Commit 7b2c36cd authored by Martin Jonáš's avatar Martin Jonáš
Browse files

Fix todos

parent bc0a4271
......@@ -150,13 +150,10 @@
year = {2015}
}
@inproceedings{BF15restarts,
@misc{BF15restarts,
author = {Armin Biere and
Andreas Fr{\"{o}}hlich},
title = {Evaluating {CDCL} Restart Schemes},
booktitle = {Workshop on Pragmatics of SAT -- {POS'15}, Austin, TX,
USA, September 24-27, 2015, Proceedings},
pages = {405--422},
year = {2015}
}
......@@ -1140,4 +1137,38 @@ year = {2005},
journal = {CoRR},
volume = {abs/1602.03050},
year = {2016}
}
@article{TZ05,
author = {Cesare Tinelli and
Calogero G. Zarba},
title = {Combining Nonstably Infinite Theories},
journal = {J. Autom. Reasoning},
volume = {34},
number = {3},
pages = {209--238},
year = {2005}
}
@inproceedings{JB10,
author = {Dejan Jovanovic and
Clark Barrett},
title = {Polite Theories Revisited},
booktitle = {Logic for Programming, Artificial Intelligence, and Reasoning - 17th
International Conference, LPAR-17, Yogyakarta, Indonesia, October
10-15, 2010. Proceedings},
pages = {402--416},
year = {2010}
}
@inproceedings{RRZ05,
author = {Silvio Ranise and
Christophe Ringeissen and
Calogero G. Zarba},
title = {Combining Data Structures with Nonstably Infinite Theories Using Many-Sorted
Logic},
booktitle = {Frontiers of Combining Systems, 5th International Workshop, FroCoS
2005, Vienna, Austria, September 19-21, 2005, Proceedings},
pages = {48--64},
year = {2005}
}
\ No newline at end of file
......@@ -122,10 +122,11 @@ For two signatures $\Sigma$ and $\Omega$, we call $\Omega$ a
$\Omega^p \subseteq \Sigma^p$. Given a $\Sigma$-structure
$\mathcal{A}$ and a signature $\Omega$ that is a sub-signature of
$\Sigma$, the \emph{reduct} of $\mathcal{A}$ to a sub-signature
$\Omega$ is an $\Omega$-structure $\mathcal{A}'$ that coincides with
$\mathcal{A}$ on all symbols from $\Omega$. For a $\Sigma_1$-theory
$\mathcal{T}_1$ and a $\Sigma_2$-theory $\mathcal{T}_2$, their
\emph{combination} $\mathcal{T}_1 + \mathcal{T}_2$ is the largest
$\Omega$ is an $\Omega$-structure $\mathcal{A}'$ that has the same
universe as $\mathcal{A}$ and coincides with $\mathcal{A}$ on all
symbols from $\Omega$. For a $\Sigma_1$-theory $\mathcal{T}_1$ and a
$\Sigma_2$-theory $\mathcal{T}_2$, their \emph{combination}
$\mathcal{T}_1 + \mathcal{T}_2$ is the largest
$(\Sigma_1 \cup \Sigma_2)$-theory that contains all
$(\Sigma_1 \cup \Sigma_2)$-structures $\mathcal{A}$ for which the
reduct of $\mathcal{A}$ to $\Sigma_1$ is in $\mathcal{T}_1$ and the
......@@ -317,8 +318,7 @@ 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}
averages~\cite{BF15restarts}.
In addition to the mentioned heuristics, an efficient implementation
of \cdcl based \sat solver relies on lazy data structures used in the
......@@ -409,7 +409,12 @@ Notable examples of decidable first-order theories include
interpreted as a value on the index $i$ of the array $a$, and a
ternary function $\arwrite(a, i, v)$ interpreted as an array $a$
modified to contain the value $v$ on the index $i$;
\item the theory of \emph{fixed-size bit-vectors}. \marginpar{TODO: describe}
\item the theory of \emph{fixed-size bit-vectors}, in which the
structure contains all bit-vectors as an universe, ordering
relations according to the corresponding integers, and interpreted
functions are bit-wise operations on the bit-vectors and modular
arithmetic operations on the corresponding integers. This theory is
in detail described in section \ref{sec:qfbv}.
\end{itemize}
For a detailed description of these theories and implementation of the
respective $\mathcal{T}$-solvers, we refer the reader for example to the book of
......@@ -426,10 +431,12 @@ 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 a
many-sorted logic, the theory is \emph{stably infinite} if every
$\mathcal{T}$-satisfiable formula has a 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.
$\mathcal{T}$-satisfiable formula has a 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. Therefore, variants of the theory
combinations in which one of the theories does not have to be stably
infinite have been proposed~\cite{TZ05, RRZ05, JB10}.
\subsection{DPLL modulo theories}
......@@ -608,6 +615,7 @@ arithmetic~\cite{CHN12}, linear integer arithmetic~\cite{KLR10, GLS11,
and uninterpreted functions~\cite{McM11}.
\section{Satisfiability of quantifier-free bit-vector formulas}
\label{sec:qfbv}
The \emph{theory of fixed sized bit-vectors (\BV)} is a many-sorted
first-order theory with infinitely many sorts $\sort{n}$ corresponding
......
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