Experimental Complexity Theory

Scott Aaronson

Theoretical physics is to this…

as theoretical computer science is to what?

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)

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~

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

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

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?

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?