General Problems Class NP ? P = NP: An Open Problem NP-Complete Problems Quantum Ideas in Economics Why Quantum Ideas . . . Beyond Quantum Quantum . . . Our Idea and What . . . Econometrics Main Ideas Behind . . . There Is a Similar Idea . . . Vladik Kreinovich 1 , Hung T. Nguyen 2 , 3 , and Home Page Songsak Sriboonchitta 3 Title Page 1 Department of Computer Science, University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas, USA, USA, vladik@utep.edu 2 Department of Mathematical Sciences, New Mexico State University ◭ ◮ Las Cruces, New Mexico 88002, USA, hunguyne@nmsu.edu Faculty of Economics, Chiang Mai University Page 1 of 25 Chiang Mai 50200, Thailand, songsakecon@gmail.com Go Back Full Screen Close Quit
General Problems Class NP 1. General Problems ? P = NP: An Open Problem • In most practical problems: NP-Complete Problems Why Quantum Ideas . . . – once we have a candidate for a solution, Quantum . . . – we can feasibly check whether this candidate is in- Our Idea and What . . . deed a solution. Main Ideas Behind . . . • For example, in mathematics, it is often difficult to find There Is a Similar Idea . . . a proof of a statement or of its negation; however: Home Page – once someone produces what intends to be a de- Title Page tailed proof, ◭◭ ◮◮ – it is feasible for a referee to check that all the steps ◭ ◮ in this text are indeed correct and thus, Page 2 of 25 – that the text does indeed constitute a proof; Go Back – we can even use a computer-based system for this checking. Full Screen Close Quit
General Problems Class NP 2. Examples of Problems ? P = NP: An Open Problem • Similarly, in physics: NP-Complete Problems Why Quantum Ideas . . . – it is often difficult to find a formula that described Quantum . . . the observed phenomena, but Our Idea and What . . . – once such a formula is proposed, one can feasibly Main Ideas Behind . . . check whether all observations satisfy it. There Is a Similar Idea . . . • In engineering, it is often difficult to come up with a Home Page design that satisfies all the given specifications; but: Title Page – once a design is produced, ◭◭ ◮◮ – we can use software packages to check that this ◭ ◮ design indeed satisfies the specifications. Page 3 of 25 Go Back Full Screen Close Quit
General Problems Class NP 3. Class NP ? P = NP: An Open Problem • For example, we can check: NP-Complete Problems Why Quantum Ideas . . . – that the designed airplane is indeed stable under Quantum . . . allowable winds, Our Idea and What . . . – that the corresponding stresses do not exceed the Main Ideas Behind . . . prescribed level, etc. There Is a Similar Idea . . . • Problems for which we can feasibly check whether a Home Page candidate is indeed a solution are known as NP. Title Page • The abbreviation NP stands for Non-deterministic ◭◭ ◮◮ Polynomial , where: ◭ ◮ – “non-deterministic” means that we are allowed to Page 4 of 25 guess, and – “polynomial” means that once a guess is produced, Go Back checking takes polynomial time; Full Screen – such polynomial bounds are a formal description of Close feasibility. Quit
General Problems Class NP 4. NP and Beyond ? P = NP: An Open Problem • Not all practical problems belong to the class NP. NP-Complete Problems Why Quantum Ideas . . . • For example: Quantum . . . – if we want to find an optimal design, Our Idea and What . . . – then, in general, it is not easy to check that a given Main Ideas Behind . . . guess is optimal: There Is a Similar Idea . . . – for that, we would need to compare it with an un- Home Page feasible number of all possible designs. Title Page • Similarly, in multi-step conflict situations: ◭◭ ◮◮ – it is not easy to check whether a given move is ◭ ◮ winning or not; Page 5 of 25 – checking it would require going over all possible Go Back counter-moves of the opposite side. Full Screen • However, many practical problem are indeed problems from the class NP. Close Quit
General Problems Class NP ? 5. P = NP: An Open Problem ? P = NP: An Open Problem NP-Complete Problems • It is still not known whether we can solve all problems Why Quantum Ideas . . . from the class NP is feasible (polynomial) time. Quantum . . . • This is the famous open problem of whether: Our Idea and What . . . – the class NP is equal to Main Ideas Behind . . . There Is a Similar Idea . . . – the class P of all the problems that can be solved Home Page feasibly (i.e., in polynomial time). Title Page • Most computer scientists believe that NP is different from P. ◭◭ ◮◮ ◭ ◮ • The fact that we do not know whether NP is different from P means that: Page 6 of 25 – there is no problem from the class NP Go Back – for which we have proven that this problem cannot Full Screen be solved in polynomial time. Close Quit
