8 ms·

Base 3 Computing Beats Binary

162 points by Tomte 2y ago

Rygian 2y ago
> Surprisingly, if you allow a base to be any real number, and not just an integer, then the most efficient computational base is the irrational number e. Now I'm left with an even more interesting question. Why e? The wikipedia page has some further discussion, hinting that the relative efficiency of different bases is a function of the ratio of their natural logarithms.
- jetrink 2y ago
  The "area" that you want to minimize is the number of digits, d, times the base, b. A = d * b d is roughly equal to the log of the number represented, N, base b. d ~= ln(N)/ln(b) Substituting, A ~= b * ln(N) / ln(b) Take the derivative of the area with respect to b and find where the derivative is zero to find the minimum. Using the quotient rule, dA/db = ln(N) * (ln(b)*1 - b/b) / ln(b)^2 0 = ln(N) * (ln(b) - 1) / ln(b)^2 0 = ln(b) - 1 ln(b) = 1 b = e I hope I got that right. Doing math on the internet is always dangerous.
  - klyrs 2y ago
    > Doing math on the internet is always dangerous. Only if you see correctness as a thing to be avoided. In my experience, being Wrong On The Internet is the fastest way to get something proofread.
  - l- 2y ago
    Although the cost function has a multiplication of a base times the floor of the log of the value with respect to that base plus one, area is a misleading analogy to describe the applicability as any geometric dimensional value has to taken with respect to a basis. For a visual, (directional) linear scaling is more in line so to say.
- vitus 2y ago
  A related point is comparing x^y vs y^x, for 1 < x < y. It can be easily shown that the "radix economy" described in the article is identical to this formulation by simply taking the logarithm of both expressions (base 10 as described in the article, but it doesn't really matter, as it's just a scaling factor; this doesn't change the inequality since log x is monotonically increasing for x > 0): y log x vs x log y. Or, if you want to rearrange the terms slightly to group the variables, y / log y vs x / log x. (This doesn't change the direction of the inequality, as when restricted to x > 1, log x is always positive.) If you minimize x / log x for x > 1, then you find that this minimum value (i.e. best value per digit) is achieved at x=e. (Choosing the base = e for calculation purposes: take a derivative and set to zero -- you get (ln x - 1) / (ln x)^2 = 0 => ln x - 1 = 0 => ln x = 1 => x = e.) For some intuition: For small x and y, you have that x^y > y^x (consider, for instance, x=1.1 and y=2 -- 1.1^2 = 1.21, vs 2^1.1 is about 2.14). But when x and y get large enough, you find the exact opposite (3^4 = 81 is larger than 4^3 = 64). You might notice that this gets really close for x=2 and y=3 -- 2^3 = 8, which is just barely smaller than 3^2 = 9. And you get equality in some weird cases (x=2, y=4 -- 2^4 = 4^2 = 16 is the only one that looks nice; if you consider 3, its pairing is roughly 2.47805). It turns out that what really matters is proximity to e (in a weird sense that's related to the Lambert W function). You can try comparing e^x to x^e, or if you want, just graph e^x - x^e and observe that's greater than 0 for x != e. https://www.wolframalpha.com/input?i=min+e%5Ex-x%5Ee https://www.wolframalpha.com/input?i=min+e%5Ex-x%5Ee
taylodl 2y ago
> Why didn’t ternary computing catch on? The primary reason was convention. Even though Soviet scientists were building ternary devices, the rest of the world focused on developing hardware and software based on switching circuits — the foundation of binary computing. Binary was easier to implement. That's not the way I've heard the story. Ternary isn't just on/off, voltage yes/no like binary - you need to know the charge of the voltage: is it positive or negative? Essentially then your circuits are -/0/+ instead of (0/+) like it is for binary. Such ternary circuits resisted miniaturization. At a certain point the - and + circuits cross and arc and create a fire. The binary circuits kept getting miniaturized. The story I've heard that goes along with this is that's how the US ultimately won the space race: the US bet on binary computing and the Soviets bet on ternary computing. The Soviets lost.
