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Jip J. Dekker 2021-05-20 16:50:54 +10:00
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5
assets/glossary.tex
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@ -169,6 +169,11 @@
description={},
}
\newglossaryentry{half-reif}{
name={half reification},
description={},
}
\newglossaryentry{indicator-var}{
name={indicator variable},
description={},

60
assets/shorthands.tex
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@ -18,7 +18,7 @@
\newcommand{\variable}{\gls{variable}\xspace{}}
\newcommand{\zinc}{\gls{zinc}\xspace{}}
% Semantic shorthands (rewriting)
\renewcommand{\phi}{\varphi}
\newcommand{\tuple}[1]{\ensuremath{\langle #1 \rangle}}
\newcommand{\Prog}{\ensuremath{\mathcal{P}}}
@ -27,54 +27,12 @@
\newcommand{\Sem}[2]{\ptinline{[\!\![#1]\!\!]\tuple{#2}}}
\newcommand{\Cbind}{\ensuremath{\wedge}}
% Context shorthands (half-reification)
\newcommand{\rootc}{\textit{root}}
\newcommand{\posc}{\textit{pos}}
\newcommand{\negc}{\textit{neg}}
\newcommand{\mixc}{\textit{mix}}
\newcommand{\changepos}{\ensuremath{+}}
\newcommand{\changeneg}{\ensuremath{-}}
\newcommand{\undefined}{\ensuremath{\bot}}
% Half-reification math things
\newcommand{\VV}{{\cal\ V}}
\newcommand{\PP}{{\cal\ P}}
\newcommand{\range}[2]{\left[\,#1\,..\,#2\,\right]}
\newcommand{\gfp}{\textup{gfp}}
\newcommand{\lfp}{\textup{lfp}}
\newcommand{\iter}{\mathit{iter}}
\newcommand{\half}{\Rightarrow}
\newcommand{\full}{\Leftrightarrow}
\renewcommand{\half}{\rightarrow}
\renewcommand{\full}{\leftrightarrow}
\newcommand{\entails}{\models}
\newcommand{\bigsqcap}{\mathop{\lower.1ex\hbox{\Large$\sqcap$}}}
\DeclareMathOperator{\rules}{\mathit{rules}}
\DeclareMathOperator{\lazy}{\mathit{lazy}}
\DeclareMathOperator{\events}{\mathit{events}}
\DeclareMathOperator{\domain}{\mathit{domain}}
\DeclareMathOperator{\bc}{\mathit{bc}}
\DeclareMathOperator{\lbc}{\mathit{lbc}}
\DeclareMathOperator{\ubc}{\mathit{ubc}}
\DeclareMathOperator{\dmc}{\mathit{dmc}}
\DeclareMathOperator{\fix}{\mathit{fix}}
\DeclareMathOperator{\event}{\mathit{event}}
\DeclareMathOperator{\minassign}{\mathit{minassign}}
\DeclareMathOperator{\assign}{\mathit{assign}}
\DeclareMathOperator{\cl}{\mathit{cl}}
\DeclareMathOperator{\UP}{\mathit{UP}}
\DeclareMathOperator{\up}{\mathit{up}}
\DeclareMathOperator{\DOM}{\mathit{DOM}}
\DeclareMathOperator{\setb}{\mathit{sb}}
\DeclareMathOperator{\bnd}{\mathit{bnd}}
\DeclareMathOperator{\dsb}{\mathit{dsb}}
\DeclareMathOperator{\vars}{\mathit{vars}}
\DeclareMathOperator{\ivars}{\mathit{input}}
\DeclareMathOperator{\ovars}{\mathit{output}}
\DeclareMathOperator{\solns}{\mathit{solns}}
\DeclareMathOperator{\solv}{\mathit{solv}}
\DeclareMathOperator{\isolv}{\mathit{isolv}}
\DeclareMathOperator{\conv}{\mathit{conv}}
\DeclareMathOperator{\ran}{\mathit{ran}}
\DeclareMathOperator{\ite}{\mathit{ite}}
\DeclareMathOperator{\VAR}{\mathit{VAR}}
\DeclareMathOperator{\dom}{\mathit{idx}}

70
chapters/4_half_reif.tex
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@ -2,14 +2,23 @@
\chapter{Context Analysis and Half Reification}\label{ch:half-reif}
%************************************************
\section{Context Analysis}%
\label{sec:half-context}
In this chapter we study the usage of \gls{half-reif}. When a constraint \mzninline{pred(...)} is reified a
\begin{itemize}
\item What (again) are the contexts?
\item Why are they important
\item When can we use half-reification
\item How does the analysis work?
\item Flattening with half reification can naturally produce the relational
semantics when flattening partial functions in positive contexts.
\item Half reified constraints add no burden to the solver writer; if they
have a propagator for constraint \mzninline{pred(....)} then they can
straightforwardly construct a half reified propagator for \mzninline{b
-> pred(...)}.
