Thick separators. Luc Jaulin and Benoît Desrochers. Lab-STICC, ENSTA Bretagne, Brest, France

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1 Thick separators Luc Jaul and Benoît Desrochers Lab-TICC, ENTA Bretagne, Brest, France Abstract. If an terval of R is an uncerta real number, a thick set is an uncerta subset of R n. More precisely, a thick set is an terval of the powerset of R n equipped with the clusion as an order relation. It can generally be defed by parameters or functions which are not known exactly, but are known to belong to some tervals. In this paper, we show how to use constrat propagation methods order to compute efficiently an ner and an er approximations of a thick set. The resultg ner/er contraction are made usg an operator which is called a thick separator. Then, we show how thick separators can be combed order to compute with thick sets. 1 Thick sets A th set is a subset of R n. It is qualified as th because its boundary is th. In this section, we present the defition of thick sets. Thick set. Denote by (P(R n ), ), the powerset of R n equipped with the clusion as an order relation. A thick set of R n is an terval of (P(R n ), ). If is a thick set of R n, there exist two subsets of R n, called the subset bound and the supset bound such that =, = P(R n ) }. (1) Another representation for the thickset is the partition,, }, where = = \ (2) = Z. 1

2 The subset is called the penumbra and plays an important role the characterization of thick sets [3]. Thick sets can be used to represent uncerta sets (such as an uncerta map [4]) or a soft constrats [1]. 2 Thick separators To characterize a th set usg a paver, we may use a separator (which is a pair of two contractors [8]) side a paver. eparators can be immediately generalized to thick sets. Now, the penumbra as a nonzero volume for thick sets. For efficiency reasons, it is important to avoid any accumulation of the pavg deep side the penumbra. This is the role of thick separators to avoid as much as possible bisections side the penumbra. Thick separators. A thick separator for the thick set is an extension of the concept of separator to thick sets. More precisely, a thick separator is a 3-uple of contractors,, } such that, for all [x] IR n ([x]) = [x] ([x]) = [x] ([x]) = [x] (3) In what follow, we defe an algebra for thick separator a similar manner than what as been done for contractors [2] or for separators [7]. 3 Algebra In this section, we show how we can defe operations for thick sets (as a union, tersection, difference, etc.). The ma motivation is to be able to compute with thick sets. Intersection. Consider two thick sets =, and Y = Y, Y with thick separators =,, } and Y = 2

3 Y, Y, } Y. A thick separator for the thick set Z = Z, Z = Y = Y, Y (4) is Z = Z, Z, } Z = Y, ( ) ( ) ( ) Y Y Y, Proof. We have Y Z = Z = Y = Y Z = Z \Z = Y \( Y ) = Y Y = Y ( Y ) = ( ) ( Y Y ) (( ) ( ( Y Y )) = Y ) ( Y ) ( Y ) Z = Z = Y = Y = Y. From the separator algebra, we get that a contractor for Z is Z = Y, a contractor for Z is Z = Y and a contractor for Z is Z = ( ) ( ( Y Y) Y). (6) 4 Usg Karnaugh map (5) This expression could have been obtaed usg Figure 1. Karnaugh map, as illustrated by Figure 2, can also can be used to get the expression for thick separators a more clear manner. For stance, if Z = Y, (7) we read from the Karnaugh map 3 }.

4 Figure 1: Intersection of two thick sets. Red means side, Blue means side and Orange means uncerta Z = Y Z = ( Y ) ( Y ) ( Y ) Z = Y. Therefore a thick separator for the thick set Z is Z = Z, Z, } Z = Y, ( ) ( ) ( Y Y Now, if we read ) Y, (8) Y (9) Z = \ Y Y \, (10) Z = ( Y ) ( Y ) Z = Y Z = ( Y ) ( Y ). Therefore a thick separator for the thick set Z is Z = Z, Z, } Z = Y, ( ) ( ) ( Y Y 4 ) Y, (11) Y (12) } }.

5 Figure 2: Karnaugh map Note that when we build such an expression from a Karnaugh map, fake boundaries may appear. They could be avoided usg the method proposed [5]. Example. Take one box [x] as Figure 3. We get ([x]) = },, ([x]) = [a], [x], } (13) where [a] the white box. Moreover, Y ([x]) = } Y, Y, Y ([x]) =, [x], }. (14) 5

6 Figure 3: Illustration of the tersection of two separators Thus Z = Z, Z, } Z ([x]) = ( ) ( Y ) ([x]), ( = Y Y Y) ([x]), = Y ([x]) } = [a], ([x] ) ([x] [x]) ([a] [x]), } =, [x], } We conclude that [x] Z. 5 Test case Interval lear system [10] [11] are lear systems of equations the coefficients of which are uncerta and belong to some tervals. Consider for stance the followg terval lear system [9]: [2, 4] x1 + [ 2, 0] x 2 [ 1, 1] (15) [ 1, 1] x 1 + [2, 4] x 2 [0, 2] For each constrat, a thick separator can be build and then combed usg Equation 5. A thick set version algorithm provides the pavg Figure 4. The solution set =, has for supset bound the 6

7 Figure 4: Thick set correspondg to the test-case. Red boxes are side the thick solution set and the blue boxes are side. The penumbra corresponds to the orange boxes. tolerable solution set (red+orange) and for subset bound the united solution set (red) [6]. Note that side the penumbra, no accumulation can be observed. References [1] Q. Brefort, L. Jaul, M. Ceberio, and V. Kreovich. If we take to account that constrats are soft,then processg constrats becomes algorithmically solvable. In Proceedgs of the IEEE eries of ymposia on Computational Intelligence CI Orlando, Florida, December 9-12,

8 [2] G. Chabert and L. Jaul. Contractor Programmg. Artificial Intelligence, 173: , [3] B. Desrochers and L. Jaul. Computg a guaranteed approximation the zone explored by a robot. IEEE Transaction on Automatic Control, [4] B. Desrochers,. Lacroix, and L. Jaul. et-membership approach to the kidnapped robot problem. In IRO 2015, [5] G. chvarcz Franco and L. Jaul. How to avoid fake boundaries contractor programmg. In WIM 16, [6] A. Goldsztejn and G. Chabert. On the approximation of lear ae-solution sets. In 12th International ymposium on cientific Computg, Computer Arithmetic and Validated Numerics, Duisburg, Germany, (CAN 2006), [7] L. Jaul and B. Desrochers. Introduction to the algebra of separators with application to path planng. Engeerg Applications of Artificial Intelligence, 33: , [8] L. Jaul and B. Desrochers. Robust localisation usg separators. In COPROD 2014, [9] V. Kreovich and. hary. Interval methods for data fittg under uncertaty: A probabilistic treatment. Reliable Computg, [10]. hary. On optimal solution of terval lear equations. IAM Journal on Numerical Analysis, 32(2): , [11]. hary. A new technique systems analysis under terval uncertaty and ambiguity. Reliable Computg, 8: ,