Albert Einstein and . . . Their Ideas Are Not As . . . Einstein – Zadeh – . . . The Main Challenge . . . Standing on the Shoulders of Is Parallelization a . . . the Giants: From Einstein’s Einstein Can Help: . . . Acausal Processes: . . . General Relativity and No Physical Theory Is . . . Conclusion: . . . Zadeh’s Fuzzy Logic to Home Page Computers of Generation Title Page Omega ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Page 1 of 22 Department of Computer Science University of Texas at El Paso Go Back El Paso, TX 79968, USA Full Screen vladik@utep.edu Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 1. Which Problems Are Most Significant? Einstein – Zadeh – . . . • In 1986, Richard Hamming, of the Hamming code The Main Challenge . . . fame, gave a talk titled You and Your Research. Is Parallelization a . . . Einstein Can Help: . . . • In this talk, he emphasized that to become a great sci- Acausal Processes: . . . entist, it is important to work on significant problems. No Physical Theory Is . . . • So which problems are most significant? Conclusion: . . . • We want to know how the world functions, what is the Home Page causal relation between different processes. Title Page • In analyzing causality, the most revolutionary results ◭◭ ◮◮ were obtained by Einstein. ◭ ◮ • We also want to know how we humans functions, how Page 2 of 22 we reason, how we make decisions. Go Back • In describing human reasoning, probably the most rev- Full Screen olutionary idea is Zadeh’s idea of fuzzy logic. Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 2. Albert Einstein and Lotfi Zadeh Einstein – Zadeh – . . . • At present, we celebrate 100 years of general relativity The Main Challenge . . . and 50 years of fuzzy. Is Parallelization a . . . Einstein Can Help: . . . • It is thus time to compare their authors. Acausal Processes: . . . • At first glance, their ideas was diametrically opposite: No Physical Theory Is . . . – Einstein was a known enemy of uncertainty, he even Conclusion: . . . Home Page objected to quantum physics, while – Zadeh emphasized uncertainty. Title Page ◭◭ ◮◮ • But on a deeper level, their ideas are similar: they both emphasize the need to challenge the prevailing dogmas. ◭ ◮ • Einstein showed that the physical world is not de- Page 3 of 22 scribed by Euclidean geometry. Go Back • Zadeh showed that our reasoning is not described by Full Screen 2-valued logic. Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 3. Their Ideas Are Not As Radical As They May Einstein – Zadeh – . . . Seem The Main Challenge . . . • Interestingly, it later turned out that their theories are Is Parallelization a . . . not as revolutionary as one might think. Einstein Can Help: . . . Acausal Processes: . . . • Einstein’s equations can be deduced from the field the- No Physical Theory Is . . . ory if we assume that energy is the field’s source. Conclusion: . . . • Zadeh’s logic can be described in traditional terms – Home Page and Lukaciewicz has done that already in 1920s. Title Page • So why are they famous? Why are their ideas widely ◭◭ ◮◮ used? ◭ ◮ • Because both were motivated by applications. Page 4 of 22 • Hilbert discovered Einstein’s equations 2 weeks after Einstein – but he did not have physical applications. Go Back Full Screen • Lukaciewicz proposed fuzzy logic 40 years before Zadeh – but he did not have applications. Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 4. Einstein – Zadeh – What Next? Einstein – Zadeh – . . . • What are significant problems now? The Main Challenge . . . Is Parallelization a . . . • In Einstein’s time, the main challenge was to come up Einstein Can Help: . . . with equations that describe the physical world. Acausal Processes: . . . • After relativity and quantum physics, we have pretty No Physical Theory Is . . . accurate equations. Conclusion: . . . • The main challenge now is how to use these equations, Home Page how to predict events using these equations. Title Page • In principle, we know the equations describing weather. ◭◭ ◮◮ • However, modern supercomputers barely have time ◭ ◮ predict tomorrow’s weather. Page 5 of 22 • In principle, we can predict whether a tornado will turn Go Back in the next 15 minutes. Full Screen • However, on modern supercomputers, these computa- tions require several days. Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 5. The Main Challenge Facing Science: How to Einstein – Zadeh – . . . Compute Faster The Main Challenge . . . • So, the main challenge now is how to compute faster. Is Parallelization a . . . Einstein Can Help: . . . • Theoretical computer science’s study of NP-hardness Acausal Processes: . . . has shown that many problems are inherently complex. No Physical Theory Is . . . • This means that we cannot decrease the number of Conclusion: . . . computational steps. Home Page • So, the only way to compute faster is to design faster Title Page computers. ◭◭ ◮◮ • How is this done now? The first natural idea is to have ◭ ◮ several processors working in parallel. Page 6 of 22 • This is why our brain can solve some problems faster than a supercomputer. Go Back Full Screen • Another idea is miniaturization: a 30 cm laptop re- quires 1 nanosecond for the signal to pass through. Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 6. Is Parallelization a Panacea? Einstein – Zadeh – . . . • If we accumulate a lot of parallel processors, maybe we The Main Challenge . . . solve exponential-time problems in polynomial time? Is Parallelization a . . . Einstein Can Help: . . . • Result: parallelism cannot reduce the computation Acausal Processes: . . . time T that drastically. No Physical Theory Is . . . • During the parallel computation time T p , we can only Conclusion: . . . access computers within a sphere of radius R = c · T p . Home Page • Within this sphere of volume V = 4 3 · π · R 3 ∼ T 3 Title Page p , we can fit ≤ V/ ∆ V ∼ T 3 p processors of given size ∆ V . ◭◭ ◮◮ • All these processors can perform T ≤ T p ◭ ◮ ∆ t · const · T 3 p = Page 7 of 22 C · T 4 p computational steps. Go Back • So, if a computation requires T sequential steps, we need T p ≥ C · T 1 / 4 steps to perform it in parallel. Full Screen Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 7. Einstein Can Help: Curved Space-Time Einstein – Zadeh – . . . • Observation: the above lower bound on parallel com- The Main Challenge . . . putation time depends on the formula V ( R ) = 4 3 · π · R 3 . Is Parallelization a . . . Einstein Can Help: . . . • Known: this formula only holds in Euclidean geometry. Acausal Processes: . . . • Idea: since the actual space-time is curved (= not Eu- No Physical Theory Is . . . clidean), we may get faster parallel computations. Conclusion: . . . Home Page • Known: in Lobachevsky space, � � R � � R � � � 2 � − R Title Page V ( R ) = 2 πk 3 · sinh · cosh ∼ exp k · R . k k k ◭◭ ◮◮ • Corollary: we can fit exponentially many processors ◭ ◮ into a sphere of radius R = c · T p . Page 8 of 22 • Conclusion: in Lobachevsky space, parallelization can reduce exponential time T = 2 n to linear time T p ∼ n . Go Back Full Screen • Lobachevsky’s idea: by measuring V ( R ), we can speed up computation of sinh( x ). Close Quit
Albert Einstein and . . . Their Ideas Are Not As . . . 8. Parallelization in Curved Space-Time (cont-d) Einstein – Zadeh – . . . • Assumption: particles are such “almost” black holes, The Main Challenge . . . entering into other “universes”. Is Parallelization a . . . Einstein Can Help: . . . • Let us show how this can help solve NP-hard problems, Acausal Processes: . . . on the example of propositional satisfiability SAT: No Physical Theory Is . . . – given a propositional formula F ( x 1 , . . . , x n ), Conclusion: . . . – find the values x 1 , . . . , x n s.t. F ( x ) is true. Home Page • To find x = ( x 1 , . . . , x n ), x i ∈ { 0 , 1 } , s.t. F ( x ), we: Title Page – find two particles (and corr. worlds); ◭◭ ◮◮ – ask World 1 to search for x = (0 , x 2 , . . . , x n ) ◭ ◮ s.t. F ( x ); Page 9 of 22 – ask World 2 to search for x = (1 , x 2 , . . . , x n ) Go Back s.t. F ( x ). Full Screen • Each of these worlds does the same split w.r.t. x 2 , etc.; in time 2 n ( ≪ 2 n ), we get an answer back. Close Quit
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