## 5.31. arith

 DESCRIPTION LINKS GRAPH AUTOMATON
Origin

Used in the definition of several automata

Constraint

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

Synonym

$\mathrm{\pi \pi \pi }$.

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

Enforce for all variables $\mathrm{\pi \pi \pi }$ of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection to have $\mathrm{\pi \pi \pi }\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.

Example
$\left(β©4,5,7,4,5βͺ,<,9\right)$

The $\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since all values of the collection $\beta ©4,5,7,4,5\beta ͺ$ are strictly less than 9.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\beta \left[=\right]$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

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

Arg. properties

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

Systems

eq in Choco, neq in Choco, geq in Choco, gt in Choco, leq in Choco, lt in Choco, rel in Gecode, #< in SICStus, #=< in SICStus, #> in SICStus, #>= in SICStus, #= in SICStus, #\= in SICStus.

Used in

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

Keywords
Cond. implications

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

Β Β Β  withΒ  $\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\beta \left[<\right]$

Β Β Β  andΒ Β  $\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)\beta ₯0$

Β Β implies $\mathrm{\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 }\right)$

Β Β Β  whenΒ  $\mathrm{\pi ²\pi \pi }\beta \left[<\right]$.

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)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }.\mathrm{\pi \pi \pi }\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.31.1 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.

Automaton

FigureΒ 5.31.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi }$ 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 ${S}_{i}$: ${\mathrm{\pi  \pi °\pi }}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}\beta {S}_{i}$. The automaton enforces for each variable ${\mathrm{\pi  \pi °\pi }}_{i}$ the condition ${\mathrm{\pi  \pi °\pi }}_{i}\mathrm{\pi \pi ΄\pi »\pi Ύ\pi Ώ}\mathrm{\pi  \pi °\pi »\pi \pi ΄}$.