L.Jaulin,B.DesrochersCoProd2016,Uppsala Thickseparators

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Thick separators Test-case

Thick separators

L. Jaulin, B. Desrochers CoProd 2016, Uppsala

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Thick sets

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A thin set is a subset of R ⁿ .

A thick set JXK of R ⁿ is an interval of ( P ( R ⁿ ),⊂).

JXK = JX ^⊂ , X ^⊃ K = { X ∈ P ( R ⁿ )| X ^⊂ ⊂ X ⊂ X ^⊃ }.

A thickset JXK is also the partition

X ⁱⁿ , X ^? , X ^out , where X ⁱⁿ = X ^⊂

X ^? = X ^⊃ \ X ^⊂ X ^out = Z ^⊃ . The subset X ^? is called the penumbra.

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Redermor DGA-TN, Brest

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Penumbra

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Contractors

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C ([x]) ⊂ [x] (contractance)

[x] ⊂ [y] ⇒ C ([x]) ⊂ C ([y]) (monotonicity)

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Properties

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Inclusion

C 1 ⊂ C 2 ⇔ ∀ [x] ∈ IR ⁿ , C 1 ([x]) ⊂ C 2 ([x]).

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A set S is consistent with C (we write S ∼ C ) if C ([x]) ∩ S = [x] ∩ S.

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C is minimal if

S ∼ C S ∼ C 1

⇒ C ⊂ C 1 .

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Separators

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A separator S is pair of contractors

S ⁱⁿ , S ^out such that S ⁱⁿ ([x]) ∪ S ^out ([x]) = [x] (complementarity).

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A set S is consistent with S (we write S ∼ S ), if S ∼ S ^out and S ∼ S ⁱⁿ .

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Properties

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Inclusion

S 1 ⊂ S 2 ⇔ S 1 ⁱⁿ ⊂ S 2 ⁱⁿ and S 1 ^out ⊂ S 2 ^out . Here ⊂ means more accurate.

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S is minimal if

S 1 ⊂ S ⇒ S 1 = S . i.e., if S ⁱⁿ and S ^out are both minimal.

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Algebra

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If S i =

S _i ⁱⁿ , S _i ^out ,i ≥ 1, are separators, we define S 1 ∩ S 2 =

S 1 ⁱⁿ ∪ S 2 ⁱⁿ , S 1 ^out ∩ S 2 ^out (intersection) S 1 ∪ S 2 =

S 1 ⁱⁿ ∩ S 2 ⁱⁿ , S 1 ^out ∪ S 2 ^out (union) S 1 \ S 2 = S 1 ∩ S 2 . (difference)

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Theorem. If S i are subsets of R ⁿ , we have (i) S 1 ∩ S 2 ∼ S 1 ∩ S 2

(ii) S 1 ∪ S 2 ∼ S 1 ∪ S 2

(iii) S i ∼ S i

(iv) S i ∼ S _i ^k , k ≥ 0 (vi) S 1 \ S 2 ∼ S 1 \ S 2 .

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Set M

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Rot( M )

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Rot( M ) ∪ M

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A thick separator JS K for JXK is a 3-uple of contractors S ⁱⁿ , S ^? , S ^out such that, for all [x] ∈ IR ⁿ

S ⁱⁿ ([x]) ∩ X ⁱⁿ = [x] ∩ X ⁱⁿ S ^? ([x]) ∩ X ^? = [x] ∩ X ^? S ôut ([x]) ∩ X ôut = [x] ∩ X ôut

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Algebra

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Intersection. Consider two thick separators JS _X K =

S _X ⁱⁿ , S _X ^? , S _X ^out and JS _Y K =

S _Y ⁱⁿ , S _Y ^? , S _Y ^out . A thick separator JS _Z K =

S _Z ⁱⁿ ,S _Z ^? , S _Z ^out for

JZK = JZ ^⊂ , Z ^⊃ K = JXK ∩ JYK

is

S _X ⁱⁿ ∩ S _Y ⁱⁿ , S _X ^? ∩ S _Y ⁱⁿ

t S _X ^? ∩ S _Y ^?

t S _X ⁱⁿ ∩ S _Y ^?

, S _X ^out t S _Y ^out .

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Intersection of two thick sets

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Illustration. Take one box [x] .

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We have

JS X K ([x]) = n

S _X ⁱⁿ , S _X ^? , S _X ^out o

([x]) = {[a],[x], /0}

where [a] the white box. Moreover, JS _Y K ([x]) = n

S _Y ⁱⁿ , S _Y ^? , S _Y ^out o

([x]) = { /0,[x], /0 } .

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JS _Z K =

S _Z ⁱⁿ , S _Z ^? , S _Z ^out ([x])

= { S _X ⁱⁿ ∩ S _Y ⁱⁿ ([x]),

= S _X ^? ∩ S _Y ⁱⁿ

t S _X ^? ∩ S _Y ^?

t S _X ⁱⁿ ∩ S _Y ^? ([x]),

= S _X ^out t S _Y ^out ([x]) }

= { [a] ∩ /0,([x] ∩ /0) t ([x] ∩ [x]) t ([a] ∩ [x]) , /0 t /0 }

= { /0, [x], /0}

We conclude that [x] ⊂ Z ⁱⁿ .

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Using Karnaugh maps

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Union. For

JZK = JXK ∪ JYK , we read from the Karnaugh map

Z ⁱⁿ = X ⁱⁿ ∪ Y ⁱⁿ Z ^? = X ^? ∩ Y ^out

∪ X ^? ∩ Y ^?

∪ X ôut ∩ Y ^? Z ôut = X ôut ∩ Y ôut .

A thick separator JS _Z K =

S _Z ⁱⁿ ,S _Z ^? , S _Z ^out for JZK is S _X ⁱⁿ t S _Y ⁱⁿ , S _X ^? ∩ S _Y ^out

t S _X ^? ∩ S _Y ^?

t S _X ^out ∩ S _Y ^?

, S _X ^out ∩ S _Y ^out

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XOR. For

JZK = JXK ⊕ JYK = JXK \ JYK ∪ JYK \ JXK ,

we read

Z ⁱⁿ = X ⁱⁿ ∩ Y ^out

∪ X ^out ∩ Y ⁱⁿ Z ^? = X ^? ∪ Y ^?

Z ^out = X ⁱⁿ ∩ Y ⁱⁿ

∪ ( X ^out ∩ Y ^out ) .

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Therefore a thick separator for the thick set JZK = JXK ⊕ JYK is S _X ⁱⁿ t S _Y ⁱⁿ , S _X ^? ∩ S _Y ^out

t S _X ^? ∩ S _Y ^?

t S _X ^out ∩ S _Y ^?

,S _X ^out ∩ S _Y ^out .

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Test-case

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Example from [Kreinovich, Shary, 2016]:

[2,4] · x 1 + [−2,0] · x 2 ∈ [−1, 1]

[−1,1] ·x ₁ + [2,4] ·x ₂ ∈ [0,2]

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Thick separators

L.Jaulin,B.DesrochersCoProd2016,Uppsala Thickseparators

Thick separators