From 64af6363418431df21f1634f7602ca82b871a2aa Mon Sep 17 00:00:00 2001 From: Guillaume Bury Date: Tue, 18 Oct 2016 19:10:31 +0200 Subject: [PATCH] [doc] Updated inference rules --- docs/msat.pdf | Bin 332781 -> 343697 bytes docs/msat.tex | 78 +++++++++++++++++++++++++++++++++++++++++++------- 2 files changed, 67 insertions(+), 11 deletions(-) diff --git a/docs/msat.pdf b/docs/msat.pdf index d1a9d0f28b154723e2dba787598adad3b6c3210b..99075567c07e1720d0690f958cff6e79171a3c55 100644 GIT binary patch delta 95295 zcmZU3Q*dB`5@l@LPA0Z(b7E_fOl;#OnPB3HZQHhO+qRwTdv9yM_OH9DtG~KVpYC%m zi!ow9F=`qh!CBZz;z6mYflCdAm~>{8&ST9NHQZ8jTp+;vT(hh)LAABgOL*skjPf5| zoz?p}pZ`9hN<(>P1xDeIKLSsY+Z|l{K^yTwKDrcs9|=r$No3?CrlcG{Ueo+h9Lw19 z)bX5ir43^F-db(RKpUF5DH^WxvR|Lm3&BiG@>_|XD1#jTBEAr!04)8Oor8!OPgPMk z(TiOd*8Jg)nZIb(c36t>TZPhzGoyBKVSn@Jd{Do!ef19H$k$kM#r&~2Z$m~^1#vk( zE2?nEUpJ;s&5=-5AU7A!=1j&!y=AnI3 z{t-FTDw%#^G!L2Ut=;bQ;#=Y!WOe;dpC7d}V~-EBLC7iz0k}1snKt8e&?MgBh4I2A 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E3NeHxZvX%Q diff --git a/docs/msat.tex b/docs/msat.tex index 87cb2d2c..0043967c 100644 --- a/docs/msat.tex +++ b/docs/msat.tex @@ -96,7 +96,7 @@ The $\text{Solve}$ state take as argument the set of hypotheses and the trail, w the $\text{Analyze}$ state also take as argument the current conflict clause. We can now formalize the \cdcl{} algorithm using the inference rules -in Figure~\ref{fig:rules}. In order to completely recover the \sat{} algorithm, +in Figure~\ref{fig:smt_rules}. In order to completely recover the \sat{} algorithm, one must apply the rules with the following precedence and termination conditions, depending on the current state: \begin{itemize} @@ -153,7 +153,7 @@ depending on the current state: \end{tabular} \end{center} \begin{center} - \begin{tabular}{c@{\hspace{1cm}}l} + \begin{tabular}{c@{\hspace{.5cm}}l} % Analyze (propagation) \AXC{$\text{Analyze}(\mathbb{S}, t :: a \leadsto_C \top, D)$} \LLc{Analyze-propagation} @@ -197,7 +197,7 @@ depending on the current state: \begin{tabular}{c@{\hspace{1cm}}l} % Conflict (theory) \AXC{$\text{Solve}(\mathbb{S}, t)$} - \LLc{Conflict-Theory} + \LLc{Conflict-Smt} \UIC{$\text{Analyze}(\mathbb{S}, t, C)$} \DP{} & \begin{tabular}{l} @@ -207,30 +207,79 @@ depending on the current state: \\ \\ \end{tabular} \end{center} + \caption{Inference rules for \sat{} and \smt{}}\label{fig:smt_rules} +\end{figure} +\begin{figure} \center{\underline{\mcsat{}}} \begin{center} - \begin{tabular}{c@{\hspace{1cm}}l} + \begin{tabular}{c@{\hspace{.2cm}}l} % Decide (assignment) \AXC{$\text{Solve}(\mathbb{S}, t)$} + \LLc{Theory-Decide} \UIC{$\text{Solve}(\mathbb{S}, t :: a \mapsto_n v)$} \DP{} & - $a \notin t, a \in \mathbb{S}, n = \text{max\_level}(t) + 1$ + \begin{tabular}{l} + $a \mapsto_k \_ \notin t$ \\ + $n = \text{max\_level}(t) + 1$ \\ + $\sigma_{t :: a \mapsto_n v} \text{ compatible with } t$ + \end{tabular} \\ \\ % Propagation (semantic) \AXC{$\text{Solve}(\mathbb{S}, t)$} + \LLc{Propagate-Theory} \UIC{$\text{Solve}(\mathbb{S}, t :: a \leadsto_n \top)$} \DP{} & - $ $ + $\sigma_t(a) = \top$ + \\ \\ + % Conflict (semantic) + \AXC{$\text{Solve}(\mathbb{S}, t)$} + \LLc{Conflict-Mcsat} + \UIC{$\text{Analyze}(\mathbb{S}, t, C)$} + \DP{} & + \begin{tabular}{l} + $\mathcal{T} \vdash