Application
AGPL	O
probabilistic	O
logic	O
,	O
arithmetic	O
.	O
</s>
<s>
relational	B-Architecture
models	I-Architecture
C++	B-Language
,	O
Scheme	B-Language
,	O
Python	B-Language
no	O
subgraph	O
isomorphism	O
OpenSMT	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
GPLv3	O
empty	B-Language
theory	I-Language
,	O
differences	O
,	O
linear	O
arithmetic	O
,	O
bitvectors	B-Data_Structure
C++	B-Language
2011	O
lazy	O
SMT	O
SolverraSATLinuxGPLv3v2.0real	O
and	O
integer	O
nonlinear	O
arithmetic2014	O
,	O
2015extension	O
of	O
the	O
Interval	O
Constraint	O
Propagation	O
with	O
Testing	O
and	O
the	O
Intermediate	O
Value	O
Theorem	O
SatEEn	O
?	O
</s>
<s>
Proprietary	O
linear	O
arithmetic	O
,	O
difference	O
logic	O
none	O
2009	O
SMTInterpol	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
LGPLv3	B-Application
uninterpreted	B-Language
functions	I-Language
,	O
linear	O
real	O
arithmetic	O
,	O
and	O
linear	O
integer	O
arithmetic	O
Java	B-Language
2012	O
Focuses	O
on	O
generating	O
high	O
quality	O
,	O
compact	O
interpolants	O
.	O
</s>
<s>
SMCHR	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
GPLv3	O
linear	O
arithmetic	O
,	O
nonlinear	O
arithmetic	O
,	O
heaps	O
C	B-Language
no	O
Can	O
implement	O
new	O
theories	O
using	O
Constraint	B-Application
Handling	I-Application
Rules	I-Application
.	O
</s>
<s>
SMT-RAT	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
MIT	B-License
linear	O
arithmetic	O
,	O
nonlinear	O
arithmetic	O
C++	B-Language
2015	O
Toolbox	O
for	O
strategic	O
and	O
parallel	O
SMT	O
solving	O
consisting	O
of	O
a	O
collection	O
of	O
SMT	O
compliant	O
implementations	O
.	O
</s>
<s>
SONOLAR	O
Linux	B-Application
,	O
Windows	B-Application
Proprietary	O
bitvectors	B-Data_Structure
C	B-Language
2010	O
SAT-solver	O
based	O
Spear	O
Linux	B-Application
,	O
Mac	B-Application
OS	I-Application
,	O
Windows	B-Application
Proprietary	O
bitvectors	B-Data_Structure
2008	O
STP	O
Linux	B-Application
,	O
OpenBSD	B-Operating_System
,	O
Windows	B-Application
,	O
Mac	B-Application
OS	I-Application
MIT	B-License
bitvectors	B-Data_Structure
,	O
arrays	B-Data_Structure
C	B-Language
,	O
C++	B-Language
,	O
Python	B-Language
,	O
OCaml	B-Language
,	O
Java	B-Language
2011	O
SAT-solver	O
based	O
SWORD	O
Linux	B-Application
Proprietary	O
bitvectors	B-Data_Structure
2009	O
UCLID	O
Linux	B-Application
BSD	B-Operating_System
empty	B-Language
theory	I-Language
,	O
linear	O
arithmetic	O
,	O
bitvectors	B-Data_Structure
,	O
and	O
constrained	O
lambda	O
(	O
arrays	B-Data_Structure
,	O
memories	O
,	O
cache	O
,	O
etc	O
.	O
)	O
</s>
<s>
There	O
are	O
multiple	O
attempts	O
to	O
describe	O
a	O
standardized	O
interface	O
to	O
SMT	B-Application
solvers	I-Application
(	O
and	O
automated	B-Application
theorem	I-Application
provers	I-Application
,	O
a	O
term	O
often	O
used	O
synonymously	O
)	O
.	O
</s>
<s>
The	O
most	O
prominent	O
is	O
the	O
SMT-LIB	O
standard	O
,	O
which	O
provides	O
a	O
language	O
based	O
on	O
S-expressions	B-Protocol
.	O
</s>
<s>
Other	O
standardized	O
formats	O
commonly	O
supported	O
are	O
the	O
DIMACS	O
format	O
supported	O
by	O
many	O
Boolean	B-Algorithm
SAT	I-Algorithm
solvers	O
,	O
and	O
the	O
CVC	O
format	O
used	O
by	O
the	O
CVC	O
automated	B-Application
theorem	I-Application
prover	I-Application
.	O
</s>
<s>
The	O
SMT-LIB	O
format	O
also	O
comes	O
with	O
a	O
number	O
of	O
standardized	O
benchmarks	O
and	O
has	O
enabled	O
a	O
yearly	O
competition	O
between	O
SMT	B-Application
solvers	I-Application
called	O
SMT-COMP	O
.	O
</s>
<s>
SMT	B-Application
solvers	I-Application
are	O
useful	O
both	O
for	O
verification	O
,	O
proving	O
the	O
correctness	O
of	O
programs	O
,	O
software	O
testing	O
based	O
on	O
symbolic	B-General_Concept
execution	I-General_Concept
