使用现有门创建 XOR 门

Question

So I'm stuck with a logic gate problem. I'm trying to create an XOR gate by combining existing gates like OR,AND,NOR or NAND.

I have 2 helper functions:

def logic_gate(w1, w2, b):  # weight_x1, weight_x2, and bias
    return lambda x1, x2: sigmoid(w1 * x1 + w2 * x2 + b)

# Helper function to test out our weight functions.
def test(gate):
    for x1, x2 in (0, 0), (0, 1), (1, 0), (1, 1):
        print(f"{x1}, {x2}: {np.round(gate(x1 , x2))}")

I have already defined the other gates:

or_gate = logic_gate(20, 20, -10)
test(or_gate)
output: 
0, 0: 0.0
0, 1: 1.0
1, 0: 1.0
1, 1: 1.0

and_gate = logic_gate(15,15,-20)
test(and_gate)
output:
0, 0: 0.0
0, 1: 0.0
1, 0: 0.0
1, 1: 1.0

nor_gate = logic_gate(-15,-15,10)
test(nor_gate)
output: 
0, 0: 1.0
0, 1: 0.0
1, 0: 0.0
1, 1: 0.0

nand_gate = logic_gate(-15,-15,20)
test(nand_gate)
output:
0, 0: 1.0
0, 1: 1.0
1, 0: 1.0
1, 1: 0.0

So the goal is to create 2 XOR functions.

1. (not a and b) or (a and not b) (¬∧)∨(∧¬) or a somewhat simplified version

2. (a or b) and not (a and b) (∨)∧¬(∧)

the functions are like this:

def xor_gate_1(a, b):
    return ...

test(xor_gate_1)



def xor_gate_2(a, b):
    return ...


test(xor_gate_2)

I'm struggling to find out what a and b means as input. Should they be like the weights of the logic_gate functions? How do I use the already created gates in the XOR function? A push into the right direction would be appreciated!

Thanks!

Answer 1

single layer perceptron can't achieve XOR op. therefore need a multi-layer perceptron

eg:

tmp1 = f(x1+x2-0.5), tmp2 = f(x1-x2+1.5)
out = f(tmp1 + tmp2 - 1.5)

where f() is sigmoid