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@ -8,10 +8,10 @@ $o$ subject to an (existentially quantified) conjunction of primitive
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constraints $c$. But modelling languages such as OPL
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\autocite{van-hentenryck-1999-opl}, Zinc/\minizinc\
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\autocite{marriott-2008-zinc, nethercote-2007-minizinc} and Essence
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\cite{frisch-2007-essence} allow much more expressive problems to be formulated.
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Modelling languages map the more expressive formulations to existentially
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quantified conjunction through a combination of loop unrolling, and flattening
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using reification.
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\autocite{frisch-2007-essence} allow much more expressive problems to be
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formulated. Modelling languages map the more expressive formulations to
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existentially quantified conjunction through a combination of loop unrolling,
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and flattening using reification.
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\begin{example}\label{ex:cons}
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Consider the following ``complex constraint'' written in \minizinc\ syntax
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@ -708,7 +708,7 @@ The flattening proceeds by evaluating fixed Boolean expressions and returning
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the value. We assume $\fixed$ checks if an expression is fixed (determined
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during \minizinc's type analysis), and $\eval$ evaluates a fixed expression. For
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simplicity of presentation, we assume that fixed expressions are never undefined
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(such as \lstinline|3 div 0|). We can extend the algorithms to handle this case,
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(such as \mzninline{3 div 0}). We can extend the algorithms to handle this case,
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but it complicates the presentation. Our implementation aborts in this
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case.\pjs{CHECK and FIX!} While this does not agree with the relational
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semantics it likely indicates a modelling error since a fixed part of the model
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@ -734,21 +734,21 @@ creates an $\mathit{element}$ constraint. If the context is root this is simple,
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otherwise it creates a new index varaible $v'$ guaranteed to only give correct
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answers, and uses this in the $\mathit{element}$ constraint to avoid
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undefinedness. Built-in predicates abort if not in the root
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context.\short{\footnote{Using half reification this can be relaxed to also
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allow positive contexts.}} They flatten their arguments and add an
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appropriate built-in constraint. \callbyvalue{ User defined predicate
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applications flatten their arguments and then flatten a renamed copy of the
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body.} \callbyunroll{ User defined predicate applications are replaced by a
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renamed and flattened copy of the body.} \emph{if-then-else} evaluates the
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condition (which must be fixed) and flattens the \emph{then} or \emph{else}
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branch appropriately. \pjs{USe the same resdtriction, but explain its not all
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required} The handling of \texttt{let} is the most complicated. The expression
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is renamed with new copies of the let variables. We extract the constraints from
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the \texttt{let} expression using function \textsf{flatlet} which returns the
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extracted constraint and a rewritten term (not used in this case, but used in
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\flatt{}). The constraints returned by function \textsf{flatlet} are then
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flattened. Finally if we are in the root context, we ensure that the Boolean $b$
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returned must be $\true$ by adding $b$ to $S$.
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context.\footnote{Using half reification this can be relaxed to also allow
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positive contexts.} They flatten their arguments and add an appropriate
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built-in constraint. \callbyvalue{ User defined predicate applications flatten
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their arguments and then flatten a renamed copy of the body.} \callbyunroll{
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User defined predicate applications are replaced by a renamed and flattened
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copy of the body.} \emph{if-then-else} evaluates the condition (which must be
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fixed) and flattens the \emph{then} or \emph{else} branch appropriately.
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\pjs{USe the same resdtriction, but explain its not all required} The handling
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of \texttt{let} is the most complicated. The expression is renamed with new
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copies of the let variables. We extract the constraints from the \texttt{let}
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expression using function \textsf{flatlet} which returns the extracted
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constraint and a rewritten term (not used in this case, but used in \flatt{}).
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The constraints returned by function \textsf{flatlet} are then flattened.
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Finally if we are in the root context, we ensure that the Boolean $b$ returned
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must be $\true$ by adding $b$ to $S$.
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{
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\begin{tabbing}
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@ -1229,10 +1229,10 @@ $S$ \cupp $\{ \new b \full (b_1 \wedge b_2)\}$
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% $b \not\in vars(c)$ is a Boolean variable naming the satisfied state of the
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% constraint $c$.
