Designing, Understanding, Is Speed Up Possible? Parallelization: - PowerPoint PPT Presentation

Main Problem: . . . Which Problems Are . . . What Is NP-Hard: . . . Propositional . . . Designing, Understanding, Is Speed Up Possible? Parallelization: Reminder and Analyzing Acausal Processes: . . . Quantum Computing Unconventional Idea: Explicit Use of . . . Computation Home Page Title Page Vladik Kreinovich ◭◭ ◮◮ ◭ ◮ Department of Computer Science University of Texas at El Paso Page 1 of 22 El Paso, TX 79968, USA vladik@utep.edu Go Back Full Screen Close Quit

Main Problem: . . . Which Problems Are . . . 1. Main Problem: Reminder and Natural Idea What Is NP-Hard: . . . • Problem: computations are often too slow. Propositional . . . Is Speed Up Possible? • Traditional approaches: Parallelization: Reminder • design faster super-computers ( hardware ); Acausal Processes: . . . • design faster algorithms . Quantum Computing Idea: Explicit Use of . . . • Limitations of the traditional approaches: Home Page • re hardware: we use the same physical processes as Title Page before; ◭◭ ◮◮ • re algorithms: we solve the same exact problem as before. ◭ ◮ Page 2 of 22 • Possible new approach: use unconventional processes: Go Back – unconventional physical processes, or Full Screen – unconventional biological processes. Close Quit

Main Problem: . . . Which Problems Are . . . 2. Which Problems Are Feasible: Brief Reminder What Is NP-Hard: . . . • In theoretical computer science: researchers usually dis- Propositional . . . tinguish between Is Speed Up Possible? Parallelization: Reminder – problems that can be solved in polynomial time, Acausal Processes: . . . i.e., in time ≤ P ( n ) where n is input length, and Quantum Computing – problems that require more computation time. Idea: Explicit Use of . . . • Terminology: Home Page – problems solvable in polynomial time are usually Title Page called feasible , ◭◭ ◮◮ – while others are called intractable . ◭ ◮ • Warning: this association is not perfect. Page 3 of 22 • Example: an algorithm that requires 10 100 · n steps is Go Back – polynomial time, but Full Screen – not practically feasible. Close Quit

Main Problem: . . . Which Problems Are . . . 3. What Problems We Are Solving: Examples What Is NP-Hard: . . . • In mathematics, we are given a statement x and we Propositional . . . want to find the proof y of either x or ¬ x . Is Speed Up Possible? Parallelization: Reminder • Once we have a detailed proof y , it is easy to check its Acausal Processes: . . . correctness, but inventing a proof is hard. Quantum Computing • A proof cannot be too long: it must be checkable. Idea: Explicit Use of . . . Home Page • In physics, we have observations x , and we want to find a law y that describes them. Title Page • Once we have y we can easily check whether it fits x , ◭◭ ◮◮ but coming up with y is often difficult. ◭ ◮ • A law cannot be too long: otherwise, we can take the Page 4 of 22 data as the law. Go Back • In engineering, we have a specification x , and we need Full Screen to find a design y that satisfies x . Close Quit

Main Problem: . . . Which Problems Are . . . 4. What Problems We Are Solving: General De- What Is NP-Hard: . . . scription Propositional . . . • In general : Is Speed Up Possible? Parallelization: Reminder – we have a string x , and Acausal Processes: . . . – we need to find y s.t. C ( x, y ) and len( y ) ≤ P ℓ (len( x )). Quantum Computing • Here, C ( x, y ) is a feasible property, i.e., a property that Idea: Explicit Use of . . . can be checked feasibly (in polynomial time). Home Page • In such problems: Title Page ◭◭ ◮◮ – once we have a guess y , – we can check its correctness in polynomial time. ◭ ◮ Page 5 of 22 • “Computations” allowing guesses are known as non- deterministic . Go Back • Thus, such problems are called Non-deterministic Poly- Full Screen nomial (NP). Close Quit

Main Problem: . . . Which Problems Are . . . 5. What Is NP-Hard: Reminder What Is NP-Hard: . . . • Ideally, we would like to call a problem hard if it cannot Propositional . . . be solved by a feasible (polynomial-time) algorithm. Is Speed Up Possible? Parallelization: Reminder • Alas, for neither of the problems from NP, we can prove Acausal Processes: . . . that this problem is hard in this sense. Quantum Computing • What we do know is that some problems are harder Idea: Explicit Use of . . . than others in the following sense: Home Page – every instance of a problem A Title Page – can be reduced to an appropriate instance of the ◭◭ ◮◮ problem B . ◭ ◮ • A problem is called NP-hard if every problem from NP Page 6 of 22 can be reduced to it. Go Back • In other words, a problem is NP-hard if it is harder Full Screen than all other problems from the class NP. Close Quit

