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Experimental Complexity Theory Scott Aaronson

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Theoretical physics is to this… as theoretical computer science is to what?

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Suppose (hypothetically) that we had the kind of money the physicists have Is there any way we could use it to advance understanding of the P vs. NP question? (Besides more students, coffee, whiteboard markers…) Idea: Use high-performance computing to find minimal circuits for hard problems (for small values of n)

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The hope: Examining the minimal circuits would inspire new conjectures about asymptotic behavior, which we could then try to prove Conventional wisdom: We wouldnt learn anything this way - There are circuits on n variables astronomical even for tiny n - Small-n behavior can be notoriously misleading about asymptotics My view: The conventional wisdom is probably right. Thats why Im talking in this session.

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Goal: Prove that when n=4, the permanent requires more arithmetic operations than the determinant A concrete challenge Fastest known algorithm for computing the determinant of an n n matrix: O(n ) For the permanent: O(n2 n ) Advantages over Boolean problems like 3SAT: More robust, less dependent on input encoding

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nBy brute force By Cramers rule By dynamic programming By Gaussian elimination n Number of arithmetic operations needed to compute n n determinant

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1. EA := E/A 2. EAB := EA+B 3. FEAB := F-EAB 4. EAC := EA C 5. GEAC := G-EAC 6. EAD := EA D 7. HEAD := H-EAD 8. IA := I/A 9. IAB := IA B 10. JIAB := J-IAB 11. IAC := IA C 12. KIAC := K-IAC 13. IAD := IA D 14. LIAD := L-IAD 15. MA := M/A 16. MAB := MA B 17. NMAB := N-MAB 18. MAC := MA C 19. OMAC := O-MAC 20. MAD := MA D 21. PMAD := P-MAD 22. JF := JIAB/FEAB 23. JFG := JF GEAC 24. KJFG := KIAC-JFG 25. JFH := JF HEAD 26. LJFH := LIAD-JFH 27. NF := NMAB/FEAB 28. NFG := NF GEAC 29. ONFG := OMAC-NFG 30. NFH := NF HEAD 31. PNFH := PMAD-NFH 32. OK := ONFG/KJFG 33. OKL := OK LJFH 34. POKL := PNFH-OKL 35. X := A FEAB 36. Y := X KJFG 37. DET := Y POKL using only 37 arithmetic operations How to compute OPTIMAL?

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To show that the 4 4 permanent cant be computed with 37 arithmetic operations, how many programs would we need to examine? Naïvely, For comparison, does floating-point operations per year How far can we cut down the search space?

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