- ComplexSystems 2y ago
  This doesn't make sense to me. You don't have to use negative voltages to encode ternary. You can just use three different positive voltages, if you like. 0V = 0, +3V = 1, +6V = 2.
  - kuroguro 2y ago
    That's my understanding as well. Voltages are relative, you are free to choose a "ground" and work with negatives or not if you want.
    - bee_rider 2y ago
      Practically it is convenient I think if your ground is third little round prong on the power cord. I wonder if this is why they suggested a negative voltage. Even though voltages are secretly relative under the hood, it seems like it could simplify things to have two directionally different voltages.
      - IWeldMelons 2y ago
        Many reasons. For example, using negative voltage will reduce DC component in the wires, that will improve reliability over long lines, as now all you need is to sense the polarity of the signal, not the level. You'd also need high power reference voltage (for "1") wire going all over the board, which will be nasty polluted with uncorrelated switching noise, will sag in uncorellated way with respect to the "2" (Vcc wire) etc.
cma 2y ago
I have heard somewhere it is theoretically better because it is much closer to e (2.718... natural logarithm base). Anyone have an explanation that includes that as to why it is better?
- vsuperpower2021 2y ago
  No, it's better because it's closer to pi (3.14...). The reason for this is that pi can be used to measure the area of a circle and circles are the most divine shape.
- ykonstant 2y ago
  Wikipedia has a nice article on this: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration https://en.wikipedia.org/wiki/Non-integer_base_of_numeration Edit: layer8 below has a much better link
  - layer8 2y ago
    This one is more to the point I think: https://en.wikipedia.org/wiki/Optimal_radix_choice https://en.wikipedia.org/wiki/Optimal_radix_choice
    - ykonstant 2y ago
      You are absolutely right, your link is the way to go.
- Qem 2y ago
  http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf http://bit-player.org/wp-content/extras/bph-publications/AmS...
  - cma 2y ago
    That one's good, the cost metric for figure 2 I think is what I had seen.
actinium226 2y ago
> A system using ternary logic can give one of three answers. Because of this, it requires only one query: “Is x less than, equal to, or greater than y?” So what does the answer to that query look like? I get a trit which is -1 for x < y, 0 for x == y, and 1 for x > y? It would make more sense for the query to be "Is x less than or equal to y?" so that I can get back a true/false value and jump to the next instruction as appropriate. This of course raises a lot of questions as to how to program a ternary computer.
- dbdr 2y ago
  The 'if' construct (or equivalently, conditional jump) is inherently tied to binary. On a ternary computer, the natural conditional would have three branches, not two.
  - saulpw 2y ago
    In some ways this seems like it might be very useful. I can easily see an if-statement with true/false/error cases, or less-than/equal-to/greater-than cases.
    - davrosthedalek 2y ago
      What percentage of real-world if cases would benefit from it, in the sense that the three branches are actually different. I would say it's a small number. Yes, there are range checks which could benefit, but more often than not, the two extreme ranges are treated the same, as an error condition. My guess is that most hot conditional jumps are loop returns. That's certainly binary.
danielvaughn 2y ago
> A binary logic system can only answer “yes” or “no.” Maybe I'm missing something, but this sounds like a silly argument for ternary. A ternary system seems like it would be decidedly harder to build a computer on top of. Control flow, bit masking, and a mountain of other useful things are all predicated on boolean logic. At best it would be a waste of an extra bit (or trit), and would also introduce ambiguity and complexity at the lowest levels of the machine, where simplicity is paramount. But again, maybe I'm missing something. I'd be super interested to read about those soviet-era ternary systems the author mentioned.
- titchard 2y ago
  its been a while since I read some of this book and enjoying it, and I remember it refering to 3bit computing in the soviet era but it might be right up your street https://mitpress.mit.edu/9780262534666/how-not-to-network-a-nation/ https://mitpress.mit.edu/9780262534666/how-not-to-network-a-...
