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\section{Preliminaries}

This section introduces the notation which is used in the rest of this

This section introduces the notation used in the rest of this

chapter. The exposition of the propositional logic is mainly based on

the work of Nieuwenhuis et al.~\cite{NOT06}. The exposition of the

first-order logic is partly based on the same source, but the

$\P = \{ x \geq y + 1, x = 4 \}$, where $x, y, 1$, and $4$ are

constant symbols, $x \geq y + 1~\vee~\neg (x = 4)$ is a clause.

A \emph{theory} $T$ is a set of first-order structures.  A formula $\varphi$

is $T$-\emph{satisfiable} or $T$-\emph{consistent} if there is a

structure $M \in T$ in which the formula $F$ holds in the standard

first-order sense. The formula $\varphi$ is $T$-\emph{unsatisfiable} or

$T$-\emph{inconsistent} otherwise. If the distinction is necessary, we

call literals over a specific background theory $T$-literals. In the

rest of this chapter, we consider only theories for which the

satisfiability of conjunctions of literals is decidable and we call

any decision procedure for conjunctions of $T$-literals over a

$T$-\emph{solver}.

A \emph{theory} $T$ is a set of first-order structures.  A formula

$\varphi$ is $T$-\emph{satisfiable} or $T$-\emph{consistent} if there

is a structure $M \in T$ in which the formula $F$ holds in the

standard first-order sense. The formula $\varphi$ is

$T$-\emph{unsatisfiable} or $T$-\emph{inconsistent} otherwise. If the

distinction is necessary, we call literals over a specific background

theory $T$-literals. In the rest of this chapter, we consider only

theories for which the satisfiability of conjunctions of literals is

decidable and we call any decision procedure for conjunctions of

$T$-literals a $T$-\emph{solver}.

Using the same notation as in the propositional logic, a partial

assignment is a set of literals and can be therefore seen as a

\section{Propositional satisfiability}

A \emph{propositional satisfiability problem} (\sat) is for a given

formula $\varphi$ in \cnf decide whether it is satisfiable. The

A \emph{propositional satisfiability problem} (\sat) is, for a given

formula $\varphi$ in \cnf, to decide whether it is satisfiable. The

restriction to formulas in \cnf is without a loss of generality, as

Tseitin transformation can be used to transform every formula to a

equisatisfiable formula in \cnf with only linear increase of its

the Tseitin transformation can be used to transform every formula to

an equisatisfiable formula in \cnf with only linear increase of its

size~\cite{Tse68}.

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

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

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

set values of variables which are implied by the decision. The

backtracking used in \dpll is \emph{chronological}, i.e. after the

backtracking used in \dpll is \emph{chronological}, i.e. after a

conflict the value of the last decided literal is changed.

%It has been observed that \dpll based solver spends the

clause $C$. Note that this literal does not have to be the last

decision literal in the search and therefore a bigger region of the

search space can be ruled out from the search, as it is known not to

contain a solution. There are multiple strategies of computing the

contain any solution. There are multiple strategies of computing the

clausal reason for the conflict, but the majority of \cdcl based \sat

solvers are using the \emph{first unique implication point} based

learning scheme, which has been shown to produce small clauses in

Another important part of modern \sat solvers are dynamic restarts,

which allow restarting the search from scratch in hope that clauses

learned during the conflict analyses will guide the search from regions

of the search space that contain no solutions~\cite{GSK98}. Although

most commonly used heuristic to decide when to restart the solver is

based on the Luby sequence~\cite{LSZ93}, the recent survey has shown

that it is outperformed by a heuristic based on the concept of

exponential moving averages~\cite{BF15restarts}.\marginpar{spravit

  citaci, BF15a nevyšlo v proceedings}

learned during the conflict analyses will guide the search from

regions of the search space that contain no

solutions~\cite{GSK98}. Although most commonly used heuristic to

decide when to restart the solver is based on the \emph{Luby

  sequence}~\cite{LSZ93}, the recent survey has shown that it is

outperformed by a heuristic based on the concept of exponential moving

averages~\cite{BF15restarts}.\marginpar{spravit citaci, BF15a nevyšlo

  v proceedings}

In addition to the mentioned heuristics, an efficient implementation

of \cdcl based \sat solver relies on lazy data structures used in the