Some changes in Objectives

\section{Objectives and Expected Results}

\subsection{Symbolic solver for quantified bit-vectors}

I plan to further develop the implemented symbolic \smt solver for

quantified bit-vecors 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 zero-suppressed 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 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.

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

  bit-vectors}

Simplifications using unconstrained variables can be extended to

$k \times x$ 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.

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

bit-vector formulas with binary-encoded 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.

\subsection{Symbolic solver for quantified bit-vectors}

We also plan to further develop the implemented symbolic \smt solver

for quantified bit-vecors 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

approximation 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 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.

\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 current

over-approximations by bit-patterns 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}

\newcommand{\EXPSPACE}{\textsf{EXPSPACE}\xspace}

\newcommand{\NEXPTIME}{\textsf{NEXPTIME}\xspace}

\newcommand{\NNEXPTIME}{\textsf{2-NEXPTIME}\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}}

\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