# A Fast Balanced Ternary Adder/Subtracter

A project log for Ternary Computing Menagerie

A place for documenting the many algorithms, data types, logic diagrams, etc. that would be necessary for the design of a ternary processor.

In this post I will describe what I consider the most important thing I've learned about balanced ternary computing thus far. A fast balanced ternary adder/subtracter.

Thus far, my research has not turned up anybody who has designed a balanced ternary adder other than the ripple-carry variety. This is fine for demonstration or testing of higher level functions, but the lack of a *fast* adder/subtracter was a total deal-breaker in terms of making balanced ternary anything other than an interesting curiosity. A ripple-carry adder has a time complexity of O(n) while a carry-lookahead adder is O(log n).

As near as I can tell, nobody has figured out how to build a balanced ternary carry-lookahead adder, meaning that we were stuck with ripple-carry adders; essentially the stone-age. I spent months of my free time bashing my head against this problem because I knew that the lack of of an O(log n) or better adder unit meant there was no point in doing any other research on the subject. Dr. Jones at the University of Iowa did figure out a carry-lookahead adder for unbalanced ternary (0,1,2) but his efforts to work one out for balanced ternary did not result in a functional design (though his work did clarify the problem quite a bit).

Ultimately, I gave up on carry-lookahead and focused my attention on a lesser-known design, the carry-select adder. A carry-select adder works by computing each possible carry-out simultaneously and passing these “carry candidates” to a multiplexer. The incoming carry from the previous digit “selects” the appropriate multiplexer input to pass along. Because the possible carry-outs from every bit position are calculated and ready before the carry-in from the previous digit arrives, each additional digit only adds a single gate delay due to the multiplexer. This type of adder has O(log n) time complexity, but takes more logic gates to build than a carry-lookahead does. These digits are put into groups with each group taking a carry-in and producing a final carry-out for the group. That group carry-out is passed on to the next group.

In a binary system, the carry-select adder requires two separate sets of logic to calculate the two possible carry candidates and the two possible sum candidates. Add two multiplexers to this and the circuit for each digit ends up being around twice the number of gates as a ripple-carry adder would need. In a ternary system, there are three possible carry candidates and three possible sum candidates, making a carry-select adder around three times size of a balanced ternary ripple-carry adder. Fortunately there are some tricks that can be used to reduce this a bit, but the end result is still pretty big.

My balanced ternary carry-select adder is composed of two types of adder circuits. They are the partial select adder and the full select adder. Each group is composed of one partial select adder in the least significant position and as many full select adders as needed to fill out the group.

As you can see below, the partial select adder takes a carry-in and produces a sum and three carry candidates. No multiplexer is required in the partial select adder. A full select adder takes three carry candidates and produces three sum candidates and three more carry candidates. It passes the sum candidates to a multiplexer. Any full select adder that is not in the most significant position of the group passes its three carry candidates to the next full select adder. If it is in the most significant position, it passes the carry candidates to a multiplexer. All multiplexers in the group are selected by the single carry-in for the group.

The only gates used in these adder units are sum, consensus, increment, decrement, multiplexers, and two new ones, the minus consensus, and plus consensus. The minus consensus can be created by passing the output of a consensus gate through a monadic C gate(-00), and the plus consensus by passing the output through an R gate (00+). There may be simpler ways to make them, but that works.

As a note on gate-delays; I am just working out a rough count of delays. Obviously some gates require the signal pass through more transistors than other gates so the time to propagate a signal is not identical from gate to gate. Those kinds of details and the trade-offs between space complexity vs. time complexity vs. power consumption vs… are better left to the implementation at the silicon layer. For my purposes, I’m just interested in getting a O(log n) adder/subtracter design that works.

If you count it out, the above three-trit carry-select group has six gate-delays from the first set of carry candidates in the partial select adder, to the final carry-select multiplexer at the end. However, the carry-in is already at the carry-select multiplexer and waiting for the group carry-out candidates to propagate down to it. This is the key to why the carry-select adder is faster than O(n).

Following is a 27-trit carry-select adder that takes advantage of that to compute 27-trits with only 11 gate-delays. Each group is calculating its internal sum candidates and carry candidates and getting them ready just in time for the carry from the previous group to arrive and select the correct candidates.

Voila!