5.321. period_vectors

DESCRIPTIONLINKS
Origin

Derived from πš™πšŽπš›πš’πš˜πš

Constraint

πš™πšŽπš›πš’πš˜πš_πšŸπšŽπšŒπšπš˜πš›πšœ(π™Ώπ™΄πšπ™Έπ™Ύπ™³,πš…π™΄π™²πšƒπ™Ύπšπš‚,π™²πšƒπšπš‚)

Types
πš…π™΄π™²πšƒπ™Ύπ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™²πšƒπšπšŠπšπš˜πš–
Arguments
π™Ώπ™΄πšπ™Έπ™Ύπ™³πšπšŸπšŠπš›
πš…π™΄π™²πšƒπ™Ύπšπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚟𝚎𝚌-πš…π™΄π™²πšƒπ™Ύπš)
π™²πšƒπšπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŒπšπš›-π™²πšƒπš)
Restrictions
|πš…π™΄π™²πšƒπ™Ύπš|β‰₯1
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™΄π™²πšƒπ™Ύπš,πšŸπšŠπš›)
π™²πšƒπšβˆˆ[=,β‰ ,<,β‰₯,>,≀]
π™Ώπ™΄πšπ™Έπ™Ύπ™³β‰₯1
π™Ώπ™΄πšπ™Έπ™Ύπ™³β‰€|πš…π™΄π™²πšƒπ™Ύπšπš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™΄π™²πšƒπ™Ύπšπš‚,𝚟𝚎𝚌)
πšœπšŠπš–πšŽ_πšœπš’πš£πšŽ(πš…π™΄π™²πšƒπ™Ύπšπš‚,𝚟𝚎𝚌)
πš›πšŽπššπšžπš’πš›πšŽπš(π™²πšƒπšπš‚,πšŒπšπš›)
|π™²πšƒπšπš‚|=|πš…π™΄π™²πšƒπ™Ύπš|
Purpose

Let us note πš…π™΄π™²πšƒπ™Ύπš 0 ,πš…π™΄π™²πšƒπ™Ύπš 1 ,β‹―,πš…π™΄π™²πšƒπ™Ύπš n-1 the vectors of the πš…π™΄π™²πšƒπ™Ύπšπš‚ collection, and d the number of components of each vector (all vectors have the same size). π™Ώπ™΄πšπ™Έπ™Ύπ™³ is the period of the sequence of vectors πš…π™΄π™²πšƒπ™Ύπš 0 ,πš…π™΄π™²πšƒπ™Ύπš 1 ,β‹―,πš…π™΄π™²πšƒπ™Ύπš n-1 according to constraints π™²πšƒπšπš‚. This means that π™Ώπ™΄πšπ™Έπ™Ύπ™³ is the smallest natural number such that βˆ€i∈[0,n-π™Ώπ™΄πšπ™Έπ™Ύπ™³-1],βˆ€j∈[0,d-1]:πš…π™΄π™²πšƒπ™Ύπš i .𝚟𝚎𝚌[j] π™²πšƒπšπš‚[j] πš…π™΄π™²πšƒπ™Ύπš i+π™Ώπ™΄πšπ™Έπ™Ύπ™³ .𝚟𝚎𝚌[j].

Example
3,𝚟𝚎𝚌-1,0,𝚟𝚎𝚌-1,5,𝚟𝚎𝚌-4,4,𝚟𝚎𝚌-1,0,𝚟𝚎𝚌-1,5,𝚟𝚎𝚌-4,4,𝚟𝚎𝚌-1,0,𝚟𝚎𝚌-1,5,=,=

The πš™πšŽπš›πš’πš˜πš_πšŸπšŽπšŒπšπš˜πš›πšœ constraint holds since its first argument π™Ώπ™΄πšπ™Έπ™Ύπ™³=3 is equal (i.e.,Β since π™²πšƒπšπš‚ is set to 〈=,=βŒͺ) to the period of the sequence 𝚟𝚎𝚌-〈1,0βŒͺ, 𝚟𝚎𝚌-〈1,5βŒͺ, 𝚟𝚎𝚌-〈4,4βŒͺ, 𝚟𝚎𝚌-〈1,0βŒͺ, 𝚟𝚎𝚌-〈1,5βŒͺ, 𝚟𝚎𝚌-〈4,4βŒͺ, 𝚟𝚎𝚌-〈1,0βŒͺ, 𝚟𝚎𝚌-〈1,5βŒͺ.

Typical
π™²πšƒπšβˆˆ[=]
|πš…π™΄π™²πšƒπ™Ύπš|>1
π™Ώπ™΄πšπ™Έπ™Ύπ™³>1
π™Ώπ™΄πšπ™Έπ™Ύπ™³<|πš…π™΄π™²πšƒπ™Ύπšπš‚|
|πš…π™΄π™²πšƒπ™Ύπšπš‚|>2
Symmetry

Items of πš…π™΄π™²πšƒπ™Ύπšπš‚ can be reversed.

Arg. properties
  • Functional dependency: π™Ώπ™΄πšπ™Έπ™Ύπ™³ determined by πš…π™΄π™²πšƒπ™Ύπšπš‚ and π™²πšƒπšπš‚.

  • Prefix-contractible wrt. πš…π™΄π™²πšƒπ™Ύπšπš‚.

  • Suffix-contractible wrt. πš…π™΄π™²πšƒπ™Ύπšπš‚.

See also

specialisation: πš™πšŽπš›πš’πš˜πšΒ (vector replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

Keywords

characteristic of a constraint: vector.

combinatorial object: periodic, sequence.

constraint arguments: pure functional dependency.

constraint type: predefined constraint.

modelling: functional dependency.