94 lines
3.3 KiB
TeX
94 lines
3.3 KiB
TeX
%************************************************
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\chapter{Context Analysis and Half Reification}\label{ch:half-reif}
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%************************************************
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In this chapter we study the usage of \gls{half-reif}. When a constraint \mzninline{pred(...)} is reified a
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\begin{itemize}
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\item Flattening with half reification can naturally produce the relational
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semantics when flattening partial functions in positive contexts.
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\item Half reified constraints add no burden to the solver writer; if they
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have a propagator for constraint \mzninline{pred(....)} then they can
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straightforwardly construct a half reified propagator for \mzninline{b
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-> pred(...)}.
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\item Half reified constraints \mzninline{b -> pred(...)} can implement fully
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reified constraints without any loss of propagation strength (assuming
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reified constraints are negatable).
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\item Flattening with half reification can produce more efficient propagation
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when flattening complex constraints.
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\item Flattening with half reification can produce smaller linear models when
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used with a mixed integer programming solver.
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\end{itemize}
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\section{Propagation and Half Reification}%
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\label{sec:half-propagation}
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\begin{itemize}
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\item custom implemented half-reified propagations.
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\item benefits
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\item downside
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\item experimental results
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\end{itemize}
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A propagation engine gains certain advantages from half-reification, but also
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may suffer certain penalties. Half reification can cause propagators to wake up
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less frequently, since variables that are fixed to true by full reification will
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never be fixed by half reification. This is advantageous, but a corresponding
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disadvantage is that variables that are fixed can allow the simplification of
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the propagator, and hence make its propagation faster.
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When full reification is applicable (where we are not using half reified
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predicates) an alternative to half reification is to implement full reification
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\mzninline{x <-> pred(...)} by two half reified propagators \mzinline{x ->
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pred(...)}, \mzninline{y \half \neg pred(...)}, \mzninline{x <-> not y}. This
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does not lose propagation strength. For Booleans appearing in a positive context
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we can make the the propagator \mzninline{y -> not pred(...)} run at the lowest
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priority, since it will never cause failure. Similarly in negative contexts we
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can make the propagator \mzninline{b -> pred(...)} run at the lowest priority.
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This means that Boolean variables are still fixed at the same time, but there is
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less overhead.
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\section{Decomposition and Half Reification}%
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\label{sec:half-decomposition}
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\section{Context Analysis}%
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\label{sec:half-context}
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\begin{tabular}{ccc}
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\(
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\begin{array}{lcl}
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\changepos \rootc & = &\posc \\
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\changepos \posc & =& \posc \\
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\changepos \negc & =& \negc \\
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\changepos \mixc & =& \mixc
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\end{array}
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\)
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&~&
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\(
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\begin{array}{lcl}
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\negop \rootc & = &\negc \\
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\negop \posc & = & \negc \\
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\negop \negc & = & \posc \\
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\negop \mixc & = & \mixc
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\end{array}
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\)
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\end{tabular}
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\begin{itemize}
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\item What (again) are the contexts?
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\item Why are they important
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\item When can we use half-reification
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\item How does the analysis work?
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\end{itemize}
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\section{Flattening and Half Reification}%
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\label{sec:half-flattening}
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\begin{itemize}
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\item cse
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\item chain compression
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\end{itemize}
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