Q: Consider a digital circuit that reads an n-bit input string and writes a 1bit output. Two circuits are considered different if they have a different output for some input string, otherwise they are considered the same (i.e. two circuits are considered the same if they have the same output bit for every possible input string). Count the number of different circuits.
Well, since there are only two outputs possible, I'd say that there are only 2 "different" circuits, regardless of the number of bits read in.
Steve,
will you please elaborate the answer.
huh? I read the problem. It says
two circuits are considered the same if they have the same output bit for every possible input string
There are only two possible output values for a single bit, so ...
To count the number of different circuits, we need to consider the different possible Boolean functions that can be implemented with n inputs and 1 output.
For each input bit, there are two possible values: 0 or 1. Therefore, for n input bits, there are 2^n possible input strings.
For each input string, there is only one corresponding output bit, which can be either 0 or 1.
Hence, there are 2^(2^n) possible different circuits, each representing a unique Boolean function.
To count the number of different circuits, we need to consider the possible combinations of outputs for each possible input string. Let's break down the problem step by step:
1. Determine the number of possible inputs: Since the input is n-bit, each bit can have two possible values (0 or 1). Therefore, the total number of possible inputs is 2^n.
2. Determine the number of possible outputs: Since the output is 1-bit, it can have two possible values as well (0 or 1).
3. Calculate the number of different circuits: For each possible input string, there are two possible output values. Therefore, the total number of different circuits is 2^(2^n).
Let's take an example to make it clearer. Suppose n=2, which means we have a 2-bit input string.
Possible inputs: There are 2^2 = 4 possible input strings: {00, 01, 10, 11}.
Possible outputs: Since the output is 1-bit, there are 2 possible output values: {0, 1}.
Different circuits: For each possible input string, we have 2 possible output values. Therefore, the total number of different circuits is 2^(2^2) = 2^4 = 16.
So, for a 2-bit input string, there are 16 different circuits.
Similarly, you can apply these steps to any value of n to calculate the number of different circuits.