![Page 1: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/1.jpg)
Experimental Complexity Theory
Scott Aaronson
![Page 2: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/2.jpg)
Theoretical physics is to this…
as theoretical computer science is to what?
![Page 3: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/3.jpg)
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)
![Page 4: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/4.jpg)
The hope: Examining the minimal circuits would inspire new conjectures about asymptotic behavior, which we could then try to prove
Conventional wisdom: We wouldn’t 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. That’s why I’m talking in this session.
n22~
![Page 5: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/5.jpg)
Goal: Prove that when n=4, the permanent requires more arithmetic operations than the determinant
A concrete challenge
n
iiiaA
1,per
n
iiiaA
1,
sgn1det
Fastest known algorithm for computing the determinant of an nn matrix: O(n2.376)
For the permanent: O(n2n)
Advantages over Boolean problems like 3SAT: More “robust,” less dependent on input encoding
![Page 6: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/6.jpg)
n By brute force
By Cramer’s rule
By dynamic programming
By Gaussian elimination
2 3 3 3 4
3 17 14 14 15
4 95 63 45 37
5 599 324 124 74
n
Number of arithmetic operations needed to compute nn determinant
n
m m
mn
2 !
12! 1
6
5
2
1
3
2 23 nnn 1! nn 121 nn n
![Page 7: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/7.jpg)
1. EA := E/A
2. EAB := EA+B
3. FEAB := F-EAB
4. EAC := EAC
5. GEAC := G-EAC
6. EAD := EAD
7. HEAD := H-EAD
8. IA := I/A
9. IAB := IAB
10. JIAB := J-IAB
11. IAC := IAC
12. KIAC := K-IAC
13. IAD := IAD
14. LIAD := L-IAD
15. MA := M/A
16. MAB := MAB
17. NMAB := N-MAB
18. MAC := MAC
19. OMAC := O-MAC
20. MAD := MAD
21. PMAD := P-MAD
22. JF := JIAB/FEAB
23. JFG := JFGEAC
24. KJFG := KIAC-JFG
25. JFH := JFHEAD
26. LJFH := LIAD-JFH
27. NF := NMAB/FEAB
28. NFG := NFGEAC
29. ONFG := OMAC-NFG
30. NFH := NFHEAD
31. PNFH := PMAD-NFH
32. OK := ONFG/KJFG
33. OKL := OKLJFH
34. POKL := PNFH-OKL
35. X := AFEAB
36. Y := XKJFG
37. DET := YPOKL
PONM
LKJI
HGFE
DCBA
using only 37 arithmetic operations
How to compute
OPTIMAL?
![Page 8: Experimental Complexity Theory Scott Aaronson. Theoretical physics is to this… as theoretical computer science is to what?](https://reader033.vdocument.in/reader033/viewer/2022061306/55146376550346b0158b4a24/html5/thumbnails/8.jpg)
To show that the 44 permanent can’t be computed with 37 arithmetic operations, how many programs would we need to examine?
Naïvely, 10123
For comparison, SETI@home does 1022 floating-point operations per year
How far can we cut down the search space?