\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{makeidx}
\usepackage{bussproofs}
\usepackage{amsmath,amssymb}
\usepackage{tabularx}
\usepackage{pgf, tikz}
\usepackage[margin=4cm]{geometry}

\usepackage{float}
\floatstyle{boxed}
\restylefloat{figure}

\usepackage{listings}
\lstset{language=[Objective]caml}

\input{macros}

\usetikzlibrary{arrows, automata}

\begin{document}

\title{\msat{}: a SAT/SMT/McSAT library}
\author{Guillaume~Bury}

\maketitle

\section{Introduction}

The goal if the \msat{} library is to provide a way to easily
create atomated theorem provers based on a \sat{} solver. More precisely,
the library, written in \ocaml{}, provides functors which, once instantiated,
provide a \sat{}, \smt{} or \mcsat{} solver.

Given the current state of the art of \smt{} solvers, where most \sat{} solvers
are written in C and heavily optimised\footnote{Some solvers now have instructions
to manage a processor's cache}, the \msat{} library does not aim to provide solvers
competitive with the existing implementations, but rather an easy way to create
reasonably eficient solvers.

\msat{} currently uses the following techniques:
\begin{itemize}
  \item 2-watch literals scheme
  \item Activity based decisions
  \item Restarts
\end{itemize}

Additionally, \msat{} has the following features:
\begin{itemize}
  \item Local assumptions
  \item Proof/Model output
  \item Adding clauses during proof search
\end{itemize}

\clearpage

\tableofcontents{}

\clearpage

\section{\sat{} Solvers: principles and formalization}

\subsection{Idea}

\subsection{Inference rules}

The SAT algorithm can be formalized as follows. During the search, the solver keeps
a set of clauses, containing the problem hypotheses and the learnt clauses, and
a trail, which is the current ordered list of assumptions and/or decisions made by
the solver.

Each element in the trail (decision or propagation) has a level, which is the number of decision
appearing in the trail up to (and including) it. So for instance, propagations made before any
decisions have level $0$, and the first decision has level $1$. Propagations are written
$a \leadsto_C \top$, with $C$ the clause that caused the propagation, and decisions
$a \mapsto_n \top$, with $n$ the level of the decision. Trails are read
chronologically from left to right.

In the following, given a trail $t$ and an atomic formula $a$, we will use the following notation:
$a \in t$ if $a \mapsto_n \top$ or $a \leadsto_C \top$ is in $t$, i.e $a \in t$ is $a$ is true
in the trail $t$. In this context, the negation $\neg$ is supposed to be involutive (i.e $\neg \neg a = a$),
so that, if $a \in t$ then $\neg \neg a = a \in t$.

The SAT algorithm has two states: first, it starts in the $\text{Solve}$ state, where propagations
and decisions are made, until a conflict is detected, at which point it enters in the $\text{Analyse}$
state, where it analyzes the conflict, backtracks, and re-enter the $\text{Solve}$ state.
The $\text{Solve}$ state take as argument the set of hypotheses and the trail, while
the $\text{Analyze}$ state also take as argument the current conflict clause.

We can now formalize the \sat{} algorithm using the inference rules
in Figure~\ref{fig: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}
  \item If the empty clause is in $\mathbb{S}$, then the problem is unsat.
    If there is no more rule to apply, the problem is sat.
  \item If we are in $\text{Solve}$ mode:
    \begin{enumerate}
      \item First is the rule \irule{Conflict};
      \item Then the try and use \irule{Propagate};
      \item Finally, is there is nothing left to be propagated, the \irule{Decide}
        rule is used.
    \end{enumerate}
  \item If we are in $\text{Analyze}$ mode, we have a choice concerning
    the order of application. First we can observe that the rules
    \irule{Analyze-Propagate}, \irule{Analyze-Decision} and \irule{Analyze-Resolution}
    can not apply simultaneously, and we will thus group them in a super-rule
    \irule{Analyze}. We now have the choice of when to apply the \irule{Backjump} rule
    compared to the \irule{Analyze} rule:
    using \irule{Backjump} eagerly will result in the first UIP strategy, while delaying it
    until later will yield the last UIP strategy, both of which are valid.
\end{itemize}

