The Effect of . . . Fuzzy Uncertainty: . . . Traditional . . . Discontinuous . . . Beyond Intervals: How Discontinuities . . . Phase Transitions Lead to Definitions and the . . . Computational . . . More General Ranges Computational . . . Computational . . . Karen Villaverde Conclusions Department of Computer Science Conclusions (cont-d) New Mexico State University Proof of NP-Hardness Las Cruces, NM 88003 Title Page Email: kvillave@cs.nmsu.edu ◭◭ ◮◮ Gilbert Ornelas Computing Sciences Corporation ◭ ◮ El Paso, TX 79912 Email: gtornelas@gmail.com Page 1 of 15 Go Back Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 1. Objectives of Science and Engineering Traditional . . . • One of the main tasks of science and engineering: Discontinuous . . . How Discontinuities . . . – use the current values of the physical quantities Definitions and the . . . x 1 , . . . , x n Computational . . . – to predict the future values y of the desired quan- Computational . . . tities. Computational . . . • To be able to perform this prediction, we must know Conclusions how y depends on x i : y = f ( x 1 , . . . , x n ). Conclusions (cont-d) • Once we know the algorithm f and the values x 1 , . . . , x n , Proof of NP-Hardness we can predict y as y = f ( x 1 , . . . , x n ). Title Page • Comment. ◭◭ ◮◮ – in this paper, we assume that we know the exact ◭ ◮ dependence f . Page 2 of 15 – In reality, often, the algorithm f represents the ac- Go Back tual physical dependence only approximately. Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 2. Measurement Inaccuracy Traditional . . . • In practice, the values of the quantities x 1 , . . . , x n usu- Discontinuous . . . ally come from measurements. How Discontinuities . . . Definitions and the . . . • Measurements are never 100% accurate. Computational . . . • The measured value � x i is different from the (unknown) Computational . . . def actual value x i : ∆ x i = � x i − x i � = 0. Computational . . . • Usually, the manufacturer of the measuring instrument Conclusions provides an upper bound ∆ i on ∆ x i : | ∆ x i | ≤ ∆ i . Conclusions (cont-d) Proof of NP-Hardness • In this case, from the measurement result � x i , we con- Title Page clude that x i is in the interval x i = [ � x i − ∆ i , � x i + ∆ i ]. ◭◭ ◮◮ • Conclusion: we usually know the current values of the ◭ ◮ physical quantities with interval uncertainty. Page 3 of 15 • Comment: sometimes, we also know the probabilities of different values x i ∈ x i . Go Back Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 3. The Effect of Measurement Inaccuracy on Predic- Traditional . . . tion Discontinuous . . . • Idealized case: How Discontinuities . . . Definitions and the . . . – we know the exact values x i of the current quanti- Computational . . . ties. Computational . . . – we can compute the exact value y = f ( x 1 , . . . , x n ) Computational . . . of the desired future quantity. Conclusions • In practice: Conclusions (cont-d) – for each i , we only know the interval x i of possible Proof of NP-Hardness values of x i ; Title Page – then, we can only conclude that y belongs to the ◭◭ ◮◮ set ◭ ◮ def y = f ( x 1 , . . . , x n ) = Page 4 of 15 { f ( x 1 , . . . , x n ) : x 1 ∈ x 1 , . . . , x n ∈ x n } . Go Back Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 4. Fuzzy Uncertainty: From the Computational View- Traditional . . . point, It Can Be Reduced to the Crisp Case Discontinuous . . . • For each possible value of x i ∈ x i , the experts describe How Discontinuities . . . the degree µ i ( x i ) to which this value is possible. Definitions and the . . . Computational . . . • Objective: compute the fuzzy number corresponding to the desired value y = f ( x 1 , . . . , x n ). Computational . . . Computational . . . • Fact: a fuzzy set can be thus viewed as a nested family Conclusions def of its α -cuts x i ( α ) = { x : µ i ( x ) ≥ α } . Conclusions (cont-d) • Meaning: α -cut is the set of values of x i which are Proof of NP-Hardness possible with degree ≥ α . Title Page ◭◭ ◮◮ • Fact: for each α , to α -cut y ( α ) can be obtained by the interval formula: y ( α ) = f ( x 1 ( α ) , . . . , x n ( α )) . ◭ ◮ • Conclusion: to find the fuzzy number for y , we can Page 5 of 15 apply an interval algorithm to the α -cuts x i ( α ). Go Back Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 5. Traditional Assumption: All Physical Dependen- Traditional . . . cies are Continuous Discontinuous . . . • Traditionally: it is assumed that all the processes are How Discontinuities . . . continuous. Definitions