General Problems Class NP 6. NP-Complete Problems ? P = NP: An Open Problem • There are problems from the class NP which are as NP-Complete Problems hard as possible within this class, in the sense that: Why Quantum Ideas . . . Quantum . . . – every other problem from the class NP Our Idea and What . . . – can be feasibly reduced to this problem. Main Ideas Behind . . . • Such problems are known as NP-complete. There Is a Similar Idea . . . • Many problems of solving non-linear equations have Home Page been proven to be NP-complete. Title Page • The 1st problem for which NP-completeness was ◭◭ ◮◮ proven was propositional satisfiability (SAT) : ◭ ◮ • Given a propositional formula F , i.e., a formula ob- Page 7 of 25 tained Go Back – from propositional (“yes”-“no”) variables v i Full Screen – by using propositional connectives & (and), ∨ (or), and ¬ (not). Close Quit
General Problems Class NP 7. NP-Complete Problems (cont-d) ? P = NP: An Open Problem • Example: F = ( v 1 ∨ v 2 ∨ ¬ v 3 ) & ( ¬ v 1 ∨ v 2 ) . NP-Complete Problems Why Quantum Ideas . . . • Find the values of the variables v i that make the for- Quantum . . . mula F true. Our Idea and What . . . • Here, a reduction of a problem A to problem B means Main Ideas Behind . . . that: There Is a Similar Idea . . . Home Page – for every instance a of the problem A , – we can feasibly compute an appropriate instance b Title Page of the problem B . ◭◭ ◮◮ • Then, ◭ ◮ – once we have a solution to the instance b , Page 8 of 25 – we can feasibly transform this solution into a solu- Go Back tion to the original instance a . Full Screen • Let us give a simple example of reduction. Close Quit
General Problems Class NP 8. Reduction ? P = NP: An Open Problem • Solving an equation p · x 4 + q · x + r = 0 can be reduced NP-Complete Problems to p · y 2 + q · y + r = 0. Why Quantum Ideas . . . Quantum . . . • If y is a solution to the quadratic equation, then x = ±√ y solves the original equation. Our Idea and What . . . Main Ideas Behind . . . • So, once we know that a problem is NP-complete, then: There Is a Similar Idea . . . Home Page – any good algorithm for solving this problem – automatically becomes a good algorithm for solving Title Page all other problems from the class NP. ◭◭ ◮◮ • This is not just a theoretical possibility: ◭ ◮ – efficient tools for solving the propositional satisfia- Page 9 of 25 bility problem (known as SAT-solvers ) Go Back – are now used to solve many problems from different Full Screen application areas. Close Quit
General Problems Class NP 9. Why Quantum Ideas in Economics ? P = NP: An Open Problem • From this viewpoint, econometrics has many complex NP-Complete Problems problems. Why Quantum Ideas . . . Quantum . . . • Sometimes, we do not have efficient algorithms for solv- Our Idea and What . . . ing these problems. Main Ideas Behind . . . • In this case, due to the above reduction, it is reason- There Is a Similar Idea . . . able: Home Page – to look for other complex (NP-complete) problem, Title Page and ◭◭ ◮◮ – see if known algorithms for solving these other ◭ ◮ problems can be applied to economics. Page 10 of 25 • Where can we find such other problems? Go Back • Most of the practical problems deal with the physical Full Screen world. Close Quit
General Problems Class NP 10. Why Quantum Ideas in Economics (cont-d) ? P = NP: An Open Problem • Thus, it is reasonable to look into physics for examples NP-Complete Problems of complex problems with known efficient algorithms. Why Quantum Ideas . . . Quantum . . . • It is known that adding quantum effects makes prob- Our Idea and What . . . lems more complex. Main Ideas Behind . . . • Thus, if we look for complex problems in physics, it is There Is a Similar Idea . . . reasonable to look for problems of quantum physics. Home Page • So, we arrive at the idea of trying to see if we can Title Page – apply known algorithms for solving complex prob- ◭◭ ◮◮ lem of quantum physics ◭ ◮ – to solve complex economics-related problems. Page 11 of 25 Go Back Full Screen Close Quit
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