## 5.26. arith_sliding

 DESCRIPTION LINKS GRAPH AUTOMATON
Origin

Used in the definition of some automaton

Constraint

$\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚁𝙴𝙻𝙾𝙿},\mathrm{𝚅𝙰𝙻𝚄𝙴}\right)$

Arguments
 $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝚁𝙴𝙻𝙾𝙿}$ $\mathrm{𝚊𝚝𝚘𝚖}$ $\mathrm{𝚅𝙰𝙻𝚄𝙴}$ $\mathrm{𝚒𝚗𝚝}$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝚁𝙴𝙻𝙾𝙿}\in \left[=,\ne ,<,\ge ,>,\le \right]$
Purpose

Enforce for all sequences of variables ${\mathrm{𝚟𝚊𝚛}}_{1},{\mathrm{𝚟𝚊𝚛}}_{2},...,{\mathrm{𝚟𝚊𝚛}}_{i}$ $\left(1\le i\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)$ of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection to have $\left({\mathrm{𝚟𝚊𝚛}}_{1}+{\mathrm{𝚟𝚊𝚛}}_{2}+...+{\mathrm{𝚟𝚊𝚛}}_{i}\right)\mathrm{𝚁𝙴𝙻𝙾𝙿}\mathrm{𝚅𝙰𝙻𝚄𝙴}$.

Example
$\left(\begin{array}{c}〈\begin{array}{c}\mathrm{𝚟𝚊𝚛}-0,\hfill \\ \mathrm{𝚟𝚊𝚛}-0,\hfill \\ \mathrm{𝚟𝚊𝚛}-1,\hfill \\ \mathrm{𝚟𝚊𝚛}-2,\hfill \\ \mathrm{𝚟𝚊𝚛}-0,\hfill \\ \mathrm{𝚟𝚊𝚛}-0,\hfill \\ \mathrm{𝚟𝚊𝚛}--3\hfill \end{array}〉,<,4\hfill \end{array}\right)$

The $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}$ constraint holds since all the following seven inequalities hold:

• $0<4$,

• $0+0<4$,

• $0+0+1<4$,

• $0+0+1+2<4$,

• $0+0+1+2+0<4$,

• $0+0+1+2+0+0<4$,

• $0+0+1+2+0+0-3<4$.

Typical
$|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1$
See also
Keywords
Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Arc generator
$\mathrm{𝑃𝐴𝑇𝐻}_\mathit{1}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}$

Arc arity
$*$
Arc constraint(s)
$\mathrm{𝚊𝚛𝚒𝚝𝚑}$$\left(\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗},\mathrm{𝚁𝙴𝙻𝙾𝙿},\mathrm{𝚅𝙰𝙻𝚄𝙴}\right)$
Graph property(ies)
$\mathrm{𝐍𝐀𝐑𝐂}$$=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|$

Automaton

Figure 5.26.1 depicts the automaton associated with the $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}$ constraint. To each item of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ corresponds a signature variable ${𝚂}_{i}$ that is equal to 0.

##### Figure 5.26.1. Automaton of the $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}$ constraint ##### Figure 5.26.2. Hypergraph of the reformulation corresponding to the automaton of the $\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}$ constraint 