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% %\newcommand{\new}{\textbf{new}~}
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% %\newcommand{\flatc}{\textsf{flatc}}
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% %\newcommand{\flatt}{\textsf{flatt}}
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% \newcommand{\safen}{\textsf{safen}}
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%\newcommand{\new}{\textbf{new}~}
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%\newcommand{\flatc}{\textsf{flatc}}
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%\newcommand{\flatt}{\textsf{flatt}}
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\newcommand{\safen}{\textsf{safen}}
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% The pseudo-code for \flatc($b$,$c$) flattens a constraint expression $c$ to be
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% equal to $b$, returning a set of constraints implementing $b \full c$. We
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@ -1376,16 +1376,15 @@ $$
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% and if so we propagate changes in variable domains $prop(c)$.
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% \begin{example}
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% Most propagators check consistency before propagating: e.g.
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% $\sum_i a_i x_i \leq a_0$ determines $L = \sum_i min_D(a_i x_i) - a_0$
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% and fails if $L > 0$ before propagating~\autocite{harveyschimpf};
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% Regin's domain propagator~\autocite{reginalldifferent} for \alldiff$([x_1, \ldots, x_n])$
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% determines a maximum matching between variables and values first,
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% if this is not of size $n$ it fails before propagating;
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% and the timetable \texttt{cumulative} constraint~\autocite{cumulative}
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% determines a profile of
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% necessary resource usage, and fails if this breaks the resource limit,
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% before considering propagation. \qed
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% Most propagators check consistency before propagating: e.g.
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% $\sum_i a_i x_i \leq a_0$ determines $L = \sum_i min_D(a_i x_i) - a_0$ and
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% fails if $L > 0$ before propagating \autocite{harveyschimpf}; Regin's domain
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% propagator \autocite{reginalldifferent} for \alldiff$([x_1, \ldots, x_n])$
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% determines a maximum matching between variables and values first, if this is
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% not of size $n$ it fails before propagating; and the timetable
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% \texttt{cumulative} constraint~\autocite{cumulative} determines a profile of
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% necessary resource usage, and fails if this breaks the resource limit,
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% before considering propagation. \qed
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% \end{example}
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% Given this separation we can improve the definition of $f_{b \half c}$ above by
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@ -1499,8 +1498,8 @@ not propagate unnecessarily.
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\end{mzn}
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which may seem to be more expensive since there are additional variables (the
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$b4[i]$), but since both $b4[i]$ and $a[i]$ are implemented by
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views~\autocite{views}, there is no additional runtime overhead. This
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$b4[i]$), but since both $b4[i]$ and $a[i]$ are implemented by views
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\autocite{schulte-2005-views}, there is no additional runtime overhead. This
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decomposition will only wake the linear constraint when some task $i$ is
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guaranteed to overlap time $s[j]$.
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\end{example}
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@ -2156,55 +2155,56 @@ the usual case for such a trivial propagator.
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% consistent propagator for $b \half b''$. Similarly define
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% $f_{b' \half \neg c}(D) =_{vars(b' \full c)} \solv(\{f_{b''' \full c}, f_{b' \half \neg b'''} \}, D)$
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\begin{theorem}
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$\forall D. \solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D) = \solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)$.
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\end{theorem}
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\begin{proof}
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Let $V = vars(c)$. First w.l.o.g. we only consider domains $D$ at a fixpoint
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of the propagators $f_{b' \full \neg b}$, i.e.
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$D(b') = \{ \neg d ~|~ d \in D(b)\}$, since both $\solv$ expressions include
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$f_{b' \full \neg b}$. Let
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$D' = \solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D)$, now since
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$f_{b \full c}$ and $f_{b' \full \neg b}$ are idempotent and $D$ is a fixpoint
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of $f_{b' \full \neg b}$, then $D' = f_{b' \full \neg b}(f_{b \full c}(D))$.
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Let
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$D'' = \solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)$.
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The proof is by cases of $D$.
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\jip{There is a proof here, but I don't have a proof environment yet}
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% \begin{theorem}
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% $\forall D. \solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D) = \solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)$.
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% \end{theorem}
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% \begin{proof}
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% Let $V = vars(c)$. First w.l.o.g. we only consider domains $D$ at a fixpoint
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% of the propagators $f_{b' \full \neg b}$, i.e.
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% $D(b') = \{ \neg d ~|~ d \in D(b)\}$, since both $\solv$ expressions include
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% $f_{b' \full \neg b}$. Let
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% $D' = \solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D)$, now since
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% $f_{b \full c}$ and $f_{b' \full \neg b}$ are idempotent and $D$ is a fixpoint
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% of $f_{b' \full \neg b}$, then $D' = f_{b' \full \neg b}(f_{b \full c}(D))$.
