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Martin Jonáš
DTEDI
Commits
e9f46bec
Commit
e9f46bec
authored
Sep 03, 2016
by
Martin Jonas
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parent
e5708327
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Bibliography.bib
View file @
e9f46bec
...
...
@@ 1133,3 +1133,11 @@ year = {2005},
pages
=
{245257}
,
year
=
{1979}
}
@article
{
Luc16
,
author
=
{Martin L{\"{u}}ck}
,
title
=
{Complete Problems of Propositional Logic for the Exponential Hierarchy}
,
journal
=
{CoRR}
,
volume
=
{abs/1602.03050}
,
year
=
{2016}
}
\ No newline at end of file
Chapters/Chapter04.tex
View file @
e9f46bec
...
...
@@ 4,66 +4,77 @@
\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 continue developing the implemented symbolic
\smt
solver for
quantified bitvecors Q3B. Besides implementing the proposed
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
I will continue developing the implemented symbolic
\smt
solver Q3B,
which is aimed for solving quantified bitvector formulas. In
particular, I plan to add support for uninterpreted functions and the
theory of arrays, which is highly desirable for applications in the
program verification. I also want to implement an extraction of an
unsatisfiable core from the intermediate
\bdds
that 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.
Additionally,
currently implemented approximations are
very simple and
could benefit from better refinement
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
Minato~
\cite
{
Min93
}
and algebraic decision diagrams introduced by
Bahar et al.~
\cite
{
BFGHMPS
}
and experimentally evaluate their effect
on the performance of the
\smt
solver.
Moreover, I want to implement other variants of decision diagrams such
as zerosuppressed decision diagrams introduced by Minato~
\cite
{
Min93
}
and algebraic decision diagrams introduced by Bahar et
al.~
\cite
{
BFGHMPS
}
and experimentally evaluate their effect on the
performance of the symbolic
\smt
solver for the quantified
bitvectors.
\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.
Although results of the symbolic
\smt
solver for quantified
bitvectors look promising, standard
\smt
solvers still perform
better on queries containing multiplication and complex
arithmetic. Therefore, I plan to develop a hybrid approach to
\smt
solving of quantified bitvector formulas that combines strengths of
both of these approaches. For example, parts of the quantified
formula, which do not contain multiplication, can be converted to the
\bdd
that can be used to guide the model search in the modelbased
quantifier instantiation. One possible way of achieving this is adding
support for
\bdds
to the solver
\mcbv
developed by Zeljić et al. The
\bdd
representation can be added to the currently used
overapproximations by bitpatterns and arithmetic
intervals. Moreover, a tighter cooperation is possible  if a
\mcsat
solver decides a value, this can be used to partially instantiate a
quantified part of the formula, for which the
\bdd
will be computed.
As the part of my PhD study,
also
an implementation of a
proposed
As the part of my PhD study, an implementation of a
solver using the
hybrid approach and its evaluation on the representative set of
benchmark is expected.
benchmark is expected. The hybrid approach to quantified bitvectors
is the main aim of my PhD study.
\subsection
{
Unconstrained variable propagation for quantified
bitvectors
}
Simplifications using unconstrained variables can be extended to
quantified formulas. However, in the quantified setting, constraints
among variables can be introduced also by the order in which the
variables are quantified. We have hypothesis that describes the
necessary condition for the quantified variable to be considered as
unconstrained. Based on this hypothesis and a partial proof of its
validity, we have implemented an proofofconcept simplification
procedure that can simplify quantified formulas which contain
unconstrained variables. Although the initial experimental results,
conducted on the formulas from the semisymbolic model checker
\SymDivine
, look promising, the formal proof of the correctness is not
yet complete.
Moreover, the simplifications of formulas with unconstrained variables
can be further extended. For example, if a formula contains a term
$
k
\times
x
$
with an odd values of
$
k
$
and the variable
$
x
$
is
unconstrained, this term can be replaced by a simpler term. In
particular, if
$
i
$
is the largest number for which
$
2
^
i
$
divides the
constant
$
k
$
and the variable
$
x
$
has bitwidth
$
n
$
, the term
$
k
\times
x
$
can be rewritten to
$
extract
_
0
^{
n

i
}
(
x
)
\cdot
0
^
i
$
. In
turn, this approach can be also extended to the multiplication of two
variables from one is unconstrained. I plan to investigate further
extensions of to the terms containing division and remainder
operations and publish all described extension to the unconstrained
variable propagation. I will also experimentally evaluate the effect
of such simplifications on our solver Q3B and on stateof the art
solvers such as Boolector, CVC4, and Z3.
\subsection
{
Complexity of BV2
}
As was explained in section
\ref
{
sec:complexity
}
, the precise
...
...
@@ 72,36 +83,38 @@ 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.
Acording to our investigation, it is probably complete for neither of
Ac
c
ording 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
inclusions is proper is not known.
inclusions is proper is not known. The
\AEXPTIMEp
hardness of the BV2
can be shown by reducing the problem of satisfiability of
\emph
{
quantified second order boolean formulas
}
, which was recently
proven to be
\AEXPTIMEp
hard by Lück~
\cite
{
Luc16
}
.
