Outline General Problem of . . . Probabilistic and . . . Interval . . . Additional Problem: . . . Quantum Versions of k-CSP The Need for . . . k -CSP problems Algorithms: a First Step Known Algorithm for . . . Sch¨ oning’s Algorithm . . . Towards Quantum The Fastest Known . . . The Fastest Algorithm . . . Analyzing Possibility . . . Algorithms for Conclusion Acknowledgments Interval-Related Constraint Title Page Satisfaction Problems ◭◭ ◮◮ ◭ ◮ Evgeny Dantsin and Alexander Wolpert Page 1 of 15 Computer Science, Roosevelt University Chicago, IL 60605, USA, { edantsin,awolpert } @roosevelt.edu Go Back Vladik Kreinovich Full Screen Computer Science, University of Texas Close El Paso, TX 79968, USA, vladik@utep.edu Quit
Outline 1. Outline General Problem of . . . Probabilistic and . . . • Data processing: Interval . . . Additional Problem: . . . – we input the results � x i of measuring easy-to-measure quantities x i , and The Need for . . . – we use these results to find estimates � y = f ( � x 1 , . . . , � x n ) for difficult- k -CSP problems to-measure quantities y which are related to x i by a known relation Known Algorithm for . . . y = f ( x 1 , . . . , x n ). Sch¨ oning’s Algorithm . . . The Fastest Known . . . • Interval uncertainty: often , we only know the bounds ∆ i on the measurement def The Fastest Algorithm . . . errors ∆ x i = � x i − x i , i.e., we only know that the actual value x i belongs to Analyzing Possibility . . . the interval [ � x i − ∆ i , � x i + ∆ i ]. Conclusion • Problem: we want to know the range of possible values of y . Acknowledgments • Why quantum computing: this problem is NP-hard; one way to speed up Title Page computations is to use quantum computing. ◭◭ ◮◮ • Quantum interval techniques have indeed been proposed. ◭ ◮ • Constraints: often, we also know some constraints on the possible values of the directly measured quantities x 1 , . . . , x n . Page 2 of 15 • Ultimate objective: extend quantum interval algorithms to such constraints. Go Back • In this paper: as a first step, we consider quantum algorithms for discrete Full Screen constraint satisfaction problems. Close Quit
Outline 2. General Problem of Data Processing under Uncer- General Problem of . . . Probabilistic and . . . tainty Interval . . . Additional Problem: . . . • Indirect measurements: way to measure y that are are difficult (or even im- The Need for . . . possible) to measure directly. k -CSP problems • Idea: y = f ( x 1 , . . . , x n ) Known Algorithm for . . . Sch¨ oning’s Algorithm . . . The Fastest Known . . . x 1 � ✲ The Fastest Algorithm . . . x 2 � ✲ Analyzing Possibility . . . y = f ( � � x 1 , . . . , � x n ) ✲ f Conclusion · · · Acknowledgments � x n ✲ Title Page ◭◭ ◮◮ • Problem: measurements are never 100% accurate: � x i � = x i (∆ x i � = 0) hence ◭ ◮ y = f ( � � x 1 , . . . , � x n ) � = y = f ( x 1 , . . . , y n ) . Page 3 of 15 def What are bounds on ∆ y = � y − y ? Go Back Full Screen Close Quit
Outline 3. Probabilistic and Interval Uncertainty General Problem of . . . Probabilistic and . . . Interval . . . ∆ x 1 ✲ Additional Problem: . . . The Need for . . . ∆ x 2 ✲ ∆ y ✲ k -CSP problems f . . . Known Algorithm for . . . Sch¨ oning’s Algorithm . . . ∆ x n ✲ The Fastest Known . . . The Fastest Algorithm . . . Analyzing Possibility . . . • Traditional approach: we know probability distribution for ∆ x i (usually Conclusion Gaussian). Acknowledgments • Where it comes from: calibration using standard MI. Title Page • Problem: sometimes we do not know the distribution because no “standard” ◭◭ ◮◮ (more accurate) MI is available. Cases: ◭ ◮ – fundamental science – manufacturing Page 4 of 15 • Solution: we know upper bounds ∆ i on | ∆ x i | hence Go Back x i ∈ [ � x i − ∆ i , � x i + ∆ i ] . Full Screen Close Quit
