Thick sets Contractors and separators Thick separators Test-case. Thick separators. L. Jaulin, B. Desrochers CoProd 2016, Uppsala.

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1 L. Jaulin, B. Desrochers CoProd 2016, Uppsala

2 Thick sets

3 A thin set is a subset of R n. A thick set X of R n is an interval of (P(R n ), ). X = X,X = {X P(R n ) X X X }. A thickset X 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.

4 Redermor DGA-TN, Brest

5 Penumbra

6 Contractors Thick sets

7 C ([x]) [x] [x] [y] C ([x]) C ([y]) (contractance) (monotonicity)

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10 Properties Thick sets

11 Inclusion Thick sets C 1 C 2 [x] IR n, C 1 ([x]) C 2 ([x]).

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

13 C is minimal if Thick sets S C S C 1 } C C 1.

14 Separators Thick sets

15 A separator S is pair of contractors { S in,s out} such that S in ([x]) S out ([x]) = [x] (complementarity).

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

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18 Properties Thick sets

19 Inclusion Thick sets S 1 S 2 S in 1 S in 2 and S out 1 S out 2. Here means more accurate.

20 S is minimal if Thick sets S 1 S S 1 = S. i.e., if S in and S out are both minimal.

21 Algebra Thick sets

22 If S i = { S in i Thick sets,si out },i 1, are separators, we define S 1 S 2 = { S1 in S 2 in,s out 1 S2 out } (intersection) S 1 S 2 = { S1 in S 2 in,s out 1 S2 out } (union) S 1 \S 2 = S 1 S 2. (difference)

23 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 Si k, k 0 (vi) S 1 \S 2 S 1 \S 2.

24 Set M

25 Rot(M)

26 Rot(M) M

27 Thick sets

28 A thick separator S for X 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 out ([x]) X out = [x] X out

29 Algebra Thick sets

30 Intersection. Consider two thick separators S X = { SX in,s X? },S out X and SY = { S in separator S Z = { SZ in,s Z? },S out Z for Y,S? Y Z = Z,Z = X Y },S out Y. A thick is { S in X SY in,( S X? S Y in ) ( S? X S Y? ) ( S in X S Y? ),S out X S Y out }.

31 Intersection of two thick sets

32 Illustration. Take one box [x].

33 We have Thick sets { } S X ([x]) = SX in,s X? out,sx ([x]) = {[a],[x], /0} where [a] the white box. Moreover, { S Y ([x]) = SY in,s Y?,S out Y } ([x]) = {/0,[x], /0}.

34 { S Z = S in Z,S Z? },S out Z ([x]) = { ( SX in S Y in([x]), = S? X SY in ) ( S? X S Y? ) ( S in X S Y)? ([x]), = SX out S Y out([x]) } = { [a] /0,([x] /0) ([x] [x]) ([a] [x]), /0 /0 } = {/0,[x], /0} We conclude that [x] Z in.

35 Using Karnaugh maps

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37 Union. For Thick sets we read from the Karnaugh map Z = X Y, Z in = X in Y in Z? = ( X? Y out) ( X? Y?) ( X out Y?) Z out = X out Y out. A thick separator S Z = { S in Z,S? Z,S out Z } for Z is { S in X SY in,( S X? S Y out ) ( S? X S Y? ) ( S out X S Y? ),S out X S Y out }

38 XOR. For Thick sets Z = X Y = X \ Y Y \ X, 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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40 Therefore a thick separator for the thick set Z = X Y is { S in X SY in,( S X? S Y out ) ( S? X S Y? ) ( S out X S Y? ),S out X S Y out }.

41 Thick sets

42 Example from [Kreinovich, Shary, 2016]: { [2,4] x1 + [ 2,0] x 2 [ 1,1] [ 1,1] x 1 + [2,4] x 2 [0,2]

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