5.24. arith

DESCRIPTIONLINKSGRAPHAUTOMATON
Origin

Used in the definition of several automata

Constraint

πšŠπš›πš’πšπš‘(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšπ™΄π™»π™Ύπ™Ώ,πš…π™°π™»πš„π™΄)

Synonym

πš›πšŽπš•.

Arguments
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›-πšπšŸπšŠπš›)
πšπ™΄π™»π™Ύπ™ΏπšŠπšπš˜πš–
πš…π™°π™»πš„π™΄πš’πš—πš
Restrictions
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšŸπšŠπš›)
πšπ™΄π™»π™Ύπ™Ώβˆˆ[=,β‰ ,<,β‰₯,>,≀]
Purpose

Enforce for all variables πšŸπšŠπš› of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection to have πšŸπšŠπš› πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄.

Example
(4,5,7,4,5,<,9)

The πšŠπš›πš’πšπš‘ constraint holds since all values of the collection 〈4,5,7,4,5βŒͺ are strictly less than 9.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>1
Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› can be replaced by any value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš›.

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

πšŠπš›πš’πšπš‘_πšœπš•πš’πšπš’πš—πš.

See also

common keyword: πšŠπš–πš˜πš—πš, πšŒπš˜πšžπš—πšΒ (value constraint).

generalisation: πšŠπš›πš’πšπš‘_πš˜πš›Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄ replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄ ∨ πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄).

system of constraints: πšŠπš›πš’πšπš‘_πšœπš•πš’πšπš’πš—πš.

Keywords

characteristic of a constraint: automaton, automaton without counters, reified automaton constraint.

constraint network structure: Berge-acyclic constraint network.

constraint type: decomposition, value constraint.

filtering: arc-consistency.

modelling: domain definition.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
π‘†πΈπΏπΉβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ)

Arc arity
Arc constraint(s)
πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ.πšŸπšŠπš› πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄
Graph property(ies)
𝐍𝐀𝐑𝐂=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.24.1 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐀𝐑𝐂 graph property, the loops of the final graph are stressed in bold.

Figure 5.24.1. Initial and final graph of the πšŠπš›πš’πšπš‘ constraint
ctrs/arithActrs/arithB
(a) (b)
Automaton

FigureΒ 5.24.2 depicts the automaton associated with the πšŠπš›πš’πšπš‘ constraint. To each variable πš…π™°πš i of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ corresponds a 0-1 signature variable πš‚ i . The following signature constraint links πš…π™°πš i and πš‚ i : πš…π™°πš i πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄β‡”πš‚ i . The automaton enforces for each variable πš…π™°πš i the condition πš…π™°πš i πšπ™΄π™»π™Ύπ™Ώ πš…π™°π™»πš„π™΄.

Figure 5.24.2. Automaton of the πšŠπš›πš’πšπš‘ constraint
ctrs/arith1
Figure 5.24.3. Hypergraph of the reformulation corresponding to the automaton of the πšŠπš›πš’πšπš‘ constraint
ctrs/arith2