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% Let
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% $D'' = \solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)$.
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% The proof is by cases of $D$.
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\textbf{(a)} Suppose $D(b) = \{\true,\false\}$. \textbf{(a-i)} Suppose
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$f_{b \full c}(D) = D$. Then clearly $f_{b \half c}(D) = D$ by definition, and
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similarly $f_{b' \half \neg c}(D) = D$. Hence $D_1 = D_2 = D$. Note that since
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$f_{b \full c}$ is reified consistent, then $f_{b \full c}(D) \neq D$ implies
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that $f_{b \full c}(D)(b) \neq \{ \false, \true \}$. \textbf{(a-ii)} Suppose
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$f_{b \full c}(D)(b) = \{ \true \}$. Let $D_1 = f_{b' \half \neg c}(D)$, then
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$D_1(b') = \{ \false \}$ by definition and $D_1(v) = D(v), v \not\equiv b'$.
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Let $D_2 = f_{b' \full \neg b}(D_1)$ then $D_2(b)= \{ \true \}$ and
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$D_2(v) = D(v), v \not\in \{b,b'\}$. Then
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$D_3 = f_{b \half c}(D_2) = f_{b \full c}(D_2)$ by definition, and by
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idempotence of $f_{b \full c}$ and since $D_2$ and $D$ differ only on $b$ and
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$b'$ this is a fixpoint of $f_{b \full c}$ and
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$D_3(v) = f_{b \full c}(D)(v), v \not \equiv b'$. $D_3$ is also a fixpoint of
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$f_{b \half c}$, $f_{b' \full \neg b}$ and $f_{b' \half \neg c}$ and hence
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$D'' = D_3$. But also $D' = f_{b' \full \neg b}(f_{b \full c}(D)) = D_3$ since
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$D_3$ and $f_{b \full c}(D)$ only differ on $b'$. \textbf{(a-iii)} Suppose
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$f_{b \full c}(D)(b) = \{ \false \}$. Let $D_1 = f_{b \half c}(D)$, then
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$D_1(b) = \{ \false \}$ by definition and $D_1(v) = D(v), v \not\equiv b$. Let
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$D_2 = f_{b' \full \neg b}(D_1)$ then $D_2(b')= \{ \true \}$ and
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$D_2(v) = D(v), v \not\in \{b,b'\}$. Then
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$D_3 = f_{b' \half \neg c}(D_2) = f_{b \full c}(D_2)$ by definition, and by
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idempotence of $f_{b \full c}$ and since $D_2$ and $D$ differ only on $b$ and
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$b'$ this is a fixpoint for $f_{b \half c}$, $f_{b' \full \neg b}$ and
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$f_{b' \half \neg c}$ and hence $D'' = D_3$. But also
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$D' = f_{b' \full \neg b}(f_{b \full c}(D)) = D_3$ since $D_3$ and
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$f_{b \full c}(D)$ only differ on $b'$.
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% \textbf{(a)} Suppose $D(b) = \{\true,\false\}$. \textbf{(a-i)} Suppose
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% $f_{b \full c}(D) = D$. Then clearly $f_{b \half c}(D) = D$ by definition, and
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% similarly $f_{b' \half \neg c}(D) = D$. Hence $D_1 = D_2 = D$. Note that since
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% $f_{b \full c}$ is reified consistent, then $f_{b \full c}(D) \neq D$ implies
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% that $f_{b \full c}(D)(b) \neq \{ \false, \true \}$. \textbf{(a-ii)} Suppose
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% $f_{b \full c}(D)(b) = \{ \true \}$. Let $D_1 = f_{b' \half \neg c}(D)$, then
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% $D_1(b') = \{ \false \}$ by definition and $D_1(v) = D(v), v \not\equiv b'$.
