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Martin Jonáš
DTEDI
Commits
34aafacc
Commit
34aafacc
authored
Sep 03, 2016
by
Martin Jonáš
Browse files
NelsonOppen
parent
2a1eb123
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3
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Bibliography.bib
View file @
34aafacc
...
...
@@ 1121,4 +1121,15 @@ year = {2005},
Design, 1993, Santa Clara, California, USA, November 711, 1993}
,
pages
=
{188191}
,
year
=
{1993}
}
\ No newline at end of file
}
@article
{
NO79
,
author
=
{Greg Nelson and
Derek C. Oppen}
,
title
=
{Simplification by Cooperating Decision Procedures}
,
journal
=
{{ACM} Trans. Program. Lang. Syst.}
,
volume
=
{1}
,
number
=
{2}
,
pages
=
{245257}
,
year
=
{1979}
}
Chapters/Chapter02.tex
View file @
34aafacc
...
...
@@ 178,7 +178,7 @@ $\mathcal{T}$\emph{solver}.
% formula without existential quantifiers by introducing uninterpreted
% functions; this process is known as \emph{Skolemization}.
\subsection
{
M
ulti
sorted logic
}
\subsection
{
M
any
sorted logic
}
For some theories, it can be convinient to distinguish several types
of objects instead of having only one universe. This can be achieved
...
...
@@ 412,12 +412,14 @@ Nelson and Oppen have shown that satisfiability of combination of
stably infinite theories with disjoint signatures can be solved by
using separate satisfiability solvers for the respective theories,
which interchange implied equalities and disequalities between shared
variables. A theory
$
\mathcal
{
T
}$
is
\emph
{
stably infinite
}
for every
$
\mathcal
{
T
}$
satisfiable formula exists a
$
\mathcal
{
T
}$
model, whose
universe is infinite. For a theory over a multisorted logic, the
theory is
\emph
{
stably infinite
}
for every
$
\mathcal
{
T
}$
satisfiable
formula exists a
$
\mathcal
{
T
}$
model, whose every sort is interpreted
as infinite. This is not a strong restriction, almost all
variables~
\cite
{
NO79
}
. A theory
$
\mathcal
{
T
}$
is
\emph
{
stably
infinite
}
if every
$
\mathcal
{
T
}$
satisfiable formula has a
$
\mathcal
{
T
}$
model whose universe is infinite. For a theory over in a
manysorted logic, the theory is
\emph
{
stably infinite
}
if every
$
\mathcal
{
T
}$
satisfiable formula has a model model, whose every sort
has an infinite domain. Although almost all practically used theories
are stably infinite, this is not true for inherently finite theories
like the theory of bitvectors.
\subsection
{
DPLL modulo theories
}
...
...
@@ 597,7 +599,7 @@ and uninterpreted functions~\cite{McM11}.
\section
{
Satisfiability of quantifierfree bitvector formulas
}
The
\emph
{
theory of fixed sized bitvectors (
\BV
)
}
is a m
ulti
sorted
The
\emph
{
theory of fixed sized bitvectors (
\BV
)
}
is a m
any
sorted
firstorder theory with infinitely many sorts
$
\sort
{
n
}$
corresponding
to bitvectors of length
$
n
$
. The only predicate symbols in the
\BV
theory are
$
=
$
,
$
\leq
_
u
$
, and
$
\leq
_
s
$
, interpreted as equality,
...
...
Chapters/Chapter04.tex
View file @
34aafacc
...
...
@@ 4,15 +4,38 @@
\section
{
Objectives and Expected Results
}
\subsection
{
Unconstrained variable propagation for quantified
bitvectors
}
Simplifications using unconstrained variables can be extended to
quantified formulas. However, in the quantified setting, constraints
can be induced also by the order of the quantified variables. We have
hypothesis of the necessary condition for the quantified variable to
be unconstrained and we have implemented an proofofconcept
simplification procedure using unconstrained variables for the
quantified bitvector formulas. Although the initial experimental
results, conducted on the formulas from the semisymbolic model
checker
\SymDivine
look promissing, the formal proof of the correctnes
is not yet complete.
Furthermore, we suggest simplifications even for term in the form
$
k
\times
x
$
with odd values of
$
x
$
. If
$
x
$
has bitwidth
$
n
$
,
$
i
$
is
the largest number such that
$
2
^
i
$
which divides the constant
$
k
$
and
the value
$
x
$
is unconstrained, the term
$
k
\times
x
$
can be rewritten
to
$
extract
_
0
^{
n

i
}
(
x
)
\cdot
0
^
i
$
. This approach can possibly be
extended to the multiplication of two variables from one is
unconstrained and further generalized. We plan to prove the
correctness of these rules and develop a formal framework to classify
such rewrite rules.
\subsection
{
Symbolic solver for quantified bitvectors
}
I plan to
further
develop the implemented symbolic
\smt
solver for
I plan to
continue
develop
ing
the implemented symbolic
\smt
solver for
quantified bitvecors Q3B. Besides implementing the proposed
simplifiactions
using unconstrained variables, I plan to add support
of uninterpreted functions and theory
of arrays, which are highly
desirable for the usage in program
verification. I also want to
implement a support for extracting
unsatisfiable cores from the
intermediate
\bdds
, which were produced
during the computation of the
solver.
simplifiactions
for quantified formulas containing unconstrained
variables, I plan to add support
of uninterpreted functions and theory
of arrays, which are highly
desirable for the usage in program
verification. I also want to
implement a support for extracting
unsatisfiable cores from the
intermediate
\bdds
, which were produced
during the computation of the
solver.
Additionally, approximations implemented are right now very simple and
could benefit from better refinement of the approximation in the case
...
...
@@ 42,29 +65,6 @@ As the part of my PhD study, also an implementation of a proposed
hybrid approach and its evaluation on the representative set of
benchmark is expected.
\subsection
{
Unconstrained variable propagation for quantified
bitvectors
}
Simplifications using unconstrained variables can be extended to
quantified formulas. However, in the quantified setting, constraints
can be induced also by the order of the quantified variables. We have
hypothesis of the necessary condition for the quantified variable to
be unconstrained and we have implemented an proofofconcept
simplification procedure using unconstrained variables for the
quantified bitvector formulas. Although the initial experimental
results, conducted on the formula from the semisymbolic model checker
\SymDivine
look promissing, the formal proof of the correctnes is not
yet complete.
Furthermore, we suggest simplifications even for term in the form
$
k
\times
x
$
with odd values of
$
x
$
. If
$
x
$
has bitwidth
$
n
$
,
$
i
$
is
the largest number such that
$
2
^
i
$
which divides the constant
$
k
$
and
the value
$
x
$
is unconstrained, the term
$
k
\times
x
$
can be rewritten
to
$
extract
_
0
^{
n

i
}
(
x
)
\cdot
0
^
i
$
. This approach can possibly be
extended to the multiplication of two variables from one is
unconstrained and further generalized. We plan to prove the
correctness of these rules and develop a formal framework to classify
such rewrite rules.
\subsection
{
Complexity of BV2
}
As was explained in section ???, the precise complexity of quantified
bitvector formulas with binaryencoded bitwidths and without
...
...
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