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

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

The depth hierarchy results for monotone circuits of Raz and McKenzie [5] are extended to the case of monotone circuits of semiunbounded fan-in. It follows that the inclusions NC ⊆ SAC ⊆ AC are proper in the monotone setting, for every i ≥ 1. Mathematics Subject Classification. 68Q17, 68Q15.

Lower Bounds for Monotone Counting Cir uits

where ce ∈ N = {0, 1, 2, . . .}, and xi = 1. Produ ts ∏n i=1 x ei i are monomials of f ; we will often omit monomials whose oe ients ce are zero. The polynomial is multilinear, if ce = 0 for all e 6∈ {0, 1}n, and is homogeneous of degree d, if e1 + · · ·+ en = d for all e with ce 6= 0. A standard model of ompa t representation of su h polynomials (with nonnegative oe ients) is that of monotone ...

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...

Quantum Cir uits: Fanout, Parity, and Counting