Nelson-Oppen

  pages     = {188--191},

  year      = {1993}

}

@article{NO79,

  author    = {Greg Nelson and

               Derek C. Oppen},

  title     = {Simplification by Cooperating Decision Procedures},

  journal   = {{ACM} Trans. Program. Lang. Syst.},

  volume    = {1},

  number    = {2},

  pages     = {245--257},

  year      = {1979}

}

% formula without existential quantifiers by introducing uninterpreted

% functions; this process is known as \emph{Skolemization}.

\subsection{Multi-sorted logic}

\subsection{Many-sorted logic}

For some theories, it can be convinient to distinguish several types

of objects instead of having only one universe. This can be achieved

stably infinite theories with disjoint signatures can be solved by

using separate satisfiability solvers for the respective theories,

which interchange implied equalities and disequalities between shared

variables. A theory $\mathcal{T}$ is \emph{stably infinite} for every

$\mathcal{T}$-satisfiable formula exists a $\mathcal{T}$ model, whose

universe is infinite. For a theory over a multi-sorted logic, the

theory is \emph{stably infinite} for every $\mathcal{T}$-satisfiable

formula exists a $\mathcal{T}$ model, whose every sort is interpreted

as infinite. This is not a strong restriction, almost all

variables~\cite{NO79}. A theory $\mathcal{T}$ is \emph{stably

  infinite} if every $\mathcal{T}$-satisfiable formula has a

$\mathcal{T}$ model whose universe is infinite. For a theory over in a

many-sorted logic, the theory is \emph{stably infinite} if every

$\mathcal{T}$-satisfiable formula has a model model, whose every sort

has an infinite domain. Although almost all practically used theories

are stably infinite, this is not true for inherently finite theories

like the theory of bit-vectors.

\subsection{DPLL modulo theories}

\section{Satisfiability of quantifier-free bit-vector formulas}

The \emph{theory of fixed sized bit-vectors (\BV)} is a multi-sorted

The \emph{theory of fixed sized bit-vectors (\BV)} is a many-sorted

first-order theory with infinitely many sorts $\sort{n}$ corresponding

to bit-vectors of length $n$. The only predicate symbols in the \BV

theory are $=$, $\leq_u$, and $\leq_s$, interpreted as equality,

\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 further develop the implemented symbolic \smt solver for

I plan to continue developing 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.

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

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

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

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}

As was explained in section ???, the precise complexity of quantified

bit-vector formulas with binary-encoded bit-widths and without