\item Half reified constraints \mzninline{b -> pred(...)} can implement fully
reified constraints without any loss of propagation strength (assuming
reified constraints are negatable).
\item Flattening with half reification can produce more efficient propagation
when flattening complex constraints.
\item Flattening with half reification can produce smaller linear models when
used with a mixed integer programming solver.
\end{itemize}
\section{Propagation and Half Reification}%
@ -22,9 +31,58 @@
\item experimental results
\end{itemize}
A propagation engine gains certain advantages from half-reification, but also
may suffer certain penalties. Half reification can cause propagators to wake up
less frequently, since variables that are fixed to true by full reification will
never be fixed by half reification. This is advantageous, but a corresponding
disadvantage is that variables that are fixed can allow the simplification of
the propagator, and hence make its propagation faster.
When full reification is applicable (where we are not using half reified
predicates) an alternative to half reification is to implement full reification
\mzninline{x <-> pred(...)} by two half reified propagators \mzinline{x ->
pred(...)}, \mzninline{y \half \neg pred(...)}, \mzninline{x <-> not y}. This
does not lose propagation strength. For Booleans appearing in a positive context
we can make the the propagator \mzninline{y -> not pred(...)} run at the lowest
priority, since it will never cause failure. Similarly in negative contexts we
can make the propagator \mzninline{b -> pred(...)} run at the lowest priority.
This means that Boolean variables are still fixed at the same time, but there is
less overhead.
\section{Decomposition and Half Reification}%
\label{sec:half-decomposition}
\section{Context Analysis}%
\label{sec:half-context}
\begin{tabular}{ccc}
\(
\begin{array}{lcl}
\changepos \rootc & = &\posc \\
\changepos \posc & =& \posc \\
\changepos \negc & =& \negc \\
\changepos \mixc & =& \mixc
\end{array}
\)
&~&
\(
\begin{array}{lcl}
\negop \rootc & = &\negc \\
\negop \posc & = & \negc \\
\negop \negc & = & \posc \\
\negop \mixc & = & \mixc
\end{array}
\)
\end{tabular}
\begin{itemize}
\item What (again) are the contexts?
\item Why are they important
\item When can we use half-reification
\item How does the analysis work?
\end{itemize}
\section{Flattening and Half Reification}%
\label{sec:half-flattening}

92
chapters/A3_temp_hr.tex
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@ -2,6 +2,56 @@
\chapter{TEMP: Half Reification Journal Paper}\label{ch:half-reif-journal}
%************************************************
%
% Half-reification math things
\newcommand{\VV}{{\cal\ V}}
\newcommand{\PP}{{\cal\ P}}
\newcommand{\range}[2]{\left[\,#1\,..\,#2\,\right]}
\newcommand{\gfp}{\textup{gfp}}
\newcommand{\lfp}{\textup{lfp}}
\newcommand{\iter}{\mathit{iter}}
\newcommand{\half}{\Rightarrow}
\newcommand{\full}{\Leftrightarrow}
\renewcommand{\half}{\rightarrow}
\renewcommand{\full}{\leftrightarrow}
\newcommand{\entails}{\models}
% \newcommand{\bigsqcap}{\mathop{\lower.1ex\hbox{\Large$\sqcap$}}}
% \DeclareMathOperator{\rules}{\mathit{rules}}
% \DeclareMathOperator{\lazy}{\mathit{lazy}}
% \DeclareMathOperator{\events}{\mathit{events}}
% \DeclareMathOperator{\domain}{\mathit{domain}}
% \DeclareMathOperator{\bc}{\mathit{bc}}
% \DeclareMathOperator{\lbc}{\mathit{lbc}}
% \DeclareMathOperator{\ubc}{\mathit{ubc}}
% \DeclareMathOperator{\dmc}{\mathit{dmc}}
% \DeclareMathOperator{\fix}{\mathit{fix}}
% \DeclareMathOperator{\event}{\mathit{event}}
% \DeclareMathOperator{\minassign}{\mathit{minassign}}
% \DeclareMathOperator{\assign}{\mathit{assign}}
% \DeclareMathOperator{\cl}{\mathit{cl}}
% \DeclareMathOperator{\UP}{\mathit{UP}}
% \DeclareMathOperator{\up}{\mathit{up}}
% \DeclareMathOperator{\DOM}{\mathit{DOM}}
% \DeclareMathOperator{\setb}{\mathit{sb}}
% \DeclareMathOperator{\bnd}{\mathit{bnd}}
% \DeclareMathOperator{\dsb}{\mathit{dsb}}
% \DeclareMathOperator{\vars}{\mathit{vars}}
% \DeclareMathOperator{\ivars}{\mathit{input}}
% \DeclareMathOperator{\ovars}{\mathit{output}}
% \DeclareMathOperator{\solns}{\mathit{solns}}
% \DeclareMathOperator{\solv}{\mathit{solv}}
% \DeclareMathOperator{\isolv}{\mathit{isolv}}
% \DeclareMathOperator{\conv}{\mathit{conv}}
% \DeclareMathOperator{\ran}{\mathit{ran}}
% \DeclareMathOperator{\ite}{\mathit{ite}}
% \DeclareMathOperator{\VAR}{\mathit{VAR}}
% \DeclareMathOperator{dom}{\mathit{idx}}
%
Discrete optimisation solvers solve constraint optimisation problems of the form
\(\min o \text{~subject to~} \wedge_{c \in C} c\), that is minimising an
objective \(o\) subject to an (existentially quantified) conjunction of
@ -244,12 +294,12 @@ correspondingly typed values, written
% PJS (not true unless \theta is in D)
% where \(d_i \in D(x_i)\) and \(A_j \in D(S_j)\).
