Commit 23460bbd authored by Martin Jonáš's avatar Martin Jonáš
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parent c9bab3f4
......@@ -473,7 +473,7 @@ and uninterpreted functions~\cite{McM11}.
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
......@@ -495,7 +495,7 @@ by a \sat solver. The transformation of a bit-vector formula to the
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
......@@ -509,7 +509,7 @@ satisfiability of the formula, theory lemmas and propagated literals
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
......@@ -535,7 +535,7 @@ heuristics for generalizing explanations of bit-vector conflicts. For
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
......@@ -549,7 +549,7 @@ detected.
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,
......@@ -562,10 +562,10 @@ necessary to satisfy randomly selected subformulas. The \sls based
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
......@@ -726,12 +726,12 @@ of the bit-vector satisfiability problem differ in allowing
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.
......@@ -790,7 +790,7 @@ original input bit-vector formula, as the number of bits may be
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
......@@ -798,7 +798,7 @@ known. BV2 is known to be in \EXPSPACE and because it contains all
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
......@@ -89,7 +89,7 @@ where $t$ is an arbitrary term that does not contain $x$.
\subsection{Variable ordering}
Ordering of \bdd variables is crucial to efficiency. The size of a
\bdd can differexponentially when choosing a different
\bdd can differ exponentially when choosing a different
ordering. Therefore, besides well known methods dynamic variable
reordering during the computation~\cite{Rud93}, Q3B precomputes
initial variable ordering, which is based on the dependencies among
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