Some changes in Objectives

Chapters/Chapter04.tex

@@ -4,6 +4,43 @@
 
 \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
@@ -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 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}

Includes/notation.tex

@@ -29,6 +29,8 @@
 \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}}
@@ -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