Main Problem: . . . Which Problems Are . . . 6. Propositional Satisfiability: Historically First What Is NP-Hard: . . . NP-Hard Problem Propositional . . . • In many applications areas, certain problems are known Is Speed Up Possible? to be NP-hard (= provably computationally intractable). Parallelization: Reminder Acausal Processes: . . . • A historically first problem proven to be NP-hard is Quantum Computing propositional satisfiability . Idea: Explicit Use of . . . • This problem is about propositional formulas , i.e., ex- Home Page pressions F like ( x 1 & x 2 ) ∨ ( x 2 & ¬ x 3 ) obtained: Title Page – from propositional (“yes”-“no”) variables x 1 , . . . , x n , ◭◭ ◮◮ – by using “and” (&), “or” ( ∨ ), and “not” ( ¬ ). ◭ ◮ • We are given a propositional formula F , we must find Page 7 of 22 values x 1 , . . . , x n that make it true. Go Back Full Screen Close Quit

Main Problem: . . . Which Problems Are . . . 7. Proof that Satisfiability Is NP-Hard: Idea What Is NP-Hard: . . . • We have an instance of an NP problem: given x find y Propositional . . . for which C ( x, y ) is true and len( y ) ≤ P ℓ (len( x )). Is Speed Up Possible? Parallelization: Reminder • We want to reduce it to propositional satisfiability. Acausal Processes: . . . • We start with a computational device that, given a Quantum Computing string x of length len( x ) = n and y , checks C ( x, y ). Idea: Explicit Use of . . . Home Page • Computing C requires polynomial time T ≤ P ( n ). Title Page • During this time, only cells at distance ≤ R = c · T from the origin can influence the result. ◭◭ ◮◮ • Let ∆ V be the smallest cell volume. ◭ ◮ • Within the sphere of volume V = 4 3 · π · R 3 ∼ T 3 , there Page 8 of 22 are ≤ V Go Back ∆ V ∼ T 3 cells, fewer than ≤ const · ( P ( n )) 3 . Full Screen • So, we have no more than polynomially many cells. Close Quit

Main Problem: . . . Which Problems Are . . . 8. Proof that Satisfiability Is NP-Hard (cont-d) What Is NP-Hard: . . . • Let ∆ t be a time quantum. Propositional . . . Is Speed Up Possible? • The state S i,t +1 cell i at moment ( t +1) · ∆ t can only be Parallelization: Reminder influenced by states S j,t of cells at distance ≤ r = c · ∆ t . Acausal Processes: . . . 3 · π · r 3 • In this vicinity, there are ≤ N neighb = 4 ∆ V cells; Quantum Computing this number does not depend on the inputs size n : Idea: Explicit Use of . . . Home Page S i,t +1 = f i,t ( S i,t , S j,t , . . . ( ≤ N neighb terms)) . Title Page • Let S be the largest number of states of each cell. ◭◭ ◮◮ • We can describe each state as 0, 1, 2, . . . ◭ ◮ def • Then we need B = ⌈ log 2 ( S ) ⌉ bits s i,b,t , 1 ≤ b ≤ B , to Page 9 of 22 describe each state S i,t , so: Go Back s i,b,t +1 = f i,b,t ( s i, 1 ,t , . . . , s i,B,t , s j, 1 ,t , . . . , s j,B,t , . . . ) . Full Screen • We can then use a truth table to transform each such Close equation to a propositional formula F i,b,t . Quit

Main Problem: . . . Which Problems Are . . . 9. Proof that Satisfiability Is NP-Hard (final steps) What Is NP-Hard: . . . • For each cell i , bit b , and moment of time t , the fact Propositional . . . that s i,b,t +1 is computed correctly can be described as Is Speed Up Possible? Parallelization: Reminder s i,b,t +1 = f i,t ( s i, 1 ,t , . . . , s i,B,t , s j, 1 ,t , . . . , s j,B,t , . . . ) . Acausal Processes: . . . • We have shown that this property can be described by Quantum Computing a propositional formulas F i,b,t . Idea: Explicit Use of . . . • By combining all these formulas, we get a long formula Home Page def Title Page F long = F 1 , 1 , 1 & F 1 , 2 , 1 & . . . & F i,b,t & . . . ◭◭ ◮◮ • Meaning of F long : that C ( x, y ) was checked correctly. ◭ ◮ • We add the formulas describing that the input was x Page 10 of 22 and that the output of checking C ( x, y ) was “true”. Go Back • The resulting propositional formula holds if and only if there exists y for which C ( x, y ) is satisfied. Full Screen • Reduction is proven, so satisfiability is indeed NP-hard. Close Quit