Depth Lower Bounds for Monotone Semi-Unbounded Fan-in Circuits

Depth Lower Bounds for Monotone Semi-Unbounded Fan-in Cir uits

Lecture 11 : Circuit Lower

There are specific kinds of circuits for which lower bounds techniques were successfully developed. One is small-depth circuits, the other is monotone circuits. For constant-depth circuits with AND,OR,NOT gates, people proved that they cannot compute simple functions like PARITY [3, 1] or MAJORITY. For monotone circuits, Alexander A. Razborov proved that CLIQUE, an NP-complete problem, has expo...

Finite Limits and Monotone Computations: The Lower Bounds Criterion

Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin non-decreasing real-valued functions as gates. The only requirement is their "local-ity". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ; x m) with small enough threshold value minfs; m ? s + 1g, are simplest examples of local gates. T...

Separation of AC0[⊕] Formulas and Circuits

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC0[⊕] basis (unbounded fan-in AND, OR, NOT and MOD2 gates). We show, for all d(n) ≤ O( logn log logn ), that there exist polynomial-size depth-d circuits that are not equivalent to depth-d formulas of size no(d) (moreover, this is optimal in that no(d) cannot be improved to nO(d)). This resul...

Circuit Complexity

Combinational circuits or shortly circuits are a model of the lowest level of computer hardware which is of interest from the point of view of computer science. Circuit complexity has a longer history than complexity theory. Complexity measures like circuit size and depth model sequential time, hardware cost, parallel time, and even storage space. This chapter contains an overview on the resear...