Computational Complexity Theory

12.

Circuit Complexity

12.1.

Boolean Circuit Model

12.1.1.

Circuit Components

12.1.1.1.

12.1.1.2.

Wires and Connections

12.1.2.

Circuit Parameters

12.1.2.1.

12.1.2.2.

Fan-in and Fan-out

12.1.3.

Circuit Evaluation

12.2.

Circuit Families

12.2.1.

Uniform Circuit Families

12.2.1.1.

Polynomial-Time Uniformity

12.2.1.2.

Logspace Uniformity

12.2.2.

Non-uniform Circuit Families

12.2.2.1.

Arbitrary Circuit Families

12.2.3.

Size and Depth Hierarchies

12.3.

Non-uniform Complexity Classes

12.3.1.

12.3.1.1.

Definition and Properties

12.3.1.2.

12.3.1.3.

Relationship to Uniform Classes

12.3.2.

Circuit Lower Bounds

12.3.2.1.

12.3.2.2.

Limitations and Barriers

12.3.3.

Natural Proofs Barrier

12.3.3.1.

Razborov-Rudich Theorem

12.3.3.2.

Implications for Lower Bounds

12.4.

Specific Circuit Classes

12.4.1.

12.4.1.1.

Constant Depth, Polynomial Size

12.4.1.2.

Parity Lower Bounds

12.4.2.

12.4.2.1.

Polylog Depth Circuits

12.4.2.2.

Parallel Computation

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13. Quantum Complexity