- __tmk__ 2y ago
  I would think there wouldn't be much of a difference because the smallest unit you can really work with on modern computers is the byte. And whether you use 8 bits to encode a byte (with 256 possible values) or 5 trits (with 243 possible values), shouldn't really matter?
  - AnotherGoodName 2y ago
    3 fewer lanes for the same computation. FWIW 8bits is the addressable unit. Computers work with 64bits today, they actually mask off computation to work with 8bits. A ternary computer equivalent would have 31trits (the difference is exponential - many more bits only adds a few trits). That means 31 conductors for the signal and 31 adders in the alu rather than 64. The whole cpu could be smaller with everything packed closer together enabling lower power and faster clock rates in general. Of course ternary computers have more states and the voltage differences between highest and lowest has to be higher to allow differentiation and then this causes more leakage which is terrible. But the actual bits vs trits itself really does matter.
alberth 2y ago
I've always wonder if our own human intelligence is limited by the language we speak and the base number system we use. E.g. Tonal languages allows individuals to express way more than Latin based languages. Sumerians used a Base-60 numbering system, and were exceedingly advanced in mathematics. EDIT: Why the downvotes?
- Thomashuet 2y ago
  > Tonal languages allows individuals to express way more than Latin based languages. Do you have any evidence of this? I've never heard this claim before.
- deleted 2y ago
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- vkazanov 2y ago
  Do you speak a tonal language? Vietnamese is one example. Having tones attached to syllables means that words and sentences are shorter. In fact, the grammar is very logical and efficient compared to baroque European ones, especially of the slavic/baltic flavour. But. The same mechanism in Indo-European languages is used for intonation. we express sarcasm, irony, etc this way be essentially using a single sentence tone. I have some good Vietnamese friends explaining how hard it is for them to use (and hear) a sentence-wide tone. So, say, some of the fluently speaking Russian Vietnamese always sound like they are sarcastic. Otoh, I always had problems ordering rice with pork in hanoi...
  - 4sak3n 2y ago
    > Vietnamese is one example. Having tones attached to syllables means that words and sentences are shorter. This does not entail that more can be expressed than other languages. Please see my other reply which goes into (admittedly only slightly) more detail.
    - vkazanov 2y ago
      Yes, sure, "expressability" is something that is hard to quantify. Otoh, there should br some connection between grammar complexity and written culture. It is my hypothesis but, say, culture with a rich novel writing tradition leads to a complication of the language grammar. A 3 page long sentence, anyone? One can see how Middle Egyptian literature made the underlying language more complex.
jerf 2y ago
"In addition to its numerical efficiency, base 3 offers computational advantages. It suggests a way to reduce the number of queries needed to answer questions with more than two possible answers. A binary logic system can only answer “yes” or “no.”" Yes... but nobody uses binary systems. We use 64 bit systems (and a number of other systems, but all larger than one bit), which has abundantly more than enough space to represent "greater than/less than/equal". The main software issue with ternary computing is that with this, the entire advantage goes up in smoke. It is quite hard to articulate an actual advantage a multi-bit system would see since we do not use 1-bit or 1-trit systems in real life. (If you've got some particular small hardware thingy that could use it, by all means go for it; it's no problem to use it in just one little place and have a conventional binary interface.) Taylodl's hardware issue with ternary circuits sounds like a reasonable one as well. If you've already minimized the voltage difference for your binary circuits to as small as it can reasonably be, the addition of ternary hardware necessarily entails doubling it the maximum voltage in the system. Is this Quanta Magazine's "Brash And Stupid Claims About Computing Week" or something? https://news.ycombinator.com/item?id=41155021 https://news.ycombinator.com/item?id=41155021 The last paragraph is outright crockery, that "trit-based security system" is well known to be from a crackpot who appears to simply not comprehend that our binary systems do not in fact explode the moment they have to represent a "3", despite repeated attempts for people to explain it to the guy. Programmers and hardware designers are not in fact just total idiots stuck in the mud about digital (vs analog) and binary (vs ternary) representations. They are the way they are for good and solid engineering reasons that aren't going anywhere, and there is basically no space here for someone to displace these things. It isn't just path dependence, these really are the best choices based on current systems.