\begin{figure}
  \center{\underline{\sat{}}}
  \begin{center}
    \begin{tabular}{c@{\hspace{1cm}}l}
      % Propagation (boolean)
      \AXC{$\text{Solve}(\mathbb{S}, t)$}
      \LLc{Propagate}
      \UIC{$\text{Sove}(\mathbb{S}, t :: a \leadsto_C \top)$}
      \DP{} &
      \begin{tabular}{l}
        $a \in C, C \in \mathbb{S}, \neg a \notin t$ \\
        $\forall b \in C. b \neq a \rightarrow \neg b \in t$ \\
      \end{tabular}
      \\ \\
      % Decide (boolean)
      \AXC{$\text{Solve}(\mathbb{S}, t)$}
      \LLc{Decide}
      \UIC{$\text{Solve}(\mathbb{S}, t :: a \mapsto_n \top)$}
      \DP{} &
      \begin{tabular}{l}
        $a \notin t, \neg a \notin t, a \in \mathbb{S}$ \\
        $n = \text{max\_level}(t) + 1$ \\
      \end{tabular}
      \\ \\
      % Conflict (boolean)
      \AXC{$\text{Solve}(\mathbb{S}, t)$}
      \LLc{Conflict}
      \UIC{$\text{Analyze}(\mathbb{S}, t, C)$}
      \DP{} &
      $C \in \mathbb{S}, \forall a \in C. \neg a \in t$
      \\ \\
    \end{tabular}
  \end{center}
  \begin{center}
    \begin{tabular}{c@{\hspace{1cm}}l}
      % Analyze (propagation)
      \AXC{$\text{Analyze}(\mathbb{S}, t :: a \leadsto_C \top, D)$}
      \LLc{Analyze-propagation}
      \UIC{$\text{Analyze}(\mathbb{S}, t, D)$}
      \DP{} &
      $\neg a \notin D$
      \\ \\
      % Analyze (decision)
      \AXC{$\text{Analyze}(\mathbb{S}, t :: a \mapsto_n \top, D)$}
      \LLc{Analyze-decision}
      \UIC{$\text{Analyze}(\mathbb{S}, t, D)$}
      \DP{} &
      $\neg a \notin D$
      \\ \\
      % Resolution
      \AXC{$\text{Analyze}(\mathbb{S}, t :: a \leadsto_C \top, D)$}
      \LLc{Analyze-Resolution}
      \UIC{$\text{Analyze}(\mathbb{S}, t, (C - \{a\}) \cup (D - \{ \neg a\}))$}
      \DP{} &
      $\neg a \in D$
      \\ \\
      % BackJump
      \AXC{$\text{Analyze}(\mathbb{S}, t :: a \mapsto_d \top :: t', C)$}
      \LLc{Backjump}
      \UIC{$\text{Solve}(\mathbb{S} \cup \{ C \}, t)$}
      \DP{} &
      \begin{tabular}{l}
        $\text{is\_uip}(C)$ \\
        $d = \text{uip\_level}(C)$ \\
      \end{tabular}
      \\ \\
    \end{tabular}
  \end{center}

  \center{\underline{\smt{}}}
  \begin{center}
    \begin{tabular}{c@{\hspace{1cm}}l}
      % Conflict (theory)
      \AXC{$\text{Solve}(\mathbb{S}, t)$}
      \LLc{Conflict-Theory}
      \UIC{$\text{Analyze}(\mathbb{S}, t, C)$}
      \DP{} &
      \begin{tabular}{l}
        $\mathcal{T} \vdash C$ \\
        $\forall a \in C, \neg a \in t$ \\
      \end{tabular}
      \\ \\
    \end{tabular}
  \end{center}

  \center{\underline{\mcsat{}}}
  \begin{center}
    \begin{tabular}{c@{\hspace{1cm}}l}
      % Decide (assignment)
      \AXC{$\text{Solve}(\mathbb{S}, t)$}
      \UIC{$\text{Solve}(\mathbb{S}, t :: a \mapsto_n v)$}
      \DP{} &
      $a \notin t, a \in \mathbb{S}, n = \text{max\_level}(t) + 1$
      \\ \\
      % Propagation (semantic)
      \AXC{$\text{Solve}(\mathbb{S}, t)$}
      \UIC{$\text{Solve}(\mathbb{S}, t :: a \leadsto_n \top)$}
      \DP{} &
      $ $
      \\ \\
    \end{tabular}
  \end{center}
  \caption{Inference rules}\label{fig:rules}
\end{figure}

\subsection{Invariants, correctness and completeness}

The following invariants are maintained by the inference rules in Figure~\ref{fig: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
    clauses in $\mathbb{S}$, and $\forall a \in C, \neg a \in t$ (i.e $C$ is entailed, yet false
    in the partial model formed by the trail $t$).
  \item[Equivalence] In any rule \AXC{$s_1$}\UIC{$s_2$}\DP{}, the set of hypotheses
    (usually written $\mathbb{S}$) in $s_1$ is equivalent to that of $s_2$.
\end{description}

These invariants are relatively easy to prove, and provide an easy proof of correctness for
the \sat{} algorithm. Termination can be proved by observing that the same trail appears
at most twice during proof search (once during propagation, and eventually a second time
right after backjumping\footnote{This could be avoided by making the \irule{Backjump} rule
directly propagate the relevant litteral of the conflict clause, but it needlessly
complicates the rule.}). Correctness and termination implies completeness of the \sat{}
algorithm.


