If Many Physicists Are Right and No What Is a Physical . . . - PowerPoint PPT Presentation

Solving NP-Complete . . . NP-Complete . . . Can Non-Standard . . . No Physical Theory Is . . . If Many Physicists Are Right and No What Is a Physical . . . Physical Theory Is Perfect, Then by What We Mean by . . . Using Physical Observations, We Can Main Result Feasibly Solve Almost All Instances of This Result Is the Best . . . Each NP-Complete Problem Proof of the Main Result Home Page Olga Kosheleva 1 , Michael Zakharevich 2 , and Title Page Vladik Kreinovich 1 ◭◭ ◮◮ 1 University of Texas at El Paso El Paso, Texas 79968, USA ◭ ◮ olgak@utep.edu, vladik@utep.edu 2 Aligh Technology Inc., ymzakharevich@yahoo.com Page 1 of 20 Go Back Full Screen Close Quit

Solving NP-Complete . . . NP-Complete . . . 1. Outline Can Non-Standard . . . • Many real-life problems are, in general, NP-complete. No Physical Theory Is . . . What Is a Physical . . . • Informally speaking, these problems are difficult to solve What We Mean by . . . on computers based on the usual physical techniques. Main Result • A natural question is: can the use of non-standard This Result Is the Best . . . physics speed up the solution of these problems? Proof of the Main Result • This question has been analyzed, e.g.: Home Page – for quantum field theory, Title Page – for cosmological solutions with wormholes and/or ◭◭ ◮◮ casual anomalies. ◭ ◮ • However, many physicists believe that no physical the- Page 2 of 20 ory is perfect; in this talk, we show that: Go Back – if such a no-perfect-theory principle is true, Full Screen – then we can feasibly solve almost all instances of each NP-complete problem. Close Quit

Solving NP-Complete . . . NP-Complete . . . 2. Solving NP-Complete Problems Is Important Can Non-Standard . . . • In practice, we often need to find a solution that sat- No Physical Theory Is . . . isfies a given set of constraints. What Is a Physical . . . What We Mean by . . . • At a minimum, we need to check whether such a solu- Main Result tion is possible. This Result Is the Best . . . • Once we have a candidate, we can feasibly check whether Proof of the Main Result this candidate satisfies all the constraints. Home Page • In theoretical computer science, “feasibly” is usually Title Page interpreted as computable in polynomial time. ◭◭ ◮◮ • The class of all such problems is called NP. ◭ ◮ • Example: satisfiability – checking whether a formula Page 3 of 20 like ( v 1 ∨ ¬ v 2 ∨ v 3 ) & ( v 4 ∨ ¬ v 2 ∨ ¬ v 5 ) & . . . can be true. Go Back • Each problem from the class NP can be algorithmically Full Screen solved by trying all possible candidates. Close Quit

Solving NP-Complete . . . NP-Complete . . . 3. NP-Complete Problems (cont-d) Can Non-Standard . . . • For example, we can try all 2 n possible combinations No Physical Theory Is . . . of true-or-false values v 1 , . . . , v n . What Is a Physical . . . What We Mean by . . . • For medium-size inputs, e.g., for n ≈ 300, the resulting time 2 n is larger than the lifetime of the Universe. Main Result This Result Is the Best . . . • So, these exhaustive search algorithms are not practi- Proof of the Main Result cally feasible. Home Page • It is not known whether problems from the class NP Title Page can be solved feasibly (i.e., in polynomial time). ◭◭ ◮◮ ? • This is the famous open problem P =NP. ◭ ◮ • We know that some problems are NP-complete : every Page 4 of 20 problem from NP can be reduced to it. Go Back • So, it is very important to be able to efficiently solve Full Screen even one NP-hard problem. Close Quit

Solving NP-Complete . . . NP-Complete . . . 4. Can Non-Standard Physics Speed Up the So- Can Non-Standard . . . lution of NP-Complete Problems? No Physical Theory Is . . . • NP-complete means difficult to solve on computers based What Is a Physical . . . on the usual physical techniques. What We Mean by . . . Main Result • A natural question is: can the use of non-standard This Result Is the Best . . . physics speed up the solution of these problems? Proof of the Main Result • This question has been analyzed for several specific Home Page physical theories, e.g.: Title Page – for quantum field theory, ◭◭ ◮◮ – for cosmological solutions with wormholes and/or ◭ ◮ casual anomalies. Page 5 of 20 • So, a scheme based on a theory may not work. Go Back Full Screen Close Quit