- ykonstant 2y ago
  I loved the penultimate paragraph; gave me a hearty laugh and a fun rabbit hole to waste time on :)
- varjag 2y ago
  Ternary is one of several crackpotry schools that were established in USSR. You'd have them write books on the subjects, rant in tech magazines… there even was an SF short story about evil alien robots defeated by ternary. Another big thing was TRIZ: a theory that you can codify invention by making a rulebook and arranging the rules in different permutations. There were plenty of smaller things too, especially in academia. It would typically start with one researcher sticking to some bizarre idea, then growing his own gang of grad students and adjuncts who all feed on that. Except the theory would be borderline batshit and all the publications are in-group, referring each other, and naturally a semester or two worth of this sectarian stuff is getting dropped into the undergrad curriculum.
asimpletune 2y ago
The benefits of ternary computing are best with demonstrated. A simple example: A waiter walks up to a table of two and asks “is everybody ready to order?”. One of them responds, “I’m not sure”. Then the other says, “Now we are”. (Edit: I didn’t really ask a question, so it may not seem like a riddle. To some people, when they imagine it, this scene makes perfect sense, so it’s not much of a riddle to them. To others, it’s sort of bizarre, and so the “question” - how is this possible, or what happened - is obvious. Then you can consider the answer. In any case, this is a very simple demonstration of ternary logic, and much more complicated riddles exist that all more or less build off of the same mechanism.)
- tromp 2y ago
  > One of them responds, “I’m not sure”. In practice this answer is hardly different from saying Yes, since it's understood that the person saying Yes is just speaking for themselves and the waiter will wait for the other person to confirm.
  - gklitz 2y ago
    Isn’t that the entire point, that it means “I’m ready but I don’t know about the other person”? If the first person was not ready they would say “no” because they know that they can’t possibly both be ready. Since the first person says “I’m not sure” and not “no” the second person can infer that the first person is ready, and since the second person is ready they can answer yes for both of them.
  - asimpletune 2y ago
    This point is brought up a lot, but it doesn’t account for all the details. The second person says “now we are”, immediately after the first said “I don’t know”. The “in practice” example explains the first statement, but not both of them.
mrob 2y ago
The concept of radix economy assumes that hardware complexity for each logic element is proportional to the number of logic levels. In practice this isn't true, and base 2 is best. >Ternary circuits: why R=3 is not the Optimal Radix for Computation https://arxiv.org/abs/1908.06841 https://arxiv.org/abs/1908.06841 Previous HN discussion: https://news.ycombinator.com/item?id=38979356 https://news.ycombinator.com/item?id=38979356
- deleted 2y ago
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- crdrost 2y ago
  I like this article but it does kind of seem like it gets to a point of “well we know how to do binary stuff in hardware real well, we don't know how to do ternary stuff that well and doing it with binary components doesn't work great.” Also ternary gets a bit weird in some other ways. The practical ternary systems that the Soviets invented used balanced ternary, digits {–, o, +} so that 25 for example is +o–+, 25 = 27 + 0*9 – 3 + 1. If you think about what is most complicated about addition for humans, it is that you have these carries that combine adjacent numbers: and in the binary system you can prove that you relax to a 50/50 state, the carry bit is 50% likely to be set, and this relaxation happens in average by the 3rd bit or so, I think? Whereas ternary full adders only have the carry trit set ¼ of the time (so ⅛ +, ⅛ –) and it takes a few more trits for it to get there. (One of those nice scattered uses for Markov chains in the back of my head, the relaxation goes as the inverse of the second eigenvalue because the first eigenvalue is 1 and it creates the steady state. I got my first summer research job by knowing that factoid!) So you start to wonder if there's something like speculative execution possible then—half-adder-+ is definitely too simple for this but full adder + chains all these bits together and for larger numbers maybe it's not! Similarly I think that binary proliferated in part because the multiplication story for binary is so simple, it's just a few bitshifts away. But for balanced ternary it's just inversions and tritshifts too, so it has always felt like maybe it has some real “teeth” there.