We extend the valuation \(\theta\) to map expressions or constraints involving the
variables in the natural way. Let \(\vars\) be the function that returns the set
variables in the natural way. Let vars be the function that returns the set
of variables appearing in an expression, constraint or valuation.
In an abuse of notation, we define a valuation \(\theta\) to be an element of a
domain \(D\), written \(\theta \in D\), if \(\theta(v) \in D(v)\) for all
\(v \in \vars(\theta)\).
\(v \in vars(\theta)\).
%\paragraph{Constraints, Propagators and Propagation Solvers}
@ -261,7 +311,7 @@ variables.
%% over integer and set variables.
We define the \emph{solutions} of a constraint \(c\) to be the set of valuations
\(\theta\) that make that constraint true, i.e.
\(\solns(c) = \{ \theta \ |\ (\vars(\theta) = \vars(c))\ \wedge\ (\entails \theta(c))\}\)
\(solns(c) = \{ \theta \ |\ (vars(\theta) = vars(c))\ \wedge\ (\entails \theta(c))\}\)
%%MC check this para that my changes are ok
We associate with every constraint \(c\) a \emph{propagator} \(f_c\). A propagator
@ -269,7 +319,7 @@ We associate with every constraint \(c\) a \emph{propagator} \(f_c\). A propagat
of the constraint. That is, for all domains \(D \sqsubseteq D_{init}\):
\(f_c(D) \sqsubseteq D\) (decreasing), and
\(\{\theta\in D \,|\ \theta \in \solns(c)\} = \{\theta \in f_c(D)\,|\ \theta \in \solns(c)\}\)
\(\{\theta\in D \,|\ \theta \in solns(c)\} = \{\theta \in f_c(D)\,|\ \theta \in solns(c)\}\)
(preserves solutions), and for domains
\(D_1 \sqsubseteq D_2 \sqsubseteq D_{init}\): \(f(D_1) \sqsubseteq f(D_2)\)
(monotonic). This is a weak restriction since, for example, the identity mapping
@ -316,11 +366,11 @@ A propagator \(f_c\) is \(X\)-consistent if \(f(D)\) is always \(X\) consistent
where \(X\) could be domain, bounds(Z) or bounds(R).
A \emph{propagation solver} for a set of propagators \(F\) and current domain \(D\),
\(\solv(F,D)\), repeatedly applies all the propagators in \(F\) starting from domain
\(D\) until there is no further change in the resulting domain. \(\solv(F,D)\) is
\(solv(F,D)\), repeatedly applies all the propagators in \(F\) starting from domain
\(D\) until there is no further change in the resulting domain. \(solv(F,D)\) is
the weakest domain \(D' \sqsubseteq D\) which is a fixpoint (i.e.~\(f(D') = D'\))
for all \(f \in F\). In other words, \(\solv(F,D)\) returns a new domain defined by
\( \solv(F,D) = \gfp(\lambda d.\iter(F,d))(D)\) where
for all \(f \in F\). In other words, \(solv(F,D)\) returns a new domain defined by
\( solv(F,D) = \gfp(\lambda d.\iter(F,d))(D)\) where
\(\iter(F,D) = \bigsqcap_{f \in F} f(D)\), where \(\gfp\) denotes the greatest
fixpoint w.r.t \(\sqsubseteq\) lifted to functions.