,	O
and	O
for	O
synthesis	B-Application
,	O
generating	O
program	O
fragments	O
by	O
searching	O
over	O
the	O
space	O
of	O
possible	O
programs	O
.	O
</s>
<s>
Outside	O
of	O
software	O
verification	O
,	O
SMT	B-Application
solvers	I-Application
have	O
also	O
been	O
used	O
for	O
type	O
inference	O
and	O
for	O
modelling	O
theoretic	O
scenarios	O
,	O
including	O
modelling	O
actor	O
beliefs	O
in	O
nuclear	O
arms	O
control	O
.	O
</s>
<s>
Computer-aided	O
verification	O
of	O
computer	B-Application
programs	I-Application
often	O
uses	O
SMT	B-Application
solvers	I-Application
.	O
</s>
<s>
There	O
are	O
many	O
verifiers	O
built	O
on	O
top	O
of	O
the	O
Z3	B-Language
SMT	I-Language
solver	I-Language
.	O
</s>
<s>
is	O
an	O
intermediate	O
verification	O
language	O
that	O
uses	O
Z3	B-Language
to	O
automatically	O
check	O
simple	O
imperative	O
programs	O
.	O
</s>
<s>
The	O
verifier	O
for	O
concurrent	O
C	B-Language
uses	O
Boogie	O
,	O
as	O
well	O
as	O
for	O
imperative	O
object-based	O
programs	O
,	O
for	O
concurrent	O
programs	O
,	O
and	O
for	O
C#	O
.	O
</s>
<s>
is	O
a	O
dependently	O
typed	O
language	O
that	O
uses	O
Z3	B-Language
to	O
find	O
proofs	O
;	O
the	O
compiler	O
carries	O
these	O
proofs	O
through	O
to	O
produce	O
proof-carrying	O
bytecode	O
.	O
</s>
<s>
The	O
encodes	O
verification	O
conditions	O
to	O
Z3	B-Language
.	O
</s>
<s>
The	O
library	O
provides	O
SMT-based	O
verification	O
of	O
Haskell	B-Language
programs	O
,	O
and	O
lets	O
the	O
user	O
choose	O
among	O
a	O
number	O
of	O
solvers	O
such	O
as	O
Z3	B-Language
,	O
ABC	O
,	O
Boolector	O
,	O
cvc5	O
,	O
MathSAT	O
and	O
Yices	O
.	O
</s>
<s>
There	O
are	O
also	O
many	O
verifiers	O
built	O
on	O
top	O
of	O
the	O
SMT	B-Application
solver	I-Application
.	O
</s>
<s>
,	O
a	O
platform	O
for	O
deductive	O
program	O
verification	O
,	O
uses	O
Alt-Ergo	B-Language
as	O
its	O
main	O
prover	O
;	O
</s>
<s>
CAVEAT	O
,	O
a	O
C-verifier	O
developed	O
by	O
CEA	O
and	O
used	O
by	O
Airbus	O
;	O
Alt-Ergo	B-Language
was	O
included	O
in	O
the	O
qualification	O
DO-178C	O
of	O
one	O
of	O
its	O
recent	O
aircraft	O
;	O
</s>
<s>
Frama-C	B-Language
,	O
a	O
framework	O
to	O
analyse	O
C-code	O
,	O
uses	O
Alt-Ergo	B-Language
in	O
the	O
Jessie	O
and	O
WP	O
plugins	O
(	O
dedicated	O
to	O
"	O
deductive	O
program	O
verification	O
"	O
)	O
;	O
</s>
<s>
SPARK	B-Language
uses	O
CVC4	O
and	O
Alt-Ergo	B-Language
(	O
behind	O
GNATprove	O
)	O
to	O
automate	O
the	O
verification	O
of	O
some	O
assertions	O
in	O
SPARK	B-Language
2014	O
;	O
</s>
<s>
Atelier-B	B-Application
can	O
use	O
Alt-Ergo	B-Language
instead	O
of	O
its	O
main	O
prover	O
(	O
increasing	O
success	O
from	O
84%	O
to	O
98%	O
on	O
the	O
)	O
;	O
</s>
<s>
Rodin	B-Application
,	O
a	O
B-method	B-Application
framework	O
developed	O
by	O
Systerel	O
,	O
can	O
use	O
Alt-Ergo	B-Language
as	O
a	O
back-end	O
;	O
</s>
<s>
Many	O
SMT	B-Application
solvers	I-Application
implement	O
a	O
common	O
interface	O
format	O
called	O
(	O
such	O
files	O
usually	O
have	O
the	O
extension	O
"	O
.smt2	O
"	O
)	O
.	O
</s>
<s>
tool	O
implements	O
a	O
refinement	O
type	O
based	O
verifier	O
for	O
Haskell	B-Language
that	O
can	O
use	O
any	O
SMTLIB2	O
compliant	O
solver	O
,	O
e.g.	O
</s>
<s>
cvc5	O
,	O
MathSat	O
,	O
or	O
Z3	B-Language
.	O
</s>
<s>
An	O
important	O
application	O
of	O
SMT	B-Application
solvers	I-Application
is	O
symbolic	B-General_Concept
execution	I-General_Concept
for	O
analysis	O
and	O
testing	O
of	O
programs	O
(	O
e.g.	O
,	O
concolic	O
testing	O
)	O
,	O
aimed	O
particularly	O
at	O
finding	O
security	O
vulnerabilities	O
.	O
</s>
<s>
SMT	B-Application
solvers	I-Application
that	O
have	O
been	O
used	O
for	O
symbolic-execution	O
applications	O
include	O
,	O
,	O
the	O
,	O
and	O
.	O
</s>