and the . . . Computational . . . • In particular: that the function y = f ( x 1 , . . . , x n ) com- puted by the algorithm f is continuous. Computational . . . Computational . . . • Known: the range of a continuous function on a bounded Conclusions connected set, e.g., on x 1 × . . . × x n , is an interval. Conclusions (cont-d) • Thus: for continuous functions f , the range y of pos- Proof of NP-Hardness sible values of the future quantity y is an interval. Title Page • So: due to inevitable measurement inaccuracy, we can ◭◭ ◮◮ only make predictions with interval uncertainty. ◭ ◮ • Computing such intervals is one of the main tasks of Page 6 of 15 interval computations . Go Back Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 6. Discontinuous Dependency: Physical Possibility Traditional . . . • Example of a discontinuous physical process: phase Discontinuous . . . transition. How Discontinuities . . . Definitions and the . . . • When a water is heated and boils, its density abruptly decreases to the density of steam ρ s . Computational . . . Computational . . . • Of course, all the molecules move continuously. Computational . . . • Theoretically, we continuously change from the density Conclusions of water ρ w to the density of steam ρ s . Conclusions (cont-d) • However, for all practical purposes, this transition is Proof of NP-Hardness very fast, practically instantaneous. Title Page • So, in practice, we can safely assume that: ◭◭ ◮◮ ◭ ◮ – the future density ρ can be equal to ρ w , – the future density ρ can be equal to ρ s , but Page 7 of 15 – the future density ρ cannot be equal to any inter- Go Back mediate value. Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 7. How Discontinuities Affect the Class of Possible Traditional . . . Ranges? Discontinuous . . . • First idea: it is sufficient to consider closed ranges. How Discontinuities . . . • Motivation: Definitions and the . . . Computational . . . – let s 1 , s 2 , . . . , s k , . . . are all possible, and s k → s , Computational . . . – then for every accuracy there is s k that is indistin- Computational . . . guishable from s (and possible); Conclusions – thus, from the practical viewpoint, s is also possi- Conclusions (cont-d) ble. Proof of NP-Hardness • Second idea: it is sufficient to consider closed classes of Title Page sets. ◭◭ ◮◮ • Motivation: similar; as a measure of closeness between ◭ ◮ sets, we use Hausdorff metric: Page 8 of 15 d H ( S, S ′ ) = min { ε > 0 : S is in the ε -neighborhood of S ′ Go Back and S ′ is in the ε -neighborhood of S } . Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 8. Definitions and the Main Result Traditional . . . Definition. A class S of closed bounded non-empty subsets Discontinuous . . . of the real line is called a class of ranges if it satisfies the How Discontinuities . . . following conditions: Definitions and the . . . (i) the class S contains an interval; Computational . . . Computational . . . (ii) the class S is closed under arbitrary continuous trans- Computational . . . formations, i.e., if S ∈ S and f ( x ) is a continuous function, then f ( S ) ∈ S ; Conclusions Conclusions (cont-d) (ii) there exist a value x 0 and a function f 0 ( x ) which is Proof of NP-Hardness continuously increasing for x < x 0 and for x > x 0 and Title Page which has a “jump” at x 0 ( f 0 ( x 0 − ) < f 0 ( x 0 +) ) such ◭◭ ◮◮ that the class S is closed under f 0 , i.e., if S ∈ S then f 0 ( S ) ∈ S ; and ◭ ◮ (iv) the class S is closed (under Hausdorff metric). Page 9 of 15 Theorem. The class of ranges coincides with the class of Go Back all bounded closed sets. Full Screen Close
The Effect of . . . Fuzzy Uncertainty: . . . 9. Computational Complexity of the Prediction Prob- Traditional . . . lem: Interval Uncertainty, Linear Functions Discontinuous . . . • Starting point: interval uncertainty, linear function How Discontinuities . . . n � Definitions and the . . . y = f ( x 1 , . . . , x n ) = a 0 + a i · x i . Computational . . . i =1 Computational . . . � n Computational . . . • Approximate value: � y = a 0 + a i · � x i . i =1 Conclusions � n Conclusions (cont-d) • Approximation error ∆ y = � y − y is ∆ y = a i · ∆ x i , Proof of NP-Hardness i =1 where ∆ x i ∈ [ − ∆ i , ∆ i ]. Title Page � n ◭◭ ◮◮ • a i · ∆ x i → max iff a i · ∆ x i → max for all i . i =1 ◭ ◮ • Conclusion: the largest possible value of the sum ∆ y Page 10 of 15 is ∆ = | a 1 | · ∆ 1 + . . . + | a n | · ∆ n . Go Back • Computational complexity: linear time (i.e., efficient). Full Screen Close
Recommend
More recommend
