Need for Quantum . . . Need for Reversible . . . Need to Take . . . When Is This Data . . . For Quantum and Reversible From Interval . . . Computing, Intervals Are Case of Fuzzy Uncertainty Intervals are Ubiquitous More Appropriate Than A Possible . . . Our Main Result General Sets, And Fuzzy Home Page Numbers Than General Title Page Fuzzy Sets ◭◭ ◮◮ ◭ ◮ Oscar Galindo and Vladik Kreinovich Page 1 of 29 Department of Computer Science Go Back University of Texas at El Paso, El Paso, Texas 79968, USA, Full Screen ogalindomo@miners.utep.edu, vladik@utep.edu Close Quit
Need for Quantum . . . Need for Reversible . . . 1. Need for Quantum Computing Need to Take . . . • Our current computers are very fast in comparison When Is This Data . . . with what was available a few years ago. From Interval . . . Case of Fuzzy Uncertainty • However, there are still computational tasks that ne- Intervals are Ubiquitous cessitate even faster computers. A Possible . . . • To speed up computers, we need to squeeze in more Our Main Result cells and into the same volume. Home Page • For that, we need to make cells as small as possible. Title Page • Already, the existing cells contain a small number of ◭◭ ◮◮ molecules. ◭ ◮ • If we decrease them further, they will contain a few Page 2 of 29 molecules. Go Back • Thus, we will need to take into account quantum ef- Full Screen fects. Close Quit
Need for Quantum . . . Need for Reversible . . . 2. Quantum Computing: Additional Advantages Need to Take . . . • There are innovative algorithms specifically designed When Is This Data . . . for quantum computing. From Interval . . . Case of Fuzzy Uncertainty • We can decrease the time needed to find an element in an unsorted array of size n from n to √ n steps. Intervals are Ubiquitous A Possible . . . • We can reduce the time needed to factor large integers Our Main Result of n digits from exponential to polynomial in n . Home Page • This task is needed to decode currently encoded mes- Title Page sages. ◭◭ ◮◮ ◭ ◮ Page 3 of 29 Go Back Full Screen Close Quit
Need for Quantum . . . Need for Reversible . . . 3. Need for Reversible Computing Need to Take . . . • One challenge in designing quantum computers is that When Is This Data . . . on the quantum level, all equations are time-reversible. From Interval . . . Case of Fuzzy Uncertainty • In the traditional algorithms, even the simplest “and”- Intervals are Ubiquitous operation a, b → a & b is not reversible: A Possible . . . – if we know its result a & b = 0 =“false”, Our Main Result – we cannot uniquely reconstruct the input ( a, b ). Home Page • Reversibility is also important because, according to Title Page statistical physics: ◭◭ ◮◮ – any irreversible process means increasing entropy, ◭ ◮ – and this leads to heat emission. Page 4 of 29 • Overheating is one of the reasons why we cannot pack Go Back too many processing units into the same volume. Full Screen • So, to pack more, it is desirable to reduce this heat emission – e.g., by using only reversible computations. Close Quit
Need for Quantum . . . Need for Reversible . . . 4. Need to Take Uncertainty into Account Need to Take . . . • We use computers mostly to process data. When Is This Data . . . From Interval . . . • When processing data, we need to take into account Case of Fuzzy Uncertainty that data comes from measurements. Intervals are Ubiquitous • Measurements are never absolutely accurate. A Possible . . . • The measurement result � x is, in general, different from Our Main Result Home Page the actual value x of the corresponding quantity. Title Page • It is therefore necessary to take this uncertainty into account when processing data. ◭◭ ◮◮ ◭ ◮ Page 5 of 29 Go Back Full Screen Close Quit
Need for Quantum . . . Need for Reversible . . . 5. Need for Interval Uncertainty Need to Take . . . • In many real life situations: When Is This Data . . . From Interval . . . – the only information that we have about the mea- Case of Fuzzy Uncertainty def surement error ∆ x = � x − x is Intervals are Ubiquitous – the upper bound ∆ on its absolute value: A Possible . . . | ∆ x | ≤ ∆ . Our Main Result Home Page • Once we have a measurement result � x , then: Title Page – the only information that we can conclude about ◭◭ ◮◮ the actual value x is that ◭ ◮ – this value is somewhere in the interval [ � x − ∆ , � x +∆]. Page 6 of 29 • Such interval uncertainty indeed appears in many prac- Go Back tical applications. Full Screen Close Quit
