How to efficiently reduce this logic sentence?

Question

I have a sentence that describes a circuit, like this:

I x (Q1 x Q0 + not Q1 x not Q0) + not I x (not Q1 x Q0 + Q1 x not Q0)

I have translated it like this:

I and ((Q1 and Q0) or (!Q1 and !Q0)) or !I and ((!Q1 and Q0) or (Q1 and !Q0)) ->
I and ((Q1 and Q0) or !(Q1 or Q0)) or !I and ((!Q1 and Q0) or (Q1 and !Q0)) ->
I and (!(Q1 xor Q0)) or !I and (Q1 xor Q0)

but I get stuck at this point, is there an easy way to make it even more compact or I'll have to solve the bit-by-bit table?

I'm not sure your transformation is correct (I have not check it), but the last formula seems to be equal to `(I xor Q1 xor Q0)` — Stanislav Kralin, Jul 09 '17 at 16:51

score 1 · Accepted Answer · answered Jul 10 '17 at 02:58

Reduction

1. I and ((Q1 and Q0) or (!Q1 and !Q0)) or !I and ((!Q1 and Q0) or (Q1 and !Q0))
   ≡
2. I and ((Q1 and Q0) or !(Q1 or Q0)) or !I and ((!Q1 and Q0) or (Q1 and !Q0))
   ≡
3. I and !(!(Q1 and Q1) and (Q1 or Q0)) or !I and ((!Q1 and Q0) or (Q1 and !Q0))
   ≡
4. I and !(Q0 xor Q1) or !I and (Q0 xor Q1)
   ≡
5. I xor (Q0 xor Q1)
   ≡
6. I xor Q0 xor Q1

Reasons

1 ≡ 2: De Morgan

2 ≡ 3: De Morgan

3 ≡ 4: Def of xor

4 ≡ 5: Def of xor

5 ≡ 6: Associativity of xor

Very helpful! Thank you for pointing out the methods you used too — Mattia Righetti, Jul 10 '17 at 07:35

How to efficiently reduce this logic sentence?

1 Answers1

Reduction

Reasons