  - dgoldstein0 2y ago
    In terms of implementing adders, the standard solution for binary logic adders is carry lookahead adders. Perhaps an equivalent could be built in ternary logic? https://en.m.wikipedia.org/wiki/Carry-lookahead_adder https://en.m.wikipedia.org/wiki/Carry-lookahead_adder
phkahler 2y ago
The metric they use for "efficiency" seems rather arbitrary and looks like a theoretical mathematicians toy idea. Unfortunately real computers have to be constructed from real physical devices, which have their own measures of efficiency. These don't match.
- NotAnOtter 2y ago
  "It's more efficient because um... the written expression is shorter" Wait until the hear about base Googol
weinzierl 2y ago
Has there ever been any real hardware that was ternary based?
- EnergyAmy 2y ago
  There were some good attempts at it back in the day: https://en.wikipedia.org/wiki/Setun https://en.wikipedia.org/wiki/Setun
Retr0id 2y ago
*in theory
dragontamer 2y ago
No discussion of tristate logic? Tristate logic is how computer busses have been made for about the last half century. I'd hardly call USB unpopular ...
- xpe 2y ago
  Good point. Claim: Three-state logic can be advantageous in a digital system with a shared bus. The third state, Hi-Z, effectively disconnects a device from the bus. It allows multiple devices to share a single line efficiently. Does the above claim overlook anything fundamental about where tristate logic shines? There are alternative ways to design buses, but my EE-fu is not particularly current (pun intended). If others want to weigh in, please go ahead... e.g. open drain, multiplexing, TDMA, etc.
  - dragontamer 2y ago
    I2C remains open collector today. The issue is that open collector is slow. It's hard to make a bus function above 1MHz and it won't go much beyond that. Sufficient for your typical temperature/ADC/whatever peripheral even today but 1Mhz is not very fast. 10kOhm pull-up will pull the bus back up to 3.3V or whatever your I2C operates at. Let's say each device on the bus adds 15pF of capacitance. At 6 devices, we are at 10kOhm * 6 * 15pF * 2.2 rise time == 2,000,000 ps rise time, aka 2us rise time aka 500kHz or somewhere in that neighborhood. (2.2 * RC constant gives the 80% rise time of a resistor-capacitor system. Capacitances in parallel add together. One 10kOhm pull-up resistor). ------- TriState buffers fix that by allowing a far greater drive of current into the bus. Current is the easiest way to fight back against fan-out * capacitance. Every chip will add parasitic capacitance and it actually adds up to real issues in practice.
- akira2501 2y ago
  The third state there is indeterminate and a read during that period is equally likely to produce a 0 or a 1. Logic /levels/ are what matters, not bus state. You can even see this in how it's implemented. You typically have an OUT line, and an OUT_ENABLE line. This is really just 2 bits.
m3kw9 2y ago
So it’s memory efficient but you’d need a new type of “transistor” to support this
IWeldMelons 2y ago
Iam feeling extremely uncomfortable seeing people in this thread being absolutely unfamiliar wrt basic electronics and basic CS fundamentals. Ternary system has very limited Energy efficiency benefit compared to binary - roughly 1.5 more efficient and a lot more difficult to trasmit over differential lines. Today the latter is a big concern.
- BuyMyBitcoins 2y ago
  I would like to become more familiar with such things, but my CS education was lacking in this regard. It was almost entirely geared towards programming, and none of these things come up in my career. I suspect this is widespread.
joshu 2y ago
except we have no space efficient transistor equivalent. what would that even look like?
aaron695 2y ago
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glii 2y ago
Base-4 (in the form of PAM-4) is already used for high speed data transmission. But IMO given how "analog" digital signals become at high speed, using anything but binary for compute logic seems like a fruitless battle. https://blog.samtec.com/post/understanding-nrz-and-pam4-signaling/ https://blog.samtec.com/post/understanding-nrz-and-pam4-sign...
kortex 2y ago
Binary is a local maximum in terms of entropy vs complexity. Going from one state per element to two states per element is an increase from 0 to 1 bits: an ∞% increase. Going from two to three states increases it to 1.585 bits per element, or a 59% increase. Cheeky math aside, getting signalling logic to work with ternary, especially when clock rates go so high the signal starts to look less like square wave pulses and almost more like a modulated sine (due to slew), you now need much more sophisticated decoders.
