Constant Time Addition

Tom Kelliher, CS 240

Feb. 26, 1999

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Assignment

Read B.4--B.7.

From Last Time

Derivation of carry lookahead adder, carry lookahead unit.

Outline

1. ``Quiz.''

2. Constant time addition.

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Latches and flip-flops, biker light example.

``Quiz''

Five minutes: Complete the 64-bit carry lookahead adder.

Constant Time Addition

The MRR4 Digit Set

Start with the standard base four digit set:

Not binary!

Throw in some ``additional'' digits:

Where:

An MRR4 number:

How do we represent the digits?

Numeric Properties

1. Sign of the leading digit is the sign of the number.

2. Conversion from non-redundant form to redundant form is trivial.

3. There are an infinite number of ways of representing a single value:

   

   Therefore, comparing two numbers is difficult.

4. Converting from redundant to non-redundant is (relatively) hard.

MRR4 Addition

One digit full adder:

Organized like this:

where:

Section of a Larger Adder

For this to work, we must have:

Can we actually accomplish this?

Addition Table:

Observations

• The carry chain is broken.

• Carries propagate at most one digit position.

• This yields a constant time adder.

What about Winograd's bound?

Conclusion

• Ordinary addition is .

• MRR4 addition is O.

• MRR4 skirts Winograd's bound by rejecting the dependency observation.

• MRR4 is quite applicable within the domain of Application Specific Systems.