C$ \\ + $\forall a \in C, \neg a \in t \lor \sigma_t(a) = \bot$ + \end{tabular} \\ \\ \end{tabular} \end{center} - \caption{Inference rules}\label{fig:rules} + \begin{center} + \begin{tabular}{c@{\hspace{.2cm}}l} + % Analyze (assign) + \AXC{$\text{Analyze}(\mathbb{S}, t :: a \mapsto_n v, C)$} + \LLc{Analyze-Assign} + \UIC{$\text{Analyze}(\mathbb{S}, t, C)$} + \DP{} & + \\ \\ + % Analyze (semantic) + \AXC{$\text{Analyze}(\mathbb{S}, t :: a \leadsto_n \top, C)$} + \LLc{Analyze-Semantic} + \UIC{$\text{Analyze}(\mathbb{S}, t, C)$} + \DP{} & + \\ \\ + \end{tabular} + \end{center} + \begin{center} + \begin{tabular}{c@{\hspace{.2cm}}l} + % Backjump (semantic) + \AXC{$\text{Analyze}(\mathbb{S}, t :: a \mapsto_n \_ :: \_, C)$} + \LLc{Backjump-Semantic} + \UIC{$\text{Solve}(\mathbb{S}, t)$} + \DP{} & + \begin{tabular}{l} + $\text{is\_semantic}(C)$ \\ + $n = \text{slevel}(C)$ \\ + \end{tabular} + \\ \\ + \end{tabular} + \end{center} + \caption{\mcsat{} specific inference rules}\label{fig:mcsat_rules} \end{figure} \subsection{Invariants, correctness and completeness} -The following invariants are maintained by the inference rules in Figure~\ref{fig:rules}: +The following invariants are maintained by the inference rules in Figure~\ref{fig:smt_rules}: \begin{description} \item[Trail Soundness] In $\text{Solve}(\mathbb{S}, t)$, if $a \in t$ then $\neg a \notin t$ \item[Conflict Analysis] In $\text{Analyze}(\mathbb{S}, t, C)$, $C$ is a clause implied by the @@ -277,9 +326,9 @@ The role of the theory $\mathcal{T}$ is to stop the proof search as soon as the of the \sat{} solver is inconsistent. A trail is inconsistent iff there exists a clause $C$, which is a tautology of $\mathcal{T}$ (thus $\mathcal{T} \vdash C$), but is not satisfied in the current trail (each of its literals has either been decided or -propagated to false). Thus, we can add the \irule{Conflict-Theory} rule (see -Figure~\ref{fig:rules}) to the \cdcl{} inference rules in order to get a \smt{} solver. -We give the \irule{Conflict-Theory} rule a slightly lower precedence than the +propagated to false). Thus, we can add the \irule{Conflict-Smt} rule (see +Figure~\ref{fig:smt_rules}) to the \cdcl{} inference rules in order to get a \smt{} solver. +We give the \irule{Conflict-Smt} rule a slightly lower precedence than the \irule{Conflict} rule for performance reason (detecting boolean conflict is faster than theory specific conflicts). @@ -544,6 +593,13 @@ set of terms of $T$ whose head symbol is uninterpreted (i.e.~not defined by the it is enough to ensure that the mapping is complete, because the theory will automatically constrain the value of terms whose head symbol is interpreted. +\subsection{Inference rules} + +In \mcsat{}, a trail $t$ contains boolean decision and propagations as well as semantic +decisions (assignments) and propagations. We can thus define $\sigma_t$ as the mapping +formed by all assignments in $t$. This lets us define the last rules in Figure~\ref{fig:mcsat_rules}, +that, together with the other rules, define the \mcsat{} algorithm. + \bibliographystyle{plain} \bibliography{biblio}