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% Let $D_2 = f_{b' \full \neg b}(D_1)$ then $D_2(b)= \{ \true \}$ and
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% $D_2(v) = D(v), v \not\in \{b,b'\}$. Then
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% $D_3 = f_{b \half c}(D_2) = f_{b \full c}(D_2)$ by definition, and by
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% idempotence of $f_{b \full c}$ and since $D_2$ and $D$ differ only on $b$ and
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% $b'$ this is a fixpoint of $f_{b \full c}$ and
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% $D_3(v) = f_{b \full c}(D)(v), v \not \equiv b'$. $D_3$ is also a fixpoint of
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% $f_{b \half c}$, $f_{b' \full \neg b}$ and $f_{b' \half \neg c}$ and hence
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% $D'' = D_3$. But also $D' = f_{b' \full \neg b}(f_{b \full c}(D)) = D_3$ since
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% $D_3$ and $f_{b \full c}(D)$ only differ on $b'$. \textbf{(a-iii)} Suppose
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% $f_{b \full c}(D)(b) = \{ \false \}$. Let $D_1 = f_{b \half c}(D)$, then
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% $D_1(b) = \{ \false \}$ by definition and $D_1(v) = D(v), v \not\equiv b$. Let
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% $D_2 = f_{b' \full \neg b}(D_1)$ then $D_2(b')= \{ \true \}$ and
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% $D_2(v) = D(v), v \not\in \{b,b'\}$. Then
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% $D_3 = f_{b' \half \neg c}(D_2) = f_{b \full c}(D_2)$ by definition, and by
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% idempotence of $f_{b \full c}$ and since $D_2$ and $D$ differ only on $b$ and
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% $b'$ this is a fixpoint for $f_{b \half c}$, $f_{b' \full \neg b}$ and
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% $f_{b' \half \neg c}$ and hence $D'' = D_3$. But also
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% $D' = f_{b' \full \neg b}(f_{b \full c}(D)) = D_3$ since $D_3$ and
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% $f_{b \full c}(D)$ only differ on $b'$.
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\textbf{(b)} If $D(b) = \{\true\}$ then clearly $f_{b \full c}$ and
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$f_{b \half c}$ act identically on variables in $vars(c)$.
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% \textbf{(b)} If $D(b) = \{\true\}$ then clearly $f_{b \full c}$ and
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% $f_{b \half c}$ act identically on variables in $vars(c)$.
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\textbf{(c)} If $D(b) = \{\false\}$ then $D(b') = \{\true\}$ and clearly
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$f_{b \full c}$ and $f_{b' \half \neg c}$ act identically on variables in
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$vars(c)$. \qed
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\end{proof}
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% \textbf{(c)} If $D(b) = \{\false\}$ then $D(b') = \{\true\}$ and clearly
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% $f_{b \full c}$ and $f_{b' \half \neg c}$ act identically on variables in
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% $vars(c)$. \qed
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% \end{proof}
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The reason for the generality of the above theorem which defines the
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half-reified propagation strength in terms of the full reified propagator is
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@ -2214,11 +2214,12 @@ fully reified propagator leads to the same propagation.
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Note that the additional variable $b'$ can be implemented as a view
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\autocite{schulte-2005-views} in the solver and hence adds no overhead.
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\begin{corollary}
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A domain (resp. bounds(Z), bounds(R)) consistent propagator for $b \full c$
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propagates identically to domain (resp. bounds(Z), bounds(R)) consistent
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propagators for $b \half c$, $b \full \neg b'$, $b' \half \neg c$. \qed
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\end{corollary}
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\jip{The corollary environment is also missing}
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% \begin{corollary}
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% A domain (resp. bounds(Z), bounds(R)) consistent propagator for $b \full c$
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% propagates identically to domain (resp. bounds(Z), bounds(R)) consistent
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% propagators for $b \half c$, $b \full \neg b'$, $b' \half \neg c$. \qed
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% \end{corollary}
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\section{Propagation and Half Reification}
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@ -2522,9 +2523,9 @@ Nodes &
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In the final experiment we compare resource constrained project scheduling
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problems (RCPSP) where the \texttt{cumulative} constraint is defined by the task
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decomposition as in Example~\ref{ex:cumul} above, using both full reification
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and half-reification. We use standard benchmark examples from
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PSPlib~\autocite{psplib}. Table~\ref{tab:rcpsp} compares RCPSP instances using
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decomposition as in \cref{ex:cumul} above, using both full reification and
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half-reification. We use standard benchmark examples from PSPlib
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\autocite{kolisch-1997-psplib}. \Cref{tab:rcpsp} compares RCPSP instances using
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\textsf{full} reification and \textsf{half} reification. We compare using J30
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instances (\textsf{J30} ) and instances due to Baptiste and Le Pape
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(\textsf{BL}). Each line in the table shows the average run time and number of
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@ -2550,7 +2551,6 @@ J30 (x 480) & 116.1 & 300 & 114.3 & 304 & 16.9 & 463 & 12.9 & 468\\
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\hline
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\end{tabular}
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\end{center}
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\tableend
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\end{table}
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\subsection{MiniZinc Challenge}
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@ -2559,8 +2559,9 @@ To verify the effectiveness of the half reification implementation within the
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\minizinc{} distribution we compare the results of the current version of the
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\minizinc{} translator (rev. \jip{add revision}) against the equivalent version
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for which we have implemented half reification. We use the model instances used
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by the \minizinc{} challenges\autocite{challenge} from 2012 until 2017. Note
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that the instances that the instances without any reifications have been
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by the \minizinc{} challenges
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\autocite{stuckey-2010-challenge,stuckey-2014-challenge} from 2012 until 2017.
