Commit e9f46bec authored by Martin Jonas's avatar Martin Jonas
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Multiple changes

parent e5708327
......@@ -1133,3 +1133,11 @@ year = {2005},
pages = {245--257},
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
......@@ -4,66 +4,77 @@
\section{Objectives and Expected Results}
\subsection{Unconstrained variable propagation for quantified
bit-vectors}
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 proof-of-concept
simplification procedure using unconstrained variables for the
quantified bit-vector formulas. Although the initial experimental
results, conducted on the formulas from the semi-symbolic 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 bit-width $n$, $i$ 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 bit-vectors}
I plan to continue developing the implemented symbolic \smt solver for
quantified bit-vecors 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 bit-vector 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 zero-suppressed 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 zero-suppressed 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
bit-vectors.
\subsection{Hybrid approach to quantified bit-vectors}
Although our results with the symbolic \smt solver for quantified
bit-vectors 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 bit-vector 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 model-based 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
over-approximations by bit-patterns and arithmetic intervals.
Although results of the symbolic \smt solver for quantified
bit-vectors 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 bit-vector 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 model-based
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
over-approximations by bit-patterns 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 bit-vectors
is the main aim of my PhD study.
\subsection{Unconstrained variable propagation for quantified
bit-vectors}
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 proof-of-concept simplification
procedure that can simplify quantified formulas which contain
unconstrained variables. Although the initial experimental results,
conducted on the formulas from the semi-symbolic 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 bit-width $n$, 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 state-of 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 @@ bit-widths 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
According 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 log-space
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 non-linear multiplication.
\item [now -- January 2019] Improvements and mantaining of the
\item [now -- January 2019] Improvements and maintaining of the
developed symbolic \smt solver Q3B.
\item [January 2017] Doctoral exam and defence of this thesis proposal.
\item [January 2017 -- May 2017] Proving a precise complexity class of
the quantfied bit-vector formulas without uninterpreted functions
and with binary encoded bit-widths.
\item [May 2017 -- January 2018] Development of a hybrid approach to
\item [January 2017] Doctoral exam and defense of this thesis proposal.
\item [February 2017 -- April 2017] Proving a precise complexity class
of the quantified bit-vector formulas without uninterpreted functions
and with binary encoded bit-widths. Preparing a paper on this topic.
\item [May 2017 -- December 2017] Development of a hybrid approach to
\smt solving of quantified bit-vector 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 PhD thesis.
\item [January 2019] The final version of the thesis.
\end{description}
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......@@ -9,28 +9,31 @@
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......@@ -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}}}
......
......@@ -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 informatics\xspace}
\newcommand{\myFaculty}{Faculty of Informatics\xspace}
%\newcommand{\myDepartment}{Put data here\xspace}
\newcommand{\myUni}{Masaryk University\xspace}
\newcommand{\myLocation}{Brno\xspace}
......@@ -185,13 +185,17 @@
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