% (We use the greatest fixpoint because we are coming down from a full domain,
@ -447,8 +497,8 @@ negative context, \mzninline{x + bool2int(b) = 5} is in a positive context, and
We assume each integer variable \(x\) is separately declared with a finite initial
set of possible values \(D_{init}(x)\). We assume each array constant is
separately declared as a mapping \(\{ i \mapsto d~|~i \in \dom(a) \}\) where its
index set \(\dom(a)\) is a finite integer range. Given these initial declarations,
separately declared as a mapping \(\{ i \mapsto d~|~i \in dom(a) \}\) where its
index set \(dom(a)\) is a finite integer range. Given these initial declarations,
we can determine the set of possible values \(poss(t)\) of any term \(t\) in the
language as \(poss(t) = \{ \theta(t)~|~\theta \in D_{init} \}\). Note also while
it may be prohibitive to determine the set of possible values for any term \(t\),
@ -1215,8 +1265,8 @@ variable \(y'\) which cannot be 0, and equates it to \(y\) if \(y \neq 0\). The
constraint \(div(x,y',z)\) never reflects a partial function application. The
new variable \(b'\) captures whether the partial function application returns a
non \undefined{} value. For \mzninline{element(v, a, x)}, it introduces a new
variable \(v'\) which only takes values in \(\dom(a)\) and forces it to equal
\(v\) if \(v \in \dom(a)\). A partial function application forces
variable \(v'\) which only takes values in \(dom(a)\) and forces it to equal
\(v\) if \(v \in dom(a)\). A partial function application forces
\(b = \mzninline{false}\) since it is the conjunction of the new variables
\(b'\). The \%HALF\% comments will be explained later.
@ -1235,10 +1285,10 @@ xx\=xx\=xx\=xx\=xx\=xx\=xx\= \kill
div(x,y,z)\}\) \\
\> \> \textbf{else if} \(c \equiv \mzninline{element(v, a, x)}\) and \(v\) can take a value outside the domain of \(a\)) \\
\> \> \> \(B\) := \(B \cup \{ \new b' \}\) \\
\> \> \> \(S\) := \(S \cup \{ \new v' \in \dom(a),
b' \full v \in \dom(a), b' \full v = v',
\> \> \> \(S\) := \(S \cup \{ \new v' \in dom(a),
b' \full v \in dom(a), b' \full v = v',
\mzninline{element(v', a, x)}\}\) \\
\> \> \> \%HALF\% \(S\) := \(S \cup \{ b' \full v \in \dom(a), b' \half
\> \> \> \%HALF\% \(S\) := \(S \cup \{ b' \full v \in dom(a), b' \half
\mzninline{element(v, a, x)}\}\) \\
\> \> \textbf{else} \(S\) := \(S \cup \{c\}\) \\
\> \> \textbf{return} \(S \cup \{b \full \wedge_{b' \in B} b'
@ -2054,27 +2104,27 @@ For the sake of simplicity of proving the following theorem we also assume
trivial propagator.
% Define
% \(f_{b \half c}(D) =_{vars(b \full c)} \solv(\{f_{b'' \full c}, f_{b \half b''} \}, D)\),
% \(f_{b \half c}(D) =_{vars(b \full c)} solv(\{f_{b'' \full c}, f_{b \half b''} \}, D)\),
% that is the half-reified version propagates on all variables only in the
% forward direction of the implication, by introducing the hidden variable
% \(b''\) which prevents propagation in the other direction. We assume a domain
% consistent propagator for \(b \half b''\). Similarly define
% \(f_{b' \half \neg c}(D) =_{vars(b' \full c)} \solv(\{f_{b''' \full c}, f_{b' \half \neg b'''} \}, D)\)
% \(f_{b' \half \neg c}(D) =_{vars(b' \full c)} solv(\{f_{b''' \full c}, f_{b' \half \neg b'''} \}, D)\)
\jip{There is a proof here, but I don't have a proof environment yet}
% \begin{theorem}
% \(\forall D. \solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D) = \solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)\).
% \(\forall D. solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D) = solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)\).
% \end{theorem}
% \begin{proof}
% Let \(V = vars(c)\). First w.l.o.g. we only consider domains \(D\) at a fixpoint
% of the propagators \(f_{b' \full \neg b}\), i.e.
% \(D(b') = \{ \neg d ~|~ d \in D(b)\}\), since both \(\solv\) expressions include
% \(D(b') = \{ \neg d ~|~ d \in D(b)\}\), since both \(solv\) expressions include
% \(f_{b' \full \neg b}\). Let
% \(D' = \solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D)\), now since
% \(D' = solv(\{ f_{b \full c}, f_{b' \full \neg b}\})(D)\), now since
% \(f_{b \full c}\) and \(f_{b' \full \neg b}\) are idempotent and \(D\) is a fixpoint
% of \(f_{b' \full \neg b}\), then \(D' = f_{b' \full \neg b}(f_{b \full c}(D))\).
% Let
% \(D'' = \solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)\).
% \(D'' = solv(\{ f_{b \half c}, f_{b' \half \neg c}, f_{b' \full \neg b}\}, D)\).
% The proof is by cases of \(D\).
% \textbf{(a)} Suppose \(D(b) = \{\mzninline{true},\mzninline{false}\}\). \textbf{(a-i)} Suppose