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Later improvements by Håstad in 1987 [6] established that any family of constant-depth circuits computing the parity function requires exponential size. Extending a result of Razborov, [7] Smolensky in 1987 [8] proved that this is true even if the circuit is augmented with gates computing the sum of its input bits modulo some odd prime p.
Parity only depends on the number of ones and is therefore a symmetric Boolean function.. The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.
In computational complexity theory, CC (Comparator Circuits) is the complexity class containing decision problems which can be solved by comparator circuits of polynomial size. Comparator circuits are sorting networks in which each comparator gate is directed, each wire is initialized with an input variable, its negation, or a constant, and one ...
In theoretical computer science, and specifically computational complexity theory and circuit complexity, TC is a complexity class of decision problems that can be recognized by threshold circuits, which are Boolean circuits with AND, OR, and Majority gates.
If the circuit C outputs A∧B for circuits A and B, join the branching programs that γ-compute A, δ-compute B, γ −1-compute A, and δ −1-compute B for a choice of 5-cycles γ and δ such that their commutator ε=γδγ −1 δ −1 is also a 5-cycle. (The existence of such elements was established in Lemma 2.)
In 1984 Furst, Saxe, and Sipser showed that calculating the parity of the input bits (unlike the aforementioned addition/subtraction problems above which had two inputs) cannot be decided by any AC 0 circuits, even with non-uniformity. [5] [1] It follows that AC 0 is not equal to NC 1, because a family of circuits in the latter class can ...
In most cases a comparator is implemented using a dedicated comparator IC, but op-amps may be used as an alternative. Comparator diagrams and op-amp diagrams use the same symbols. A simple comparator circuit made using an op-amp without feedback simply heavily amplifies the voltage difference between Vin and VREF and outputs the result as Vout.
Circuit diagram for threshold detector with hysteresis. The threshold detector with hysteresis consists of an operational amplifier and a series of resistors that provide hysteresis. [1]: 5 [2]: 7 [3] Like other detectors, this device functions as a voltage switch, but with an important difference. The state of the detector output is not ...