Realization of Incompletely Specified Reversible Functions

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Realization of Incompletely Specified Reversible Functions. Manjith Kumar Ying Wang Natalie Metzger Bala Iyer Marek Perkowski Portland Quantum Logic Group Portland State University, Oregon RM 2007, Oslo, Norway. Objectives.

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### Realization of Incompletely Specified Reversible Functions

Manjith Kumar

Ying Wang

Natalie Metzger

Bala Iyer

Marek Perkowski

Portland Quantum Logic Group

Portland State University, Oregon

RM 2007, Oslo, Norway

Objectives

To improve the capabilities of the original Miller-Maslov-Dueck algorithm (MMD):

• to synthesize incompletely specified multi-output functions as reversible circuits

2) to synthesize non-reversible Boolean functions by converting them to reversible functions

Don’t Care Algorithm for Reversible Logic (DCARL)

Step 1:

Assign values to the “don't cares” outputs, and map the outputs according to the assigned input values, creating thus a completely specified reversible Boolean function specification.

Step 2:

Apply the MMD algorithm to this specification to synthesize the network.

Step 3:

Compare the cost in terms of the number of Toffoli gates and keep track of the “don't cares” values with the minimal cost.

Backtrack to find K solutions or until no more backtracking is possible.

The Code-How It works.

Pseudo code:

For each of the output with don’t cares (e.g. 10xx10x)

{

for set of all don’t care’s (i.e. “xxx”)

{

assign_values_for_don’t_cares;

}

if (find_conflict)

{

try the next assignment;

}

}

Backtrack_and_reassign_for_N_solutions() ;

Capabilities of the system

1.It can accept any reversible function with don’t cares as input and produce a fully specified reversible output.

2. It can accept any non reversible function and give a reversible output.

3. Functions with fewer outputs than inputs can also be accepted.

4. Capable of running in sync with the standard MMD code.

5. Makes use of cost function in the standard MMD code to compare solutions.

6. The number of solutions needed can be configured.

Limitations of the system
• Huge complexity in worst case input scenarios limits the number of solutions possible.
• Very fast for small number of variables (n). But slows down considerably for n>10.
• Complete backtracking is not feasible for large number of don’t cares. So the code is optimized to give a limited number of solutions.
Function creation – intermediate steps S6 – S10

101 cannot be selected

100 cannot be selected

101 cannot be selected

Function creation – intermediate steps S11 – S15

Backtrack and try counting from last used value

Backtrack and try counting from zero again

000 cannot be selected

001 cannot be selected

Function creation – intermediate steps S21 – S24

100 cannot be selected

101 cannot be selected

000 cannot be selected

Function creation – intermediate steps S25 – S29

001 cannot be selected

010 cannot be selected

Backtrack, start from next value

Backtrack, start from next value

Function creation – intermediate steps S30 – S34

101 cannot be selected

100 cannot be selected

Function creation – intermediate steps S35 – S38

101 cannot be selected

001 can be assigned and table is successfully completed

000 cannot be selected

Original incomplete reversible function versus final complete reversible function after DCARL
Sample Solution:

A circuit with size=17 and Cost=17

This is not the minimum solution!! But a good starting point for minimization.
DCARL Testing
• 6-bit and 9-bit randomly generated functions were used.
• 20%, 40%, 60% , 80% “don’t cares” were included.
• Increased number of “don’t cares” increases the ability of the method to find assignments with low MMD cost
DCARL Results
• DCARL is not designed to find the best solution.
• Does DCARL allow MMD to handle incompletely specified functions?
• YES
Number of output patterns as a function of percentage of “don’t cares” for the 8-bit Gray code benchmark from Maslov WWW Page.
Future Work