- creer 2y ago
  Yes and / but you now have "long distance" cross-chip busses where driving and receiving from that internal network is part of the cost of doing business.
daft_pink 2y ago
Soon scientists will decide that base 10 beats base 3 and those ancient Egyptians were right the whole time.
dangirsh 2y ago
Base e is optimal under a certain metric, and 3 is closest to e. https://en.m.wikipedia.org/wiki/Non-integer_base_of_numeration https://en.m.wikipedia.org/wiki/Non-integer_base_of_numerati...
paulsmith 2y ago
> To see why, consider an important metric that tallies up how much room a system will need to store data. You start with the base of the number system, which is called the radix, and multiply it by the number of digits needed to represent some large number in that radix. For example, the number 100,000 in base 10 requires six digits. Its “radix economy” is therefore 10 × 6 = 60. In base 2, the same number requires 17 digits, so its radix economy is 2 × 17 = 34. And in base 3, it requires 11 digits, so its radix economy is 3 × 11 = 33. For large numbers, base 3 has a lower radix economy than any other integer base. I thought that was interesting so I made (well, Claude 3.5 Sonnet made) a little visualization, plotting the radix efficiency of different bases against a range of numbers: https://paulsmith.github.io/radix-efficiency/radix_effciency.html https://paulsmith.github.io/radix-efficiency/radix_effciency...
- Manabu-eo 2y ago
  Base 4 is surprisingly competitive, but of course never better than base 2. Base 5 is the highest base that could stand at the pareto frontier, but just once and then never more.
shoggouth 2y ago
Thought I would link to the Wikipedia page for ternary computers[0]. [0]: https://en.wikipedia.org/wiki/Ternary_computer https://en.wikipedia.org/wiki/Ternary_computer
throwawaymaths 2y ago
Not in the article but (pulling this out of my ass here) I wonder if ternary logic is ever so slightly more quantum resistant since modeling ternary systems using a "natural" spin-1 (-1, 0, 1) system is way less accessible vs spin-1/2 systems which "naturally" model binary operations
kerkeslager 2y ago
This article reads as something trying to make the case for ternary without knowing anything about it. Being able to store 0-8 numbers in 2 trits instead of 0-7 numbers in 3 bits is not a value added. The comparison (</=/>) is the only real advantage they mentioned, but the way that might provide an advantage is if it's a single trit register within a primarily binary system. I.e. your (binary) comparison instruction (comparing two binary numbers) drops the result in the single-trit register, and then your (binary) jump instructions jump to a (binary) location based on that trit register. Point is, there's no benefit to having the entire system be ternary just for one ternary value. This is a semantic benefit, because the operation really does have 3 possible outcomes, but there are potential energy efficiency downsides to this. A bigger benefit might be representing unknowns or don't-cares--notably numbers in this system are still binary, but some bits are unknown or don't matter. In this case, you can actually make some energy efficiency gains, especially in the context of don't cares, because you can simply jam whatever voltage is most convenient into that "bit"--it doesn't even have to match a voltage range. But I'm not entirely sure that it's accurate to call those systems "ternary".
creer 2y ago
Is you want more insight in what happens these days see also crazy modulation schemes like XPIC (cross-polarization interference cancelling) which is something like polarization-division multiplexing, and countless phase and amplitude transmission modulation schemes (MLT-3, up to 1024-QAM). Yes, "computation" is not done in that medium (except, really, for the complex demodulation process) but still in that field. Everyone is pushing hard to use what physical substrate and ingenuity they have on hand. Binary computing if anything has a massive advantage in the accumulation of technology underlying it. Hard to beat. But still worth considering it now and then as it might be bringing you an advantage in one function or another. Currently within a chip, obviously storage density and transmission density - which doesn't mean you shouldn't keep it in mind for other functions.
