Alenka

equisatisfiable formula in \cnf with only linear increase of its

equisatisfiable formula in \cnf with only linear increase of its

size~\cite{Tse68}.

size~\cite{Tse68}.

\subsection{DPLL}

\subsection{Davis--Putnam--Logemann--Loveland algorithm}

Historically, the first procedure to solve \sat without explicitly

Historically, the first procedure to solve \sat without explicitly

computing the truth table of the formula was proposed by Davis and

computing the truth table of the formula was proposed by Davis and

refinement of \dppr algorithm was introduced in 1962 by Davis, Putnam,

refinement of \dppr algorithm was introduced in 1962 by Davis, Putnam,

Logemann and Loveland~\cite{DPLL62}.

Logemann and Loveland~\cite{DPLL62}.

The Davis--Putnam--Logemann--Loveland algorithm (\dpll) iterativelly

The Davis--Putnam--Logemann--Loveland \quotegraffito{If you don't know

tries to build a satisfying assignment by deciding values of the

  where you are going any road can take you there.}{Alice in

propositional variables and it backtracks if any of the input clauses

  Wonderland} algorithm (\dpll) iterativelly tries to build a

becomes false in the current assignment. A key procedure guiding the

satisfying assignment by deciding values of the propositional

\dpll search is the \emph{unit clause rule}, which is based on the

variables and it backtracks if any of the input clauses becomes false

observation that if a formula contains a clause $C$ that is unit in

in the current assignment. A key procedure guiding the \dpll search is

the current assignment, the only way to build a satisfying assignment

the \emph{unit clause rule}, which is based on the observation that if

is to add the sole undefined literal of $C$ to $M$. The iterated

a formula contains a clause $C$ that is unit in the current

application of the unit clause rule is called \emph{unit propagation}

assignment, the only way to build a satisfying assignment is to add

or \emph{Boolean constraint propagation} (\bcp)~\cite{ZM88}. The \dpll

the sole undefined literal of $C$ to $M$. The iterated application of

the unit clause rule is called \emph{unit propagation} or

\emph{Boolean constraint propagation} (\bcp)~\cite{ZM88}. The \dpll

search consists of decision and propagation steps. In decision steps,

search consists of decision and propagation steps. In decision steps,

a variable and its new value are chosen and added to the current

a variable and its new value are chosen and added to the current

partial assignment. After each decision step, the \bcp is performed to

partial assignment. After each decision step, the \bcp is performed to

  all structures from $\teuf$ that are isomorphic to real numbers and

  all structures from $\teuf$ that are isomorphic to real numbers and

  interpret the function $+$ as addition, and the predicate $\leq$ as

  interpret the function $+$ as addition, and the predicate $\leq$ as

  the real comparison;

  the real comparison;

\item the theory of \emph{real arithmetic}, which consists of all

\item the theory of \emph{real arithmetic}\marginpar{In contrast to

  structures from $\teuf$ that are isomorphic to real numbers and

    real arithmetic, integer arithmetic with multiplication was shown

  interpret the function $+$ as addition, $\times$ as multiplication,

    to be undecidable by Gödel.}, which consists of all structures

  and the predicate $\leq$ as the real comparison;

  from $\teuf$ that are isomorphic to real numbers and interpret the

  function $+$ as addition, $\times$ as multiplication, and the

  predicate $\leq$ as the real comparison;

\item the theory of \emph{arrays}, which consists of all structures

\item the theory of \emph{arrays}, which consists of all structures

  from $\teuf$ isomorphic to the set of arrays with a binary function

  from $\teuf$ isomorphic to the set of arrays with a binary function

  $read(a, i)$ interpreted as a value in the index $i$ of the array

  $read(a, i)$ interpreted as a value in the index $i$ of the array

respective $T$-solvers, we refer the reader for example to the book of

respective $T$-solvers, we refer the reader for example to the book of

Bradley and Manna~\cite{BM07}.\marginpar{TODO: paragraph about combination of theories}

Bradley and Manna~\cite{BM07}.\marginpar{TODO: paragraph about combination of theories}

