1. Fuzzy Reversible Logic using Toffoli-like gates

2. Building reversible fuzzy Toffoli gate
We have to design such gate that knowing P,Q,R we will be able to find unique values of A,B and C.
If we know values of P and Q we can find uniquely the value of X.
We need now to find a fuzzy operator that can find in an uniqe way C from X and R and that will make the whole gate reversible.
Observe that this must be continuous operator, but similar in operation to modulo, XOR or Latin Square. * OP A B C P Q R X
Assume arithmetic addition
What should be the OP?

3. Solution to OP
Several solutions to OP are discussed in R. Rovatti, G. Baccarani, “Fuzzy Reversible Logic”, Proc IEEE International Conference on Fuzzy Systems (FUZZ-IEEE'98) - + * 1 X1 N X2
N(X)
Operations * , - and + are arithmetic X
1-X
Generalized inverter N needs discontinuity

4. a a * + a 1 1
Now we can generate all fuzzy functions of a single variable

5. The same circuit for fuzzy reversible functions of two variables
(c0 b+c1)a +(c2 b+c3) = c0 ba +c1 a +c2 b +c3
c0 b+c1 c0 a * +
c2 b+c3 c2 a b * + * + b c0 c1 c2 c3
Realization of Positive Polarity Reed-Muller for functions of two variables
The same circuit for fuzzy reversible functions of two variables

6. fuzzy reversible lattice for three variables * +
c2 b+c3 c2 a b * + * + b c
c0 c+c1 * + * + * + c c0 c1 c2 c3 c4 c5
Explanation of
fuzzy reversible lattice for three variables * + c

7. Graphical method to calculate decision diagram from Toffoli gates* + A B f g X
Y= C 1 C
 C 00 01 11 10
0 1
AB C
f A’
X=f A’ f A
f B’
f B’  f B
Graphical method to calculate decision diagram from Toffoli gates

8. C 1 + C * + A B f g X
Y= C 1 C
 C
 C f + B * + X
Y= C A * + A

9. Realizations of binary logic with Toffoli and reversible logic with Toffoli-like circuits
Kronecker functional Diagram
Kronecker function-driven Diagram
ESOP
Kronecker Lattice Diagram
PPRM-like forms
other canonical
Reversible Fuzzy Diagrams