Circuit Lower Bounds (and More) via the Fusion Method

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Fusion method reframes lower bounds as a static cover problem: pick a subset Omega of functions and lower bound circuit size by the minimum number of abstract gate-sets needed to cover Omega_F. Meta-theorems show this cover number matches circuit size for many models, reproving Razborov’s monotone clique lower bound, capturing monotone P, P/nonuniform, NP (via self‑dual filters) and parity‑L (via affine Omegas), and yielding span programs as the explicit model; a greedy assignment builds a circuit and often gives tight (even linear) characterizations.

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" chapters (31)

1. 0:00Introduction to fusion method
2. 1:09Historical context of bounds
3. 3:33Approximation method origins
4. 5:57Limits of approximation
5. 8:38General approximation method
6. 11:53Introduction to the fusion method
7. 15:18Progress measures in proofs
8. 18:00Circuit approximation mechanics
9. 20:43Monotone vs general cases
10. 23:12Algebraic approximation method
11. 26:03Fusion method framework
12. 28:54Crux of the fusion method
13. 31:49Magical properties of fusion
14. 34:33Proving lower bounds
15. 37:00Covering problems
16. 39:08Meta theorem for bounds
17. 41:47Tightness of bounds
18. 44:10Static nature of fusion
19. 45:50Greedy construction algorithm
20. 46:53Solving the covering problem
21. 49:21Definition of cover
22. 50:35Scope of the method
23. 52:04Functional domains
24. 52:17Methodology constraints
25. 53:28Relative complexity classes
26. 54:20Omega choices and models
27. 55:30Branching program results
28. 57:47NP characterization
29. 59:06Affine functions and parity
30. 1:00:26Open questions and models
31. 1:01:32Conclusion and summary