# Guest column: a casual tour around a circuit complexity bound

@article{Williams2011GuestCA, title={Guest column: a casual tour around a circuit complexity bound}, author={Ryan Williams}, journal={ArXiv}, year={2011}, volume={abs/1111.1261} }

I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.

## 23 Citations

Some ways of thinking algorithmically about impossibility

- Computer ScienceSIGL
- 2017

A central question on the minds of today's complexity theorists is how will the authors find better ways to reason about all efficient programs.

Randomness in completeness and space-bounded computations

- Mathematics
- 2015

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER

New algorithms and lower bounds for circuits with linear threshold gates

- Computer ScienceSTOC
- 2014

An algorithm for evaluating an arbitrary symmetric function of 2no(1) ACC o THR circuits of size 2 no(1), on all possible inputs, in 2n · poly(n) time is given, evidence that non-uniform lower bounds for THR o THR are within reach.

Non-uniform ACC Circuit Lower Bounds

- Computer Science2011 IEEE 26th Annual Conference on Computational Complexity
- 2011

The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms can be applied to obtain the above lower bounds.

NEXP is not in ACC 0

- Mathematics, Computer Science
- 2014

C is a circuit with n inputs, and tt(C) is the class of all problems computable by constant depth, unbounded fan-in circuits with ∧,∨,¬ and MODm gates for any constant m.

Thinking Algorithmically About Impossibility

- Computer Science
- 2015

It is argued that some progress can be made by (very deliberately) thinking algorithmically about lower bounds, and to prove a lower bound against some class C of programs, to start by treating C as a set of inputs to another process, which is intended to perform some basic analysis of programs in C.

New lower bounds for probabilistic degree and AC0 with parity gates

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

The first progress is made on probabilistic-degree lower bounds and correlation bounds for polynomials since the papers by Razborov and Smolensky in the 80’s and the proofs build on Williams’ “guess-and-SAT” method.

Revisiting Cook-Levin theorem using NP-Completeness and Circuit-SAT

- Mathematics, Computer ScienceInternational Journal of Advanced Engineering Research and Science
- 2020

The Cook-Levin Theorem is revisited but using a completely different approach to prove the theorem, which showed that Boolean satisfiability problem is NP-complete through the reduction of polynomial time algorithms for NP-completeness and circuit-SAT.

Parameterized Graph Modification Algorithms

- Mathematics, Computer Science
- 2015

This thesis shows that editing towards trivially perfect graphs, threshold graphs, and chain graphs are all NP-complete, resolving 15 year old open questions and provides several new results in classical complexity, kernelization complexity, and subexponential parameterized complexity.

Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds

- Computer ScienceMFCS
- 2018

A deterministic algorithm for counting the number of satisfying assignments of any AC0[⊕] circuit C of size s and depth d over n variables in time 2n−f(n,s,d), which beats the lower bound of 2Ω(n) due to Razborov and Smolensky for large d.

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