\section{\smt{} solver architecture}

\subsection{Idea}

We can represent a simplified version of the information flow (not taking into
account backtracking) of usual SMT Solvers, using the graph in fig~\ref{fig:smt_flow}.
In a \smt{} solver, after each propagation and decision, the solver sends the newly
assigned literals to the theory. The theory then has the possibility to declare the
current set of literals incoherent, and give the solver a tautology in which all
literals are currently assigned to $\bot$\footnote{or rather for each literal, its negation
is assigned to $\top$}, thus prompting the solver to backtrack.

\subsection{Formalization and Theory requirements}

An \smt{} solver is the combinaison of a \sat{} solver, and a theory $\mathcal{T}$.
The role of the theory $\mathcal{T}$ is to stop the proof search as soon as the trail
of the \sat{} solver is inconsistent. Thus, we can add the \irule{Conflict-Theory} rule
to the \sat{} ifnerence rules in order to get a \smt{} solver. We give a slightly lower
precedence than the \irule{Conflict} rule for performance reason (detecting boolean
conflict is faster than theory specifi conflicts).

So, what is the minimum that a theory must implement to be used in a \smt{} solver~?
Given its precedence, all a theory has to do in a \smt{} sovler, is to ensure that
the current trail is consistent (when seens as a conjunction of literals). This
reflects the fact that the \sat{} core takes care of the purely propositional content
of the input problem, and leaves the theory specific reasoning to the theory, which
then does not have to deal with propositional content such as disjunction. The theory,
however, must have a global reasoning to ensure that the whole trail is consistent.

\begin{figure}
  \begin{center}
    \begin{tikzpicture}[
        ->, % Arrow style
        > = stealth, % arrow head style
        shorten > = 1pt, % don't touch arrow head to node
        node distance = 2cm, % distance between nodes
        semithick, % line style
        auto
      ]

      \tikzstyle{state}=[rectangle,draw=black!75]

      \node (sat) {SAT Core};
      \node (th) [right of=sat, node distance=6cm] {Theory};
      \node[state] (d) [below of=sat, node distance=1cm] {Decision (boolean)};
      \node[state] (bp) [below of=d, node distance=2cm] {Boolean propagation};
      \node[state] (tp) [right of=bp, node distance=6cm] {Theory propagation};

      \draw (d) edge [bend left=30] (bp);
      \draw (bp) edge [bend left=30] (d);
      \draw (bp) edge (tp);

      \draw[black!50] (-2,1) rectangle (2,-4);
      \draw[black!50] (4,1) rectangle (8,-4);

    \end{tikzpicture}
  \end{center}
  \caption{Simplified SMT Solver architecture}\label{fig:smt_flow}
\end{figure}



\section{McSat: An extension of SMT Solvers}

\subsection{Introduction}


Mcsat is an extension of usual SMT solvers, introduced in~\cite{VMCAI13} and~\cite{FMCAD13}.
In usual SMT Solvers, interaction between the core SAT Solver and the Theory is pretty limited~:
the SAT Solver make boolean decision, and sends them to the theory, whose role is in return to
stop the SAT Solver as soon as the current set of assumptions is incoherent. This means
that the information that theories can give the SAT Solver is pretty limited, and furthermore
it limits the ability of theories to guide the proof search.

While it appears to leave a reasonably simple job to the theory, since it completely
hides the propositional structure of the problem, this simple interaction between the
SAT Solver and the theory makes it harder to combine multiple theories into one. Usual
techniques for combining theories in SMT solvers typically require to keep track of
equality congruence classes and require of theories to propagate any equality they discover.

McSat extends the SAT paradigm by allowing more exchange of information between the theory
and the SAT Solver. This is achieved by allowing the solver to not only decide on the truth value
of atomic propositions, but also to decide assignments for terms that appear in the problem.
For instance, if the SAT Solver assumes a formula $x = 0$,
an arithmetic theory could propagate to the SAT Solver that the formula $x < 1$ must also hold,
instead of waiting for the SAT Solver to guess the truth value of $x < 1$ and then
inform the SAT Solver that the conjunction~: $x = 0 \land \neg x < 1$ is incoherent.