Need for Quantum . . . Need for Reversible . . . 6. Data Processing under Interval Uncertainty Need to Take . . . • In a data processing algorithm: When Is This Data . . . From Interval . . . – we take several inputs x 1 , . . . , x n , and Case of Fuzzy Uncertainty – we apply an appropriate algorithm to generate the Intervals are Ubiquitous result y depending on these inputs. A Possible . . . • Let us denote this dependence by f ( x 1 , . . . , x n ). Our Main Result Home Page • For each input i , we only know the interval X i = [ � x i − ∆ i , � x i + ∆ i ] of possible values of x i . Title Page • Then, the only information that we can have about y ◭◭ ◮◮ is that y belongs to the set ◭ ◮ def Y = f ( X 1 , . . . , X n ) = Page 7 of 29 { f ( x 1 , . . . , x n ) : x 1 ∈ X 1 , . . . , x n ∈ X n } . Go Back Full Screen • When the sets X i are intervals and the function f ( x 1 , . . . , x n ) is continuous, the resulting set Y is also an interval. Close Quit
Need for Quantum . . . Need for Reversible . . . 7. Interval Uncertainty (cont-d) Need to Take . . . • In most practical situations, the measurement errors When Is This Data . . . are relatively small. From Interval . . . Case of Fuzzy Uncertainty • So, we can expand the function f ( x 1 , . . . , x n ) in Taylor Intervals are Ubiquitous series and retain only linear terms. A Possible . . . • Then, we get Our Main Result f ( x 1 , . . . , x n ) = f ( � x 1 − ∆ x 1 , . . . , � x n − ∆ x n ) ≈ Home Page n � = ∂f Title Page def def � y − c i · ∆ x i , y � = f ( � x 1 , . . . , � x n ) , c i . ∂x i | x i = � ◭◭ ◮◮ x i i =1 • In other words, f ( x 1 , . . . , x n ) becomes a linear function: ◭ ◮ n n � � Page 8 of 29 def f ( x 1 , . . . , x n ) = c 0 + c i · x n , c 0 = � y − c i · � x i . Go Back i =1 i =1 Full Screen • In other words, data processing can be, in effect, re- duced to multiplication by a constant c i and addition. Close Quit
Need for Quantum . . . Need for Reversible . . . 8. When Is This Data Processing Reversible? Need to Take . . . • Multiplication by a constant is always reversible. When Is This Data . . . From Interval . . . • Indeed, if we know the interval Y = c · X , then, we can reconstruct X as X = c − 1 · Y . Case of Fuzzy Uncertainty Intervals are Ubiquitous • Addition y = x 1 + x 2 is also reversible. A Possible . . . • Indeed, if we know that x 1 ∈ [ x 1 , x 1 ] and x 2 ∈ [ x 1 , x 2 ], Our Main Result Home Page then Y = [ y, y ] has the form Title Page Y = [ x 1 + x 2 , x 1 + x 2 ] . ◭◭ ◮◮ • If we know Y = [ y, y ] and X 1 = [ x 1 , x 1 ], then we can ◭ ◮ reconstruct X 2 = [ x 2 , x 2 ] as Page 9 of 29 x 2 = y − x 1 and x 2 = y − x 1 . Go Back Full Screen Close Quit
Need for Quantum . . . Need for Reversible . . . 9. From Interval Uncertainty to a More General Need to Take . . . Set Uncertainty When Is This Data . . . • In some cases: From Interval . . . Case of Fuzzy Uncertainty – in addition to knowing that values of x are within Intervals are Ubiquitous a certain interval [ x, x ], A Possible . . . – we also know that some values from this interval Our Main Result are not possible. Home Page • In this case, the set X of possible values of x is different Title Page from an interval. ◭◭ ◮◮ • No matter how crude the measurements are, there is ◭ ◮ always an upper bound ∆ on the measurement error. Page 10 of 29 • Thus, all possible values of x are in the interval Go Back [ � x − ∆ , � x + ∆] . Full Screen • Thus, the set X is bounded. Close Quit
Need for Quantum . . . Need for Reversible . . . 10. Set Uncertainty (cont-d) Need to Take . . . • In general, we can safely assume that the set X is When Is This Data . . . closed. From Interval . . . Case of Fuzzy Uncertainty • Indeed, suppose that x 0 is a limit point of the set. Intervals are Ubiquitous • Then, for every ε > 0, there are elements x ∈ X is any A Possible . . . ε -neighborhood ( x 0 − ε, x 0 + ε ) of this value x 0 . Our Main Result • This means that: Home Page – no matter how accurately we measure the corre- Title Page sponding value, ◭◭ ◮◮ – we will not be able to distinguish between the limit ◭ ◮ value x 0 and a sufficient close value x ∈ X . Page 11 of 29 • It is therefore reasonable to simply assume that x 0 is Go Back possible. Full Screen • Thus, we conclude that the set of possible values of x contains all its limit points, i.e., is closed. Close Quit
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