## 5.393. sum_squares_ctr

Origin

Arithmetic constraint.

Constraint

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi },\mathrm{\pi  \pi °\pi }\right)$

Synonyms

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$.

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

Constraint the sum of the squares of a set of domain variables. More precisely, let $\mathrm{\pi }$ denote the sum of the squares of the variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection (when the collection is empty the corresponding sum is equal to 0). Enforce the following constraint to hold: $\mathrm{\pi }\mathrm{\pi ²\pi \pi }\mathrm{\pi  \pi °\pi }$.

Example
$\left(β©1,1,4βͺ,=,18\right)$

The $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ constraint holds since the condition ${1}^{2}+{1}^{2}+{4}^{2}=18$ is satisfied.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi ²\pi \pi }\beta \left[=,<,\beta ₯,>,\beta €\right]$
Symmetry

Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

Arg. properties
• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ when $\mathrm{\pi ²\pi \pi }\beta \left[<,\beta €\right]$.

• Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ when $\mathrm{\pi ²\pi \pi }\beta \left[\beta ₯,>\right]$.

• Aggregate: $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left(\mathrm{\pi \pi \pi \pi \pi }\right)$, $\mathrm{\pi ²\pi \pi }\left(\mathrm{\pi \pi }\right)$, $\mathrm{\pi  \pi °\pi }\left(+\right)$.

Keywords
Cond. implications

$\beta ’$ $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi },\mathrm{\pi  \pi °\pi }\right)$

Β Β Β  withΒ  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta ₯-1$

Β Β Β  andΒ Β  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta €1$

Β Β implies $\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi },\mathrm{\pi  \pi °\pi }\right)$

Β Β Β  whenΒ  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta ₯-1$

Β Β Β  andΒ Β  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta €1$.

$\beta ’$ $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi },\mathrm{\pi  \pi °\pi }\right)$

Β Β Β  withΒ  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta ₯-1$

Β Β Β  andΒ Β  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta €1$

Β Β implies $\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi },\mathrm{\pi  \pi °\pi }\right)$

Β Β Β  whenΒ  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta ₯-1$

Β Β Β  andΒ Β  $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\beta €1$.