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26 \begin{definition}[Polyhedral Relation]
27 A {\em polyhedral relation}\index{polyhedral relation}
47 \begin{definition}[Parameter Domain of a Relation]
48 Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
53 \begin{definition}[Domain of a Relation]
54 Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
63 \begin{definition}[Range of a Relation]
64 Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
91 \begin{definition}[Difference Set of a Relation]
92 Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation.
108 basic set or basic relation that contains a given set or relation.
520 the constraints of a set or relation in disjunctive normal form.
839 \begin{definition}[Power of a Relation]
840 Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation and
842 a positive number, then power $k$ of relation $R$ is defined as
855 \begin{definition}[Transitive Closure of a Relation]
856 Let $R \in \Z^n \to 2^{\Z^{d+d}}$ be a relation,
872 Since the transitive closure of a polyhedral relation
873 may no longer be a polyhedral relation \parencite{Kelly1996closure},
878 That is, given a relation $R$, we will compute a relation $T$
887 out the parameter $k$ from the resulting relation.
890 As a trivial example, consider the relation
918 of $R^k$ as a polyhedral relation.
924 That is, we will only consider the difference set $\Delta\,R$ of the relation.
1000 We therefore need to take the union with the identity relation
1005 Taking the union with the identity relation means that
1008 disjuncts in the input relation, then a direct application
1009 of the composition operation may therefore result in a relation
1158 Consider the relation
1163 The difference set of this relation is
1239 Consider the relation
1245 The difference set of this relation is
1319 Consider the relation
1424 If the input relation $R$ is a union of several basic relations
1506 \caption{The relation from \autoref{ex:closure4}}
1511 Consider the relation in example {\tt closure4} that comes with
1522 This relation is shown graphically in \autoref{f:closure4}.
1586 \caption{The relation from \autoref{ex:decomposition}}
1591 Consider the relation on the right of \textcite[Figure~2]{Beletska2009},
1593 The relation
1617 The figure shows this relation for $n = 7$.
1649 The algorithm of \autoref{s:power} assumes that the input relation $R$
1656 in a single relation, as is done by, e.g.,
1674 \Output{Updated relations $R_{pq}$ such that each relation
1695 Let the input relation $R$ be a union of $m$ basic relations $R_i$.
1752 \caption{The relation (solid arrows) on the right of Figure~1 of
1757 Consider the relation on the right of Figure~1 of
1760 This relation can be described as
1768 Note that the domain of the upward relation overlaps with the range
1769 of the rightward relation and vice versa, but that the domain
1770 of neither relation overlaps with its own range or the domain of
1771 the other relation.
1799 The transitive closure of the original relation is then equal to
1826 as a single basic relation, i.e., without a union.
1831 to relax the constraints of $R_i^+$ to include part of the identity relation,
1979 The input relation $R$ is first overapproximated by a ``d-form'' relation
1994 The transitive closure of such a ``d-form'' relation is
2006 intersected with those of the input relation.
2010 use the above algorithm as a substep on the disjuncts in the relation.
2014 similar to that of a single conjunct [sic] relation.
2017 of the whole input relation should be used.