Outline 4. Interval Computations: A Problem General Problem of . . . Probabilistic and . . . Interval . . . x 1 ✲ Additional Problem: . . . The Need for . . . x 2 ✲ y = f ( x 1 , . . . , x n ) ✲ k -CSP problems f · · · Known Algorithm for . . . Sch¨ oning’s Algorithm . . . x n ✲ The Fastest Known . . . The Fastest Algorithm . . . Analyzing Possibility . . . • Given: Conclusion Acknowledgments • an algorithm y = f ( x 1 , . . . , x n ) that transforms n real numbers x i into a number y ; Title Page • n intervals x i = [ x i , x i ]. ◭◭ ◮◮ • Compute: the corresponding range of y : ◭ ◮ [ y, y ] = { f ( x 1 , . . . , x n ) | x 1 ∈ [ x 1 , x 1 ] , . . . , x n ∈ [ x n , x n ] } . Page 5 of 15 • Fact: even for quadratic f , the problem of computing the exact range y is Go Back NP-hard. • Practical challenge: speed up interval computations. Full Screen Close Quit
Outline 5. Additional Problem: Constraints General Problem of . . . Probabilistic and . . . • Traditional interval computations: Interval . . . Additional Problem: . . . – we know the intervals x i of possible values of different parameters x i , The Need for . . . and k -CSP problems – we assume that an arbitrary combination of these values is possible. Known Algorithm for . . . • In geometric terms: the set of possible combinations x = ( x 1 , . . . , x n ) is a Sch¨ oning’s Algorithm . . . box x = x 1 × . . . × x n . The Fastest Known . . . The Fastest Algorithm . . . Analyzing Possibility . . . Conclusion Acknowledgments Title Page • In practice: we also know additional restrictions on the possible combinations of x i . ◭◭ ◮◮ • Example: in geosciences, in addition to intervals for velocities v i at different ◭ ◮ points, we know that | v i − v j | ≤ ∆ for neighboring points: � Page 6 of 15 � � � � Go Back � � � Full Screen • Example: in nuclear engineering, experts often state that combinations of Close extreme values are impossible, we have an ellipsoid, not a box. Quit
Outline 6. The Need for Quantum Algorithms in Interval Com- General Problem of . . . Probabilistic and . . . putations and in CSPs Interval . . . Additional Problem: . . . • Problem: interval computation problems are difficult to solve (NP-hard). The Need for . . . • In plain words: computation time grows exponentially with the number n of k -CSP problems inputs. Known Algorithm for . . . Sch¨ oning’s Algorithm . . . • Result: For large n , the resulting computation time is unrealistically long. The Fastest Known . . . • Quantum algorithms: a way to speed up computations. The Fastest Algorithm . . . Analyzing Possibility . . . • Example: Grover’s algorithm searches an unsorted list of N elements in time √ Conclusion O ( N ). Acknowledgments • What is known: quantum algorithms for (pure) interval computation. Title Page • Ultimate objective: efficient quantum algorithms for solving interval-related ◭◭ ◮◮ continuous CSP problems. ◭ ◮ • In this paper: we show how quantum computing can speed up the simplest constraint satisfaction problems (CSP): discrete CSPs. Page 7 of 15 Go Back Full Screen Close Quit
Outline 7. k -CSP problems General Problem of . . . Probabilistic and . . . • Discrete CSP: Interval . . . Additional Problem: . . . – each of n variables x 1 , . . . , x n can take d ≥ 2 possible values, and The Need for . . . – the goal is to find the values x i which satisfy given constraints. k -CSP problems • Exhaustive search: solves this problem in time ∼ d n ( ∼ means equality mod- Known Algorithm for . . . Sch¨ oning’s Algorithm . . . ulo a term which is polynomial in the length of the input formula). The Fastest Known . . . • Important case: k -CSP problems, in which every constraint contains ≤ k The Fastest Algorithm . . . variables. Analyzing Possibility . . . Conclusion • SAT: another important case of CSP is the satisfiability problem (SAT): Acknowledgments – We are given a Boolean formula F in conjunctive normal form C 1 & . . . & C m , Title Page where each clause C j is a disjunction l 1 ∨ . . . ∨ l k of literals, i.e., variables or their negations. ◭◭ ◮◮ – We need to find a truth assignment x 1 = a 1 , . . . , x n = a n that makes F true. ◭ ◮ • Here, clauses C j are constraints. Page 8 of 15 • A simple exhaustive search can solve this problem in time ∼ 2 n . Go Back • k -CSP leads to k -SAT, a restricted version of SAT where each clause has at Full Screen most k literals. Close Quit
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