This exchange of information between the SAT Solver and the theories results in
the construction of a model throughout the proof search (which explains the name
Model Constructing SAT).

The main addition of McSat is that when the solver makes a decision, instead of
being restricted to making boolean assignment of formulas, the solver now can
decide to assign a value to a term belonging to one of the literals. In order to do so,
the solver first chooses a term that has not yet been assigned, and then asks
the theory for a possible assignment. Like in usual SMT Solvers, a McSat solver
only exchange information with one theory, but, as we will see, combination
of theories into one becomes easier in this framework.

Using the assignments on terms, the theory can very easily do efficient
propagation of formulas implied by the current assignments: is suffices to
evaluate formulas using the current partial assignment.
The information flow then looks like fig~\ref{fig:mcsat_flow}.

\begin{figure}
  \begin{center}
    \begin{tikzpicture}[
        ->, % Arrow style
        > = stealth, % arrow head style
        shorten > = 1pt, % don't touch arrow head to node
        node distance = 2cm, % distance between nodes
        semithick, % line style
        auto
      ]

      \tikzstyle{state}=[rectangle,draw=black!75]

      \node (sat) {SAT Core};
      \node (th) [right of=sat, node distance=6cm] {Theory};
      \node[state] (d) [below of=sat, node distance=1cm] {Decision};
      \node[state] (ass) [right of=d, node distance=6cm] {Assignment};
      \node[state] (bp) [below of=d, node distance=2cm] {Boolean propagation};
      \node[state] (tp) [right of=bp, node distance=6cm] {Theory propagation};

      \draw (bp)[right] edge [bend left=5] (tp);
      \draw (tp) edge [bend left=5] (bp);
      \draw (bp) edge [bend left=30] (d);
      \draw (ass) edge [bend left=5] (d);
      \draw (d) edge [bend left=5] (ass);
      \draw (d) edge [bend left=30] (bp);
      \draw[<->] (ass) edge (tp);

      \draw[black!50] (-2,1) rectangle (2,-4);
      \draw[black!50] (4,1) rectangle (8,-4);

    \end{tikzpicture}
  \end{center}
  \caption{Simplified McSat Solver architecture}\label{fig:mcsat_flow}
\end{figure}

\subsection{Decisions and propagations}

In this document, semantic propagations are a bit different from the propagations
used in traditional SMT Solvers. In the case of McSat (or at least the version presented here),
semantic propagations strictly correspond to the evaluation of formulas in the
current assignment. Moreover, in order to be able to correctly handle these semantic
propagations during backtrack, they are assigned a level: each decision is given a level
(using the same process as in a \sat{} solvers: a decision level is the number of decision
previous to it, plus one), and a formula is propagated at the maximum level of the decisions
used to evaluate it.

For instance, if the current trail is $\{x \mapsto_1 0, x + y + z = 0 \mapsto_2 \top, y\mapsto_3 0\}$,
then $x + y = 0$ can be propagated at level $3$, but $z = 0$ can not be propagated (at least not
given to the sat solver, however nothing prevents the theory from propagating and using it internally).
\subsection{First order terms}

In the following, we use the following notations:
\begin{itemize}
  \item $\mathbb{V}$ is an infinite set of variables
  \item $\mathbb{C}$ is a possibly infinite set of constants defined
    by a problem's logic\footnote{For instance, the theory of arithmetic
    defines the usual operators $+, -, *, /$ as well as the constants
    $0, -5, \frac{1}{2}, -2.3, \ldots$}
  \item $\mathbb{S}$ is a finite set of constants defined by a problem's type definitions
  \item $\mathbb{T}(\mathbb{X})$ for the (infinite) set of first-order terms over $\mathbb{X}$
    (for instance $a, f(0), x + y, \ldots$)
  \item $\mathbb{F}(\mathbb{T})$ for the (infinite) set of first order quantified formulas
    over the terms in $\mathbb{T}$
\end{itemize}

\subsection{Models}

A model traditionally is a triplet which comprises a domain, a signature and an interpretation function.
Most problems define their signature using type definitions, and builtin theories such as arithmetic
usually have canonic models, so in the following we consider the domain and siganture constant, and
talk about the interpretation, more specifically, about the interpretation of symbols not defined
by the theory\footnote{Since theory-defined symbols, such as addition, already have an intepretation
in the canonical domain.}, i.e non-interpreted functions. An intepretation $\mathcal{I}$ can easily
be extended to a function from ground terms to model value by recursively applying it:
\[
  \mathcal{I}( f ( {(e_i)}_{1\leq i \leq n})) = \mathcal{I}_f ( {( \mathcal{I}(e_i) )}_{1 \leq i \leq n} )
\]