\subsection{DPLL(T)}

\subsection{DPLL modulo theories}

Most of the \smt approaches can be classified as \emph{eager} or

Most of the \smt approaches can be classified as \emph{eager} or

\emph{lazy}~\cite{BSST09}. The eager \smt approach consists in

\emph{lazy}~\cite{BSST09}. The eager \smt approach consists in

\label{ssec:natDomainSat}

\label{ssec:natDomainSat}

Although the separation of the Boolean and theory reasoning in the

Although the separation of the Boolean and theory reasoning in the

\dpll approach allows the solver to be modular, it can be also

\dpll approach allows the solver to be modular, it can be also

restricting in some cases. In particular, the \dpllt based solvers can

restricting in some cases. In particular, \dpllt based solvers can not

not directly reason about values of first-order variables, but have to

directly reason about values of first-order variables, but have to

rely on the $T$-solver guiding the search over Boolean

rely on the $T$-solver guiding the search over Boolean

valuations. While there are some techniques like \emph{splitting on

valuations. While there are some techniques like \emph{splitting on

  demand}~\cite{BNOT06} , which allow the $T$-solver to add new atoms

  demand}~\cite{BNOT06} , which allow the $T$-solver to add new atoms

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

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

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

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

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

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

theory are $=$, $\leq_u$, and $\leq_s$, representing equality,

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

unsigned inequality of binary-encoded natural numbers, and signed

unsigned inequality of binary-encoded natural numbers, and signed

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

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

respectively. Function symbols in the theory are

respectively. Function symbols in the theory are

$+, \times, \div, \&, \mid, \oplus, \ll, \gg, \cdot, \extract{n}{p}$,

$+, \times, \div, \&, \mid, \oplus, \ll, \gg, \cdot, \extract{n}{p}$,

representing addition, multiplication, unsigned division, bit-wise

interpreted as addition, multiplication, unsigned division, bit-wise

and, bit-wise or, bit-wise exclusive or, left-shift, right-shift,

and, bit-wise or, bit-wise exclusive or, left-shift, right-shift,

concatenation, and extraction of $n$ bits starting from the position

concatenation, and extraction of $n$ bits starting from the position

$p$, respectively. For detailed description of the \BV theory syntax

$p$, respectively. For the detailed description of the \BV theory

and semantics, see for example Hadarean's PhD

syntax and semantics, see for example Hadarean's PhD

thesis~\cite{Had15}. This section focuses on the problem of

thesis~\cite{Had15}. This section focuses on the problem of

satisfiability of the quantifier-free fragment of the \BV theory,

satisfiability of the quantifier-free fragment of the \BV theory,

denoted \QFBV. %The the full \BV logic is treated in the next section.

denoted \QFBV. The the full \BV logic is treated in the next section.

Current state-of-the-art \smt solvers for the \QFBV logic rely on

Current state-of-the-art \smt solvers for the \QFBV logic rely on

rewriting techniques, used to simplify the formula during the

rewriting techniques, used to simplify the formula during the

reasoning and if the sub-solvers are not sufficient for deciding the

reasoning and if the sub-solvers are not sufficient for deciding the

satisfiability of the formula, theory lemmas and propagated literals

satisfiability of the formula, theory lemmas and propagated literals

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

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

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

\dpllt bit-blasting solver is

employed~\cite{HBJBT14}. \marginpar{TODO: Nepřidat odstavec o

  propagating-complete \cnf encodings?}

\subsection{Word-level techinques}

\subsection{Word-level techinques}

Although bit-blasting is highly efficient for most of practical

Although bit-blasting is highly efficient for most of practical

problems, it can exhaust memory if the input formula contains complex

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

arithmetic or variables with large bit-width. Several techniques that

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

which avoid the bit-blasting have been proposed to alleviate this

techniques that avoid the bit-blasting have been proposed to alleviate

problem.

this problem.

Some useful fragments of the bit-vector theory can be solved by

Some useful fragments of the bit-vector theory can be solved by

specialized algorithms for deciding satisfiability. For example,

specialized algorithms for deciding satisfiability. For example,

only arithmetic and no bit-wise operations~\cite{BM05}.

only arithmetic and no bit-wise operations~\cite{BM05}.

