## 5.155. exactly

Origin
Constraint

$\mathrm{\pi \pi ‘\pi \pi \pi \pi \pi ’}\left(\mathrm{\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi  \pi °\pi »\pi \pi ΄}\right)$

Synonym

$\mathrm{\pi \pi \pi \pi \pi }$.

Arguments
 $\mathrm{\pi ½}$ $\mathrm{\pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\mathrm{\pi ½}\beta ₯0$ $\mathrm{\pi ½}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

Exactly $\mathrm{\pi ½}$ variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection are assigned value $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

Example
$\left(2,β©4,2,4,5βͺ,4\right)$

The $\mathrm{\pi \pi ‘\pi \pi \pi \pi \pi ’}$ constraint holds since exactly $\mathrm{\pi ½}=2$ variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }=\beta ©4,2,4,5\beta ͺ$ collection are assigned value $\mathrm{\pi  \pi °\pi »\pi \pi ΄}=4$.

Typical
 $\mathrm{\pi ½}>0$ $\mathrm{\pi ½}<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• An occurrence of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ that is different from $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ can be replaced by any other value that is also different from $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

Arg. properties
• Functional dependency: $\mathrm{\pi ½}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

• Aggregate: $\mathrm{\pi ½}\left(+\right)$, $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi  \pi °\pi »\pi \pi ΄}\left(\mathrm{\pi \pi }\right)$.

Systems

generalisation: $\mathrm{\pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ and $\mathrm{\pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi }$ of $\mathrm{\pi \pi \pi \pi \pi \pi }$).

implies: $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$Β ($=\mathrm{\pi ½}$ replaced by $\beta ₯\mathrm{\pi ½}$), $\mathrm{\pi \pi \pi \pi \pi \pi }$Β ($=\mathrm{\pi ½}$ replaced by $\beta €\mathrm{\pi ½}$).

Keywords
Arc input(s)

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

Arc generator
$\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }=\mathrm{\pi  \pi °\pi »\pi \pi ΄}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½}$

Graph model

Since each arc constraint involves only one vertex ($\mathrm{\pi  \pi °\pi »\pi \pi ΄}$ is fixed), we employ the $\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$ arc generator in order to produce a graph with a single loop on each vertex.

PartsΒ (A) andΒ (B) of FigureΒ 5.155.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property, the loops of the final graph are stressed in bold. The $\mathrm{\pi \pi ‘\pi \pi \pi \pi \pi ’}$ constraint holds since exactly two variables are assigned value 4.

Automaton

FigureΒ 5.155.2 depicts the automaton associated with the $\mathrm{\pi \pi ‘\pi \pi \pi \pi \pi ’}$ constraint. To each variable ${\mathrm{\pi  \pi °\pi }}_{i}$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a 0-1 signature variable ${S}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$ and ${S}_{i}$: ${\mathrm{\pi  \pi °\pi }}_{i}=\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta {S}_{i}$.