Before formalizing the SAT, SMT and McSat algorithms as inference rules, we need to formalize the notion
of partial interpretation. Indeed, during the proof search, the McSat algorithm maintains a partial mapping from
expressions to model values. The intention of this mapping is to represent a partial interpretation of the input
problem, so that if the mapping is complete (i.e all variables are assigned), then it directly gives an interpretation
of the input problem (quantified formula notwithstanding).

More than simply an incomplete interpretation, we also want to be able to give partial function (instead of complete functions) as interpretation of
constants with positive arity. And even further, we'd like to specify these partial interpretations in a somewhat
abstract way, using mappings from expressions to model values instead of a function from model values to model
values. For instance, given a function symbol $f$ of type $\text{int} \rightarrow \text{int}$ and an integer constant $a$, we'd like to specify that in our mapping,
$f(a) \mapsto 0$, regardless of the values that $f$ takes on other argument, and also regardless
of the value mapped to $a$. To that end we introduce a notion of abstract partial interpretation.

An abstract partial interpretation $\sigma$ is a mapping from ground expressions to model values. To each abstract partial
interpretation correspond a set of complete models that realize it. More precisely, any mapping $\sigma$ can be completed
in various ways, leading to a set of potential interpretations:
\[
  \text{Complete}(\sigma) =
    \left\{
      \mathcal{I}
      \; | \;
      \forall f( {(e_i)}_{1 \leq i \leq n} ) \mapsto v \in \sigma ,
        \mathcal{I}_f ( {( \mathcal{I}(e_i) )}_{1 \leq i \leq n} ) = v
    \right\}
\]

We can then consider that the natural interpretation corresponding to a given mapping is the
partial interpretation on which all completion of the mapping agrees, i.e the intersection of
all potential candidates:
\[
  \sigma_\mathcal{I} = \bigcap_{ \mathcal{I} \in \text{Complete}(\sigma) } \mathcal{I}
\]

Of course, it might happen that a mapping does not admit any potential interpretation,
and thus has no natural interpretation, for instance there is no possible completion of the
following mapping:
\[
  \sigma = \left\{
    \begin{matrix}
      a \mapsto 0 \\
      b \mapsto 0 \\
      f(a) \mapsto 0 \\
      f(b) \mapsto 1 \\
    \end{matrix}
  \right.
\]


\subsection{Expected theory invariants}

TODO: rewrite this section.

During proof search are maintained a set of assertions $\mathcal{S}$
and a partial assignment $\sigma$ as a partial function from $\mathbb{T}(\mathbb{V, C, S})$
to $\mathbb{T}(\mathbb{C})$. $\sigma$ is easily extended to a complete substitution function of
$\mathbb{T}(\mathbb{V, C, S})$.

We say that $\sigma$ is compatible with $\mathcal{S}$ iff for every $\varphi \in \mathcal{S}$,
$\varphi\sigma$ is satisfiable (independently from the rest of the formulas in $\mathcal{S}$).
Intuitively, this represents the fact that the substitution $\sigma$ does not imply any
ground contradictions.

A theory then has to ensure that for every expression $e \in \mathbb{T}(\mathbb{V,C,S})$
there exists $v \in \mathbb{T}(\mathbb{C})$ such that $\sigma' = \sigma \cup \{ e \rightarrow v \}$
is coherent, and compatible with $\mathcal{S}$\footnote{Note that in particular, this implies
that $\sigma$ is coherent with $\mathcal{S}$}. As soon as the current assignment is not coherent,
compatible with the current assertions, or that there is a term $e$ with no viable assignment, the theory
should inform the SAT Solver to backtrack, since the current branch is clearly not satisfiable.
If we reach the point where all expressions of the problem have been assigned, then we
have a ground model for the current set of assertions, which is then also a model
of the input problem.

Of interest if the fact that theories that respect this invariant can easily be combined
to create a theory that also respect this invariant, as long as for any expression $e$,
there is exactly one theory responsible for finding an assignment for $e$\footnote{this
is similar to the usual criterion for combining SMT theories which is that they do not
share symbols other than equality.}.


\clearpage

\bibliographystyle{plain}
\bibliography{biblio}

\clearpage
\appendix

\end{document}