Recently, an instance of the model-constructing satisfiability

Recently, an instance of the model-constructing satisfiability

calculus introduced in subsection \ref{ssec:natDomainSat} was

calculus, which was introduced in subsection \ref{ssec:natDomainSat},

implemented by Zeljić et al. in the solver \mcbv~\cite{ZWR16}. They

was implemented by Zeljić et al. in the solver

have extended the \mcsat framework by the ability to perform partial

\mcbv~\cite{ZWR16}. Zeljić et al. have extended the \mcsat framework

assignments and proposed heuristics for generalizing explanations of

by the ability to perform partial assignments and have proposed

bit-vector conflicts. For example, the solver \mcsat can perform the

heuristics for generalizing explanations of bit-vector conflicts. For

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

example, the solver \mcsat can perform the partial assignment

two least significant bits of $x$ are $10$. To be able to efficiently

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

use such partial assignments, the solver \mcbv mantains an

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

over-approximation of the set of models that are compatible with the

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

current partial assignment using bit-patterns and arithmetic

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

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

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

represent undefined bits, and constrain the values of particular bits

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

in the assignment. Arithmetic intervals are pairs of bit-vector values

represents undefined bit, and constrain the values of particular bits

representing lower and upper bounds and constrain integral values of

in the assignment. On the other hand, arithmetic intervals are pairs

bit-vectors. Both bit-patterns and arithmetic intervals can be ordered

of bit-vector values representing lower and upper bounds and constrain

to form a lattice, in which the solver performs a search if a conflict

integral values of bit-vectors. Both bit-patterns and arithmetic

is detected in order to generalize from the conflict.

intervals can be ordered to form a lattice in which the solver

performs a search for a more general explanation if a conflict is

detected.

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

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

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

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

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

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

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

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

solver randomly chooses initial values of bit-vector variables and

solver randomly chooses initial values of bit-vector variables and

tries to find a satisfying assignment by performing random bit flips

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

guided by the scoring function based on the satisfaction of

which are guided by the scoring function based on the satisfaction of

subformulas of the input formula. Niemetz et al. have improved the

subformulas of the input formula. Niemetz et al. have improved the

\sls technique with the path-based propagation, which, instead of

\sls technique with the \emph{path-based propagation}, which, instead

relying solely on random modifications guided by the scoring function,

of relying solely on random bit modifications guided by the scoring

allows computing values of bit-vector variables that are necessary to

function, allows computing values of bit-vector variables that are

satisfy randomly selected subformulas. The \sls based solver has been

necessary to satisfy randomly selected subformulas. The \sls based

shown to be able to decide several formulas not decided by

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

bit-blasting solvers. To combine benefit of bit-blasting and \sls

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

approaches, the latest version of Boolector, which have won the 2016

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

SMT competition in category of unquantified bit-vectors, uses a

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

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

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

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

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

the \sls solver fails to solve the formula~\cite{BoolectorComp}.

solver if the \sls solver fails to solve the

formula~\cite{BoolectorComp}.

\subsection{Preprocessing}

\subsection{Preprocessing}

For both bit-blasting and word-level approaches, a preprocessing of

For both bit-blasting and word-level approaches, a preprocessing of

the input formula is necessary for the efficiency of the

the input formula is crucial for the efficiency of the

solver. Therefore, modern SMT solvers employ hundreds of rewrite rules

solver. Therefore, modern \smt solvers employ hundreds of rewrite

in order to simplify the input formula. Franzén describes several

rules in order to simplify the input formula~\cite{Fra10}. Franzén

simplifications implemented in the solver MathSAT5: canonization,

describes several classes of simplification methods implemented in the

unconstrained variables propagation, packet splitting, and disjunctive

solver MathSAT5: canonization, unconstrained variables propagation,

partitioning~\cite{Fra10}.

packet splitting, and disjunctive partitioning~\cite{Fra10}.