Expected result is a paper published at an international conference or
in a journal.
\newpage
\section
{
Progression Schedule
}
The plan of my future study and research activities is following:
\begin{description}
[style=nextline,leftmargin=0.8cm]
\item
[now  January 2017] Extending unconstrained variable
propagation to quantified formulas and to nonlinear multiplication.
\item
[now  January 2019] Improvements and mantaining of the
\item
[now  January 2019] Improvements and ma
i
ntaining of the
developed symbolic
\smt
solver Q3B.
\item
[January 2017] Doctoral exam and defen
c
e of this thesis proposal.
\item
[
Jan
uary 2017 
May
2017] Proving a precise complexity class
of
the quantfied bitvector formulas without uninterpreted functions
and with binary encoded bitwidths.
\item
[May 2017 
January
201
8
] Development of a hybrid approach to
\item
[January 2017] Doctoral exam and defen
s
e of this thesis proposal.
\item
[
Febr
uary 2017 
April
2017] Proving a precise complexity class
of
the quant
i
fied bitvector formulas without uninterpreted functions
and with binary encoded bitwidths.
Preparing a paper on this topic.
\item
[May 2017 
December
201
7
] Development of a hybrid approach to
\smt
solving of quantified bitvector formulas combining symbolic
representation of formulas with
\dpllt
or
\mcsat
.
\item
[January 2018  August 2018] Implementing a hybrid
\smt
solver
\item
[
August
2018  January 2019] Text of the Ph
.D.
thesis.
\item
[
September
2018  January 2019] Text of the Ph
D
thesis.
\item
[January 2019] The final version of the thesis.
\end{description}
...
...
FrontBackmatter/Titlepage.tex
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e9f46bec
...
...
@@ 9,28 +9,31 @@
\hfill
\
vfill
\
bigskip
\bigskip
\begingroup
\color
{
Maroon
}
\spacedallcaps
{
\myTitle
}
\\
\bigskip
\Large
{
\color
{
Maroon
}
\spacedallcaps
{
\myTitle
}
}
\\
\bigskip
\endgroup
\spacedlowsmallcaps
{
\myName
}
\vfill
\includegraphics
[width=6cm]
{
gfx/fi
_
logo
}
\\
\bigskip
\includegraphics
[width=5cm]
{
gfx/fi
_
logo
}
\vfill
\mySubtitle
\\
\medskip
\mySubtitle
\\
%\myDegree \\
%\myDepartment \\
%\myFaculty \\
%\myUni \\ \bigskip
\smallskip
\myTime
\vfill
\end{center}
\end{center}
\noindent
\myFaculty
,
\myUni
\\
\noindent
Supervisor:
\myProf
\end{addmargin}
\end{titlepage}
Includes/notation.tex
View file @
e9f46bec
...
...
@@ 18,6 +18,7 @@
\newcommand
{
\sls
}{
\textsc
{
sls
}
\xspace
}
\newcommand
{
\mcbv
}{
\textsc
{
mcbv
}
\xspace
}
\newcommand
{
\nnf
}{
\textsc
{
nnf
}
\xspace
}
\newcommand
{
\qsbf
}{
\textsc
{
qsbf
}
\xspace
}
\newcommand
{
\true
}{
\ensuremath
{
\texttt
{
true
}}}
\newcommand
{
\false
}{
\ensuremath
{
\texttt
{
false
}}}
...
...
classicthesisconfig.tex
View file @
e9f46bec
...
...
@@ 49,7 +49,7 @@
\newcommand
{
\myProf
}{
doc. RNDr. Jan Strejček, Ph.D.
\xspace
}
%\newcommand{\myOtherProf}{Put name here\xspace}
%\newcommand{\mySupervisor}{Put name here\xspace}
\newcommand
{
\myFaculty
}{
Faculty of
i
nformatics
\xspace
}
\newcommand
{
\myFaculty
}{
Faculty of
I
nformatics
\xspace
}
%\newcommand{\myDepartment}{Put data here\xspace}
\newcommand
{
\myUni
}{
Masaryk University
\xspace
}
\newcommand
{
\myLocation
}{
Brno
\xspace
}
...
...
@@ 185,13 +185,17 @@
% ********************************************************************
% Using PDFLaTeX
% ********************************************************************
%\usepackage{tikz}
%\usetikzlibrary{backgrounds,calc}
%\usepackage{graphicx,lipsum}
\PassOptionsToPackage
{
pdftex,hyperfootnotes=false,pdfpagelabels
}{
hyperref
}
\usepackage
{
hyperref
}
% backref linktocpage pagebackref
\pdfcompresslevel
=9
\pdfadjustspacing
=1
\PassOptionsToPackage
{
pdftex
}{
graphicx
}
\usepackage
{
graphicx
}
% \usepackage{esopic}
% ********************************************************************
% Hyperreferences
...
...
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