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Third Base

Brian Hayes

Trit by Trit by Trit

This special property of base 3 attracted the notice of early computer designers. On the hypothesis that a computer's component count would be roughly proportional both to the width and to the depth of the numbers being processed, they suggested that rw might be a good predictor of hardware cost, and so ternary notation would make the most efficient use of hardware resources. The earliest published discussion of this idea I've been able to find appears in the 1950 book High-speed Computing Devices, a survey of computer technologies compiled on behalf of the U.S. Navy by the staff of Engineering Research Associates.

Figure 1. Most economical radix for a numbering system . . .Click to Enlarge Image

At about the same time as the ERA survey, Herbert R. J. Grosch proposed a ternary architecture for the Whirlwind computer project at MIT. Whirlwind evolved into the control system for a military radar network, which stood vigil over North American airspace through 30 years of the Cold War. Whirlwind was also the proving ground for several novel computer technologies—including magnetic core memory—but ternary arithmetic was not among the innovations tested; Whirlwind and its successors were binary machines.

As it happens, the first working ternary computer was built on the other side of the Iron Curtain. The machine was designed by Nikolai P. Brusentsov and his colleagues at Moscow State University and was named Setun, for a river that flows near the university campus. Some 50 machines were built between 1958 and 1965. Setun operated on numbers composed of 18 ternary digits, or trits, giving the machine a numerical range of 387,420,489. A binary computer would need 29 bits to reach this capacity; in terms of rw, the ternary design wins 54 to 58.

Unfortunately, Setun did not realize the potential of base 3 to reduce component counts. Each trit was stored in a pair of magnetic cores, wired in tandem so that they had three stable states. A pair of cores could have held two binary bits, which amounts to more information than a single trit, and so the ternary advantage was squandered.

Along with ternary arithmetic, a computer built of base-3 hardware can also exploit ternary logic. Consider the task of comparing two numbers. In a machine based on binary logic, comparison is often a two-stage process. First you ask, "Is x less than y?"; depending on the answer, you may then have to ask a second question, such as "Is x equal to y?" Ternary logic simplifies the process: A single comparison can yield any of three possible outcomes: "less," "equal" and "greater."

Ternary computers were a fad that faded, though not quickly. In the 1960s there were several more projects to build ternary logic gates and memory cells, and to assemble these units into larger components such as adders. In 1973 Gideon Frieder and his colleagues at the State University of New York at Buffalo designed a complete base-3 machine they called ternac, and created a software emulator of it. Since then the idea of ternary computing has had occasional revivals, but you're not going to find a ternary minitower in stock at CompUSA.

Figure 2. Most economical integer radix is almost always 3Click to Enlarge Image

Why did base 3 fail to catch on? One easy guess is that reliable three-state devices just didn't exist or were too hard to develop. And once binary technology became established, the tremendous investment in methods for fabricating binary chips would have overwhelmed any small theoretical advantage of other bases. Furthermore, it's only a hypothesis that such an advantage exists. Everything hinges on the assumption that rw is a proper measure of hardware complexity, or in other words that the incremental cost of increasing the radix is the same as the incremental cost of increasing the number of digits.

But even if ternary circuits don’t find a home in computer hardware, the Goldilocks argument favoring base 3 may apply in other contexts. Suppose you are creating one of those dreadful telephone menu systems—Press 1 to be inconvenienced, Press 2 to be condescended to, and so forth. If there are many choices, what is the best way to organize them? Should you build a deep hierarchy with lots of little menus that each offer just a few options? Or is it better to flatten the structure into a few long menus? In this situation a reasonable goal is to minimize the number of options that the wretched caller must listen to before finally reaching his or her destination. The problem is analogous to that of representing an integer in positional notation: The number of items per menu corresponds to the radix r, and the number of menus is analogous to the width w. The average number of choices to be endured is minimized when there are three items per menu.

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