The aim of the \emph{canonization} is converting subterms of the

The aim of the \emph{canonization} is converting subterms of the

formula to their canonical forms. For example $x + x = 0$ can be

formula to their canonical forms. For example, the term $x - x$ can be

rewritten to $2*x = 0$. Although efficient algorithms exist for

rewritten to $0$. Although efficient algorithms exist for several

several fragments of the \BV theory, like the core theory of

fragments of the \BV theory, like the core theory of bit-vectors, the

bit-vectors, the problem of computing the canonical form for the full

problem of computing the canonical form for the full \BV theory is

\BV theory is NP-hard~\cite{BDL98}. Therefore, SMT solvers usually do

NP-hard~\cite{BDL98}. Therefore, SMT solvers usually do not compute

not compute the canonical form, but rely on multiple heuristically

the canonical form, but rely on multiple heuristically chosen rewrite

chosen rewrite rules that produce the canonical form for simple

rules that produce the canonical form for simple terms, but are not

terms, but are not required to produce the canonical form in general.

required to produce the canonical form of all terms.

From the remaining simplification techniques presented by Franzén, we

From the remaining simplification techniques presented by Franzén, we

describe propagation of unconstrained variables in more detail, as it

focus on propagation of unconstrained variables, as it is highly

is highly relevant for software verification.

relevant for software verification.

A variable $x$ in the given formula is called \emph{unconstrained} if

A variable $x$ in a formula is called \emph{unconstrained} if it

it occurs only once in the formula. Brummayer~\cite{Brum10} and

occurs only once in the formula. Brummayer~\cite{Brum10} and

Bruttomesso~\cite{Bru08} independently observed that if an

Bruttomesso~\cite{Bru08} have independently observed that if an

unconstrained variable occurs as an argument to a function symbol,

unconstrained variable occurs as an argument to a function symbol,

which can be \emph{inverted} with respect to this argument, replacing

which can be \emph{inverted} with respect to this argument, replacing

this function with a fresh variable yields an equi-satisfiable

this function with a fresh variable yields an equisatisfiable

formula. Moreover, unconstrained variables often occur in the

formula. Moreover, unconstrained variables often occur in the

industrial benchmarks and especially in benchmarks comming from a

industrial benchmarks and especially in benchmarks produced during a

verification of programs in a single static assignment form, such as

verification of programs in a single static assignment form, such as

LLVM bit-code.

LLVM bit-code.

With a slight abuse of notation, which identifies interpreted function

With a slight abuse of notation which identifies interpreted function

symbols with their intended interpretations, a simple definition of

symbols with their intended interpretations, a simple definition of

invertibility for binary function symbols, which can be easily

invertibility for binary function symbols, which can be easily

generalized, is as follows. A binary function $f$ can be inverted with

generalized, is as follows. A binary function $f$ can be inverted with

respect to its first argument, if for every two values $a_i, a_o$,

respect to its first argument if for every two values $a_i, a_o$ there

there exists a value $b$ such that $f(b,a_i) = a_o$. For example, as

exists a value $b$ such that $f(b,a_i) = a_o$. For example, as the

the addition is invertible with respect to its first argument, the

addition is invertible with respect to its first argument, the formula

formula $\varphi \equiv x + (y * y - 30 * z * y) = 0$ can be

$\varphi \equiv x + (y * y - 30 * z * y) = 0$ can be transformed to an

transformed to an equi-satisfiable formula $v = 0$, where $v$ is a

equisatisfiable formula $v = 0$, where $v$ is a fresh variable,

fresh variable, because $x$ is unconstrained in $\varphi$. Note that in

because $x$ is unconstrained in $\varphi$. Note that in contrast to

contrast with the theory of integers, the function $f(x) = k \times x$

the theory of integers, the function $f(x) = k \times x$ is invertible

is invertible precisely if $k$ is odd.

in bit-vectors precisely if $k$ is odd.