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Note that the instances that the instances without any reifications have been
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excluded from the experiment as they would not experience any change. The
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performance of the generated \flatzinc{} models will be tested using Gecode,
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flattened with its own library, and Gurobi and CBC, flattened with the linear
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@ -2581,7 +2582,6 @@ library.
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\hline
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\end{tabular}
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\end{center}
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\tableend
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\end{table}
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\jip{Better headers for table \ref{tab:flat_results} and update with current
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results}
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@ -2608,24 +2608,25 @@ of memory running the Ubuntu 16.04.3 LTS operating system.
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Half reification on purely Boolean constraints is well understood, this is the
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same as detecting the \emph{polarity} of a gate, and removing half of the
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clausal representation of the circuit (see e.g.~\autocite{polarity}). The
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flattening of functions (partial or total) and the calculation of polarity for
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Booleans terms inside \texttt{bool2int} do not arise in pure Boolean
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constraints.
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clausal representation of the circuit (see \eg\
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\autocite{plaisted-1986-polarity}). The flattening of functions (partial or
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total) and the calculation of polarity for Booleans terms inside
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\texttt{bool2int} do not arise in pure Boolean constraints.
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Half reified constraints have been used in constraint modeling but are typically
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not visible as primitive constraints to users, or produced through flattening.
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Indexicals~\autocite{indexicals} can be used to implement reified constraints by
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specifying how to propagate a constraint $c$, %($prop(c)$),
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Indexicals \autocite{van-hentenryck-1992-indexicals} can be used to implement
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reified constraints by specifying how to propagate a constraint
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$c$, %($prop(c)$),
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propagate its negation,
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%$prop(\neg c)$,
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check disentailment, % $check(c)$,
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and check entailment, %$check(\neg c)$,
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and this is implemented in SICstus Prolog~\autocite{sicstus}. A half reified
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propagator simply omits entailment and propagating the negation.
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%$prop(\neg c)$ and $check(\neg c)$.
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%Half reification was briefly discussed in our earlier work~\autocite{cp2009d},
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%in terms of its ability to reduce propagation.
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and this is implemented in SICstus Prolog \autocite{carlsson-1997-sicstus}. A
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half reified propagator simply omits entailment and propagating the negation.
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% $prop(\neg c)$ and $check(\neg c)$. Half reification was briefly discussed in
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% our earlier work \autocite{cp2009d}, in terms of its ability to reduce
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% propagation.
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Half reified constraints appear in some constraint systems, for example SCIP
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\autocite{gamrath-2020-scip} supports half-reified real linear constraints of
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the form $b \half \sum_i a_i x_i \leq a_0$ exactly because the negation of the
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@ -2639,11 +2640,11 @@ strongly. Schulte proposes a generic implementation of $b \full c$ propagating
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the original variables \autocite*{schulte-2000-deep}; entailment and
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disentailment of $c$ fix the $b$ variable appropriately, although when $b$ is
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made $\false$ the implementation does not propagate $\neg c$. This can also be
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implemented using propagator groups~\autocite{groups}. Brand and Yap define an
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approach to propagating complex constraint formulae called controlled
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propagation which ensures that propagators that cannot affect the satisfiability
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are not propagated \autocite*{brand-2006-propagation}. They note that for a
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formula without negation, they could omit half their control rules,
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implemented using propagator groups \autocite{lagerkvist-2009-groups}. Brand and
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Yap define an approach to propagating complex constraint formulae called
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controlled propagation which ensures that propagators that cannot affect the
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satisfiability are not propagated \autocite*{brand-2006-propagation}. They note
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that for a formula without negation, they could omit half their control rules,
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corresponding to the case for half reification of a positive context. Jefferson
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et al. similarly define an approach to propagating positive constraint formulae
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by using watch literal technology to only wake propagators for reified
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