Guest column: a casual tour around a circuit complexity bound

  title={Guest column: a casual tour around a circuit complexity bound},
  author={Ryan Williams},
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. 
Some ways of thinking algorithmically about impossibility
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
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New algorithms and lower bounds for circuits with linear threshold gates
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
  • Ryan Williams
  • Computer Science
    2011 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
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
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
  • Emanuele Viola
  • Computer Science
    Electron. 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
  • E. E. Ogheneovo
  • Mathematics, Computer Science
    International 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
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
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.


Separating the polynomial-time hierarchy by oracles
  • A. Yao
  • Computer Science, Mathematics
  • 1985
We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all the
A Uniform Circuit Lower Bound for the Permanent
The authors show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the
Worst-Case Upper Bounds
The chapter is a survey of ideas and techniques behind satisfiability algorithms with the currently best asymptotic upper bounds on the worst-case running time. The survey also includes related
Parity, circuits, and the polynomial-time hierarchy
A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function and connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
A Note on Succinct Representations of Graphs
A full derandomization of schöning's k-SAT algorithm
A deterministic version ofchopping in 1999 presented a simple randomized algorithm for k-SAT with running time a n+o(n) for a = 2(k-1)/k is given.
ON ACC and threshold circuits
  • A. Yao
  • Computer Science
    Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
  • 1990
It is proved that any language in ACC can be approximately computed by two-level circuits of size 2 raised to the (log n)/sup k/ power, with a symmetric-function gate at the top and only AND gates on the first level, giving the first nontrivial upper bound on the computing power of ACC circuits.
Some connections between nonuniform and uniform complexity classes
This work aims to understand when nonuniform upper bounds can be used to obtain uniform upper bounds, and how to relate it to more common notions.
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.