alan-crowe 2y ago
A basic exercise in mathematical logic involves looking for minimal sets of connectives. You can base logic on just NAND. But you get NOT by joining two inputs together, so you might refuse to count that and say it is NOT and AND. Another possibility for binary logic is implication, =>, and NOT. Because (NOT A) => B is equivalent to A OR B and obviously (OR and NOT) work just as well as (AND and NOT). What though are minimal sets for ternary? The familiar 74181 has control inputs to let the circuit summon any of the 16 operations on two bits. (A two bit operation has four possible inputs, and two possible outputs for each input so 2 ^ 4 = 16 possible operations). A similar ALU part for ternary would have two trits as input so nine possible inputs. Each would have three possible values, so 3 ^ 9 = 19683. It would need nine control inputs, each a trit. But 19683 is so much more than 16. How many gates do you need to be able to combine them to produce all possible (trit,trit)->trit gates? Here are three gates that get you a long way mathematically. First a swap gate -1 -> 0 0 -> -1 1 -> 1 Then a permute gate -1 -> 0 0 -> 1 1 -> -1 This lets your generate all permutations Then you need a "spot" gate that spots one input, say (0 0) and output 1, and 0 otherwise. Since you have all permutations you have all spot gates. And inverse spot gates. Does that solve the problem? It is late and I'm tired. And I can see that the mathematically minimal set is impractical. https://commons.wikimedia.org/wiki/File:Balanced_ternary_operation_tables.svg https://commons.wikimedia.org/wiki/File:Balanced_ternary_ope... has six operations. There are three for doing arithmetic, the units from multiplication, the units from addition, and the carry from addition (it is part of the attraction of balanced ternary that there is no carry out from multiplying digits). There are also three for doing logic, based on -1 is false, +1 is true, and basically degenerating into binary logic. Is that how this is done? The Instruction Set Architecture (ISA) has two kinds of words, numbers in balanced ternary, and bit-rows in degenerate binary. The hardware is built with a pragmatic mixture of binary and ternary logic gates, with the ternary gates heavily used in the Arithmetic Unit, barely at all in the Logic unit, and only opportunistically in the rest of the hardware, that is mostly binary gates as usual? I have a bizarre mad science interest in this because I see how to build AND gates with mechanical linkages, http://alan.sdf-eu.org/linkage-logic/now-with-labels.html http://alan.sdf-eu.org/linkage-logic/now-with-labels.html and wonder if the same technique works for ternary gates. But what are the operation tables for ternary gates? I should attempt the specialized ones for arithmetic first.
NotAnOtter 2y ago
> Why didn’t ternary computing catch on? The primary reason was convention .. No This article is honestly worthless, it doesn't address the actual challenges & strengths of moving to ternary or a different base system. Binary is superior mostly because the components are way more reliable and therefor can be made much much smaller for the same price. Trust me if Nvidia or Snapdragon could make a chip or even a whole prebuild computer working in base 3 that was 5% more efficient for the same price they ABSOLUTLY would and would get rich doing it.
mmooss 2y ago
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DrNosferatu 2y ago
Like DNA? (CTG)

162 points by Tomte 2y ago

> Surprisingly, if you allow a base to be any real number, and not just an integer, then the most efficient computational base is the irrational number e. Now I'm left with an even more interesting question. Why e? The wikipedia page has some further discussion, hinting that the relative efficiency of different bases is a function of the ratio of their natural logarithms.

