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Martin Jonáš
DTEDI
Commits
baf0af47
Commit
baf0af47
authored
Sep 02, 2016
by
Martin Jonáš
Browse files
Some changes in Objectives
parent
a2b78be4
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2
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Chapters/Chapter04.tex
View file @
baf0af47
...
...
@@ 4,6 +4,43 @@
\section
{
Objectives and Expected Results
}
\subsection
{
Symbolic solver for quantified bitvectors
}
I plan to further develop 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.
Additionally, approximations implemented are right now very simple and
could benefit from better refinement of the approximation in the case
that the current approximation is too coarse.
Moreover, we want to implement other variants of decision diagrams
such as zerosuppressed decision diagrams introduced by ??? and
algebraic decision diagrams introduced by ??? and experimentally
evaluate their effect on the performance of the
\smt
solver.
\subsection
{
Hybrid approach to quantified bitvectors
}
Although our results with the symbolic
\smt
solver for quantified
bitvectors look promissing, standard
\smt
solvers still perform
better on simple queries and on queries containing
multiplication. Therefore, I want to develop a hybrid approach to
\smt
solving of quantified bitvector formulas, which combines strengths of
both of these approaches. For example, a part of the quantified
formula without multiplication can be converted to the
\bdd
, which can
be used to guide the model search in the modelbased quantifier
instantiation. One possible way of achieving this is adding
\bdd
based
representation of sets of assignments to the
\mcbv
solver developed by
Zeljić et al. The
\bdd
representation can be added to the current
overapproximations by bitpatterns and arithmetic intervals.
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
...
...
@@ 21,39 +58,30 @@ 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.
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
uninterpreted functions is not known. It is known to be in
\EXPSPACE
and to be
\NEXPTIME
hard. However, a class for which the problem is
complete is not known.
\subsection
{
Symbolic solver for quantified bitvectors
}
We also plan to further develop the implemented symbolic
\smt
solver
for quantified bitvecors Q3B. Besides implementing the proposed
simplifiactions using unconstrained variables, we plan to add support
of uninterpreted functions and theory of arrays to the Q3B. Also used
approximations are right now very simple and could benefit from better
refinement of the approximation in the case that the current
approximat
ion is
too coarse
.
Acording to our investigation, it is probably complete for neither of
those complexity classes. We are working on a proof which shows that
BV2 is complete for the class of problems solvable by the
\emph
{
alternating Turing machine
}
(
\atm
) with the exponential space
and
\emph
{
polynomial number of alternations
}
with respect to logspace
reduction. This class is usually denoted as
\AEXPTIMEp
and is known to
be in between
\EXPSPACE
and
\NEXPTIME
. However, whether any of the
inclus
ion
s
is
proper is not known
.
\subsection
{
Hybrid approach to quantified bitvectors
}
Although our results with the symbolic
\smt
solver for quantified
bitvectors look promissing, standard
\smt
solvers still perform
better on simple queries and on queries containing
multiplication. Therefore, I want to develop a hybrid approach to
\smt
solving of quantified bitvector formulas, which combines strengths of
both of these approaches. For example, a part of the quantified
formula without multiplication can be converted to the
\bdd
, which can
be used to guide the model search in the modelbased quantifier
instantiation. One possible way of achieving this is adding
\bdd
based
representation of sets of assignments to the
\mcbv
solver developed by
Zeljić et al. The
\bdd
representation can be added to current
overapproximations by bitpatterns and arithmetic intervals.
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.
Expected result is a paper published at an international conference or
in a journal.
\newpage
\section
{
Progression Schedule
}
...
...
Includes/notation.tex
View file @
baf0af47
...
...
@@ 29,6 +29,8 @@
\newcommand
{
\EXPSPACE
}{
\textsf
{
EXPSPACE
}
\xspace
}
\newcommand
{
\NEXPTIME
}{
\textsf
{
NEXPTIME
}
\xspace
}
\newcommand
{
\NNEXPTIME
}{
\textsf
{
2NEXPTIME
}
\xspace
}
\newcommand
{
\AEXPTIMEp
}{
\textsf
{
AEXPTIME(poly)
}
\xspace
}
\newcommand
{
\atm
}{
\textsc
{
atm
}
\xspace
}
\newcommand
{
\state
}
[2]
{
\ensuremath
{
#1
\;

\;
#2
}
\xspace
}
\newcommand
{
\dec
}
[1]
{
\ensuremath
{
#1
^
\bullet
}}
...
...
@@ 42,7 +44,7 @@
\newcommand
{
\sort
}
[1]
{
\ensuremath
{
[#1]
}}
\newcommand
{
\extract
}
[2]
{
\ensuremath
{
\texttt
{
extract
}^{
#1
}_{
#2
}}}
\newcommand
{
\SymDivine
}{
\textsf
{
SymDIVINE
}}
\newcommand
{
\SymDivine
}{
\textsf
{
SymDIVINE
}
\xspace
}
\newcommand
{
\der
}{
\textsc
{
der
}
\xspace
}
\newcommand
{
\teuf
}{
\ensuremath
{
T
_
\mathit
{
EUF
}}
\xspace
}
\ No newline at end of file
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