## 5.25. arith_or

Origin

Used in the definition of several automata

Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ},\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1},\mathrm{\pi \pi \pi }\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2},\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}|$
Purpose

Enforce for all pairs of variables $\mathrm{\pi \pi \pi }{\mathtt{1}}_{i},\mathrm{\pi \pi \pi }{\mathtt{2}}_{i}$ of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collections to have $\mathrm{\pi \pi \pi }{\mathtt{1}}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta ¨\mathrm{\pi \pi \pi }{\mathtt{2}}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

Example
$\left(\begin{array}{c}β©0,1,0,0,1βͺ,\hfill \\ β©0,0,0,1,0βͺ,=,0\hfill \end{array}\right)$

The constraint $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi }$ holds since, for all pairs of variables $\mathrm{\pi \pi \pi }{\mathtt{1}}_{i},\mathrm{\pi \pi \pi }{\mathtt{2}}_{i}$ of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collections, there is at least one variable that is equal to 0.

Typical
$|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|>0$
Symmetry

Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ are permutable (same permutation used).

specialisation: $\mathrm{\pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ $\beta ¨$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$).

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$

Arc generator
$\mathrm{\pi \pi  \pi \pi ·\pi \pi Ά\pi }$$\left(=\right)\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta ¨\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}|$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.25.1 respectively show the initial and final graphs associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the arcs of the final graph are stressed in bold.

Automaton

FigureΒ 5.25.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi }$ constraint. Let $\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i}$ and $\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}$ be the ${i}^{th}$ variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{1}$ and $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\mathtt{2}$ collections. To each pair of variables $\left(\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i},\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}\right)$ corresponds a signature variable ${\mathrm{\pi }}_{i}$. The following signature constraint links $\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i}$, $\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}$ and ${\mathrm{\pi }}_{i}$: $\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta ¨\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta {\mathrm{\pi }}_{i}$. The automaton enforces for each pair of variables $\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i}$,$\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}$ the condition $\mathrm{\pi  \pi °\pi }{\mathtt{1}}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta ¨\mathrm{\pi  \pi °\pi }{\mathtt{2}}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.