Spell check

The \emph{theory of fixed sized bit-vectors (\BV)} is a multi-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 $\leq_s$, interpretead as equality,

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

unsigned inequality of binary-encoded natural numbers, and signed

inequality of integers in $2$'s complement representation,

respectively. Function symbols in the theory are

equivalent propositional formula is traditionally called

\emph{bit-blasting}~\cite{Kro08}.  \marginpar{TODO: Nepřidat odstavec

  o propagating-complete \cnf encodings?} More lazy approach to the

bit-blasting is benefitial when the theory combination is

bit-blasting is beneficial when the theory combination is

required. For example, solvers Z3 and Yices apply bit-blasting to all

operations except for the equality, which is handled by a specialized

solver, and dynamically add axioms of the array

generated by the sub-solvers are added to the formula and a lazy

\dpllt bit-blasting solver is employed~\cite{HBJBT14}.

\subsection{Word-level techinques}

\subsection{Word-level techniques}

Although bit-blasting is highly efficient for most of practical

problems, it can exhaust memory of the solver if the input formula

contains complex arithmetic or variables with large bit-width. Several

example, the solver \mcsat can perform the partial assignment

$\extract{2}{0}(x) \mapsto 10$, denoting that the two least

significant bits of $x$ are $10$. To be able to efficiently use such

partial assignments, the solver \mcbv mantains two over-approximations

partial assignments, the solver \mcbv maintains two over-approximations

of the set of models that are compatible with the current partial

assignment -- using \emph{bit-patterns} and \emph{arithmetic

  intervals}. Bit-patterns are sequences of $0$, $1$ and $u$, which

Another word-level approach for the full bit-vector theory is

\emph{stochastic local search} (\sls), proposed for solving

bit-vectors by Frohlich et al.~\cite{FBWH15} and subsequently improved

bit-vectors by Fröhlich et al.~\cite{FBWH15} and subsequently improved

by Niemetz et al.~\cite{NPBF15,NPB16}. In the \sls approach, the

solver randomly chooses initial values of bit-vector variables and

tries to find a satisfying assignment by performing random bit flips,

solver has been shown to be able to decide several formulas not

decided by bit-blasting solvers. To combine benefit of bit-blasting

and \sls approaches, the latest version of Boolector, which have won

the 2016 SMT competition in category of unquantified bit-vectors, uses

a protfolio approach, which consists in first running a \sls based

solver for a short period of time and then running a bit-blasting

solver if the \sls solver fails to solve the

the 2016 SMT competition in category of quantifier-free bit-vectors,

uses a portfolio approach, which consists in first running a \sls

based solver for a short period of time and then running a

bit-blasting solver if the \sls solver fails to solve the

formula~\cite{BoolectorComp}.

\subsection{Preprocessing}

uninterpreted functions, allowing quantifiers, and in encoding of the

bit-widths (unary vs. binary). In the following, we follow the

notation of Kováznai et al~\cite{KFB16} -- decision problems for

quantifer-free fragments are dentoted by the prefix QF\_, the

quantifier-free fragments are denoted by the prefix QF\_, the

combination with the theory of uninterpreted functions is denoted by

the prefix UF, and the problems with unary and binary encoded

bit-widths are denoted by suffixes 1 and 2, respectively. For example,

QF\_UFBV2 is the decision problem for quantifier free formulas with

uninterpreted functions and binary encoded bit-withs. The completeness

uninterpreted functions and binary encoded bit-widths. The completeness

results for these classes are summarized in table

\ref{tbl:complexity}, and briefly explained in the rest of this section.

exponential with respect to the size of the formula. Therefore,

bit-blasting shows that QF\_BV2 is in \NEXPTIME. On the other hand,

Kovásznai et al. have presented a polynomial time reduction of

satisfiability of \emph{dependent quantified boolean formulas} (\dqbf)

satisfiability of \emph{dependent quantified Boolean formulas} (\dqbf)

to QF\_BV2. Since \dqbf is well known to be \NEXPTIME-complete, this

reduction shows \NEXPTIME-hardness of QF\_BV2~\cite{KFB12}. In

contrast, the precise complexity after adding quantifiers is not

formulas from QF\_BV2, it is also \NEXPTIME-hard.

Similarly to the case with the unary encoding, the complexity of the

quantifier-fre fragment stays the same when the uninterpreted

quantifier-free fragment stays the same when the uninterpreted

functions are added -- QF\_UFBV2 can be shown to be in \NEXPTIME by

the Ackermann reduction and \NEXPTIME-hard by the simple reduction

from QF\_UFBV2. The complexity of the problem after adding quantifiers