Question

Implement the parity function of N variables using threshold gates

Answer 1

Threshold gates are based on the called majority or threshold decision principle, which means that the output value depends on whether the arithmetic sum of values of its inputs exceeds a threshold. The threshold principle is general itself and conventional simple logic gates, such as AND and OR gates, are special cases of threshold gates. Thus, threshold logic can treat conventional gates as well as threshold gates in general, in a unified manner.

The parity function, which is 1 if and only if an even number of its inputs are logical 1. No logic circuit with polynomial (in n) number of unbounded fan-in AND-OR-NOT gates can compute it in constant-depth.

In Figure A) depth-2 logic network implementing parity for n = 4 is depicted. For an arbitrary n, its size is 2ⁿ – 1 + 1. A depth-2 threshold network for of four variables is shown in Figure B. For an arbitrary n only (n+1) gates are required.

Figure A

Figure B