First draft of Objectives

\section{Objectives and Expected Results}

\subsection{Unconstrained variable propagation for quantified bit-vectors}

\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 formula 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{Complexity of BV2}

\subsection{Hybrid approach to quantified bit-vectors}

\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.

\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.

\newpage

\section{Progression Schedule}

\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

  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