\section{Satisfiability of quantified bit-vector formulas}

\section{Satisfiability of quantified bit-vector formulas}

was shown to be \NP-complete. The complexity of the satisfiability

was shown to be \NP-complete. The complexity of the satisfiability

problems for various variants of bit-vector theory have been recently

problems for various variants of bit-vector theory have been recently

studied and was shown to range from \NP-complete to

studied and was shown to range from \NP-complete to

2-NEXPTIME-complete~\cite{KFB16}. In particular, interesting variants

\NNEXPTIME-complete~\cite{KFB16}. In particular, interesting variants

of the bit-vector satisfiability problem differ in allowing

of the bit-vector satisfiability problem differ in allowing

uninterpreted functions, allowing quantifiers, and in encoding of the

uninterpreted functions, allowing quantifiers, and in encoding of the

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

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

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

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

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

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

combination with the theory of uninterpreted functions is denoted by

combination with the theory of uninterpreted functions is denoted by

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

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

polynomial time reduction from QF\_BV1 to \sat, showing that QF\_BV1

polynomial time reduction from QF\_BV1 to \sat, showing that QF\_BV1

is in NP. A similar reduction from BV1 to \qbf can show that BV is in

is in NP. A similar reduction from BV1 to \qbf can show that BV is in

\PSPACE. For lower bounds, \NP-hardness of QF\_BV1 follows from a

\PSPACE. For lower bounds, \NP-hardness of QF\_BV1 follows from a

simple reduction from \sat, which encodes each propositional variable

simple reduction from \sat, by encoding each propositional variable as

as a bit-vector of bit-width 1, and similarly, BV1 can be shown to be

a bit-vector of bit-width 1, and similarly, BV1 can be shown to be

\PSPACE-hard.

\PSPACE-hard.

In quantifier-free formulas, uninterpreted functions can be eliminated

In quantifier-free formulas, uninterpreted functions can be eliminated

the formula. Therefore, QF\_UFBV1 is in \NP. As the set of QF\_UFBV1

the formula. Therefore, QF\_UFBV1 is in \NP. As the set of QF\_UFBV1

formulas contains all QF\_BV1 formulas, QF\_UFBV1 is also NP-hard. The

formulas contains all QF\_BV1 formulas, QF\_UFBV1 is also NP-hard. The

case of UFBV1 is more involved. Wintersteiger et al. have shown that

case of UFBV1 is more involved. Wintersteiger et al. have shown that

UFBV1 is \NEXPTIME-complete by using \emph{effectively propositonal}

UFBV1 is \NEXPTIME-complete by using \emph{effectively propositional}

fragment of the first-order logic, which is well known to be

fragment of the first-order logic, which is well known to be

\NEXPTIME-complete~\cite{WHD13}. The class of effectively

\NEXPTIME-complete~\cite{WHD13}. The class of effectively

propositional formulas, also known as the Bernays--Schönfinkel class,

\marginpar{The class of effectively propositional formulas is also

  known the Bernays--Schönfinkel class.} propositional formulas

consists only of formulas in form $\exists^*\forall^*\varphi$, where

consists only of formulas in form $\exists^*\forall^*\varphi$, where

$\varphi$ does not contain any quantifiers or function symbols.

$\varphi$ does not contain any quantifiers or function symbols.

\paragraph{Binary encoded bit-widths}

\paragraph{Binary encoded bit-widths}

With binary encoded bit-widths, the bit-blasting may yield

For formulas with binary encoded bit-widths, the bit-blasting may

propositional formula that is exponentially larger than the original

yield propositional formula that is exponentially larger than the

input bit-vector formula, as the number of bits may be exponential

original input bit-vector formula, as the number of bits may be

with respect to the size of the formula. Therefore, bit-blasting shows

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

that QF\_BV2 is in \NEXPTIME. On the other hand, Kovásznai et al. have

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

presented a polynomial time reduction of satisfiability of

Kovásznai et al. have presented a polynomial time reduction of

\emph{dependent quantified boolean formulas} (\dqbf) to QF\_BV2. Since

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

\dqbf is well known to be \NEXPTIME-complete, this reduction shows

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

\NEXPTIME-hardness of QF\_BV2~\cite{KFB12}. In contrast, the precise

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

complexity after adding quantifiers is not known. BV2 is known to be

contrast, the precise complexity after adding quantifiers is not

in \EXPSPACE and hence it contains all formulas from QF\_BV2, it is

known. BV2 is known to be in \EXPSPACE and because it contains all

also \NEXPTIME-hard.

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

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

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

quantifier-fre fragment stays the same when the uninterpreted

quantifier-fre fragment stays the same when the uninterpreted

\theoremstyle{definition}

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\newtheorem{exmp}{Example}[section]

\newtheorem{exmp}{Example}[section]

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