Solving NP-Complete . . . NP-Complete . . . 5. No Physical Theory Is Perfect Can Non-Standard . . . • If a speed-up is possible within a given theory, is this No Physical Theory Is . . . a satisfactory answer? What Is a Physical . . . What We Mean by . . . • In the history of physics, Main Result – always new observations appear This Result Is the Best . . . – which are not fully consistent with the original the- Proof of the Main Result ory. Home Page • For example, Newton’s physics was replaced by quan- Title Page tum and relativistic theories. ◭◭ ◮◮ • Many physicists believe that every physical theory is ◭ ◮ approximate. Page 6 of 20 • For each theory T , inevitably new observations will Go Back surface which require a modification of T . Full Screen • Let us analyze how this idea affects computations. Close Quit

Solving NP-Complete . . . NP-Complete . . . 6. No Physical Theory Is Perfect: How to Formal- Can Non-Standard . . . ize This Idea No Physical Theory Is . . . • Statement: for every theory, eventually there will be What Is a Physical . . . observations which violate this theory. What We Mean by . . . Main Result • To formalize this statement, we need to formalize what This Result Is the Best . . . are observations and what is a theory . Proof of the Main Result • Most sensors already produce observation in the computer- Home Page readable form, as a sequence of 0s and 1s. Title Page • Let ω i be the bit result of an experiment whose de- ◭◭ ◮◮ scription is i . ◭ ◮ • Thus, all past and future observations form a (poten- tially) infinite sequence ω = ω 1 ω 2 . . . of 0s and 1s. Page 7 of 20 • A physical theory may be very complex. Go Back Full Screen • All we care about is which sequences of observations ω are consistent with this theory and which are not. Close Quit

Solving NP-Complete . . . NP-Complete . . . 7. What Is a Physical Theory? Can Non-Standard . . . • So, a physical theory T can be defined as the set of all No Physical Theory Is . . . sequences ω which are consistent with this theory. What Is a Physical . . . What We Mean by . . . • A physical theory must have at least one possible se- Main Result quence of observations: T � = ∅ . This Result Is the Best . . . • A theory must be described by a finite sequence of Proof of the Main Result symbols: the set T must be definable. Home Page • How can we check that an infinite sequence ω = ω 1 ω 2 . . . Title Page is consistent with the theory? ◭◭ ◮◮ • The only way is check that for every n , the sequence ◭ ◮ ω 1 . . . ω n is consistent with T ; so: ∀ n ∃ ω ( n ) ∈ T ( ω ( n ) . . . ω ( n ) Page 8 of 20 = ω 1 . . . ω n ) ⇒ ω ∈ T. 1 n Go Back • In mathematical terms, this means that T is closed in def the Baire metric d ( ω, ω ′ ) = 2 − N ( ω,ω ′ ) , where Full Screen def N ( ω, ω ′ ) = max { k : ω 1 . . . ω k = ω ′ 1 . . . ω ′ Close k } . Quit

Solving NP-Complete . . . NP-Complete . . . 8. What Is a Physical Theory: Definition Can Non-Standard . . . • A theory must predict something new. No Physical Theory Is . . . What Is a Physical . . . • So, for every sequence ω 1 . . . ω n consistent with T , there What We Mean by . . . is a continuation which does not belong to T . Main Result • In mathematical terms, T is nowhere dense . This Result Is the Best . . . • By a physical theory , we mean a non-empty closed Proof of the Main Result Home Page nowhere dense definable set T . Title Page • A sequence ω is consistent with the no-perfect-theory principle if it does not belong to any physical theory. ◭◭ ◮◮ • In precise terms, ω does not belong to the union of all ◭ ◮ definable closed nowhere dense set. Page 9 of 20 • There are countably many definable set, so this union Go Back is meager (= Baire first category ). Full Screen • Thus, due to Baire Theorem, such sequences ω exist. Close Quit

Solving NP-Complete . . . NP-Complete . . . 9. How to Represent Instances of an NP-Complete Can Non-Standard . . . Problem No Physical Theory Is . . . • For each NP-complete problem P , its instances are se- What Is a Physical . . . quences of symbols. What We Mean by . . . Main Result • In the computer, each such sequence is represented as This Result Is the Best . . . a sequence of 0s and 1s. Proof of the Main Result • We can append 1 in front and interpret this sequence Home Page as a binary code of a natural number i . Title Page • In principle, not all natural numbers i correspond to ◭◭ ◮◮ instances of a problem P . ◭ ◮ • We will denote the set of all natural numbers which Page 10 of 20 correspond to such instances by S P . Go Back • For each i ∈ S P , we denote the correct answer (true or false) to the i -th instance of the problem P by s P ,i . Full Screen Close Quit