jetrink 2y ago
The "area" that you want to minimize is the number of digits, d, times the base, b. A = d * b d is roughly equal to the log of the number represented, N, base b. d ~= ln(N)/ln(b) Substituting, A ~= b * ln(N) / ln(b) Take the derivative of the area with respect to b and find where the derivative is zero to find the minimum. Using the quotient rule, dA/db = ln(N) * (ln(b)*1 - b/b) / ln(b)^2 0 = ln(N) * (ln(b) - 1) / ln(b)^2 0 = ln(b) - 1 ln(b) = 1 b = e I hope I got that right. Doing math on the internet is always dangerous.
- klyrs 2y ago
  > Doing math on the internet is always dangerous. Only if you see correctness as a thing to be avoided. In my experience, being Wrong On The Internet is the fastest way to get something proofread.
- l- 2y ago
  Although the cost function has a multiplication of a base times the floor of the log of the value with respect to that base plus one, area is a misleading analogy to describe the applicability as any geometric dimensional value has to taken with respect to a basis. For a visual, (directional) linear scaling is more in line so to say.
vitus 2y ago
A related point is comparing x^y vs y^x, for 1 < x < y. It can be easily shown that the "radix economy" described in the article is identical to this formulation by simply taking the logarithm of both expressions (base 10 as described in the article, but it doesn't really matter, as it's just a scaling factor; this doesn't change the inequality since log x is monotonically increasing for x > 0): y log x vs x log y. Or, if you want to rearrange the terms slightly to group the variables, y / log y vs x / log x. (This doesn't change the direction of the inequality, as when restricted to x > 1, log x is always positive.) If you minimize x / log x for x > 1, then you find that this minimum value (i.e. best value per digit) is achieved at x=e. (Choosing the base = e for calculation purposes: take a derivative and set to zero -- you get (ln x - 1) / (ln x)^2 = 0 => ln x - 1 = 0 => ln x = 1 => x = e.) For some intuition: For small x and y, you have that x^y > y^x (consider, for instance, x=1.1 and y=2 -- 1.1^2 = 1.21, vs 2^1.1 is about 2.14). But when x and y get large enough, you find the exact opposite (3^4 = 81 is larger than 4^3 = 64). You might notice that this gets really close for x=2 and y=3 -- 2^3 = 8, which is just barely smaller than 3^2 = 9. And you get equality in some weird cases (x=2, y=4 -- 2^4 = 4^2 = 16 is the only one that looks nice; if you consider 3, its pairing is roughly 2.47805). It turns out that what really matters is proximity to e (in a weird sense that's related to the Lambert W function). You can try comparing e^x to x^e, or if you want, just graph e^x - x^e and observe that's greater than 0 for x != e. https://www.wolframalpha.com/input?i=min+e%5Ex-x%5Ee https://www.wolframalpha.com/input?i=min+e%5Ex-x%5Ee

taylodl 2y ago

> Why didn’t ternary computing catch on? The primary reason was convention. Even though Soviet scientists were building ternary devices, the rest of the world focused on developing hardware and software based on switching circuits — the foundation of binary computing. Binary was easier to implement. That's not the way I've heard the story. Ternary isn't just on/off, voltage yes/no like binary - you need to know the charge of the voltage: is it positive or negative? Essentially then your circuits are -/0/+ instead of (0/+) like it is for binary. Such ternary circuits resisted miniaturization. At a certain point the - and + circuits cross and arc and create a fire. The binary circuits kept getting miniaturized. The story I've heard that goes along with this is that's how the US ultimately won the space race: the US bet on binary computing and the Soviets bet on ternary computing. The Soviets lost.

ComplexSystems 2y ago
This doesn't make sense to me. You don't have to use negative voltages to encode ternary. You can just use three different positive voltages, if you like. 0V = 0, +3V = 1, +6V = 2.
- kuroguro 2y ago
  That's my understanding as well. Voltages are relative, you are free to choose a "ground" and work with negatives or not if you want.
  - bee_rider 2y ago
    Practically it is convenient I think if your ground is third little round prong on the power cord. I wonder if this is why they suggested a negative voltage. Even though voltages are secretly relative under the hood, it seems like it could simplify things to have two directionally different voltages.
    - IWeldMelons 2y ago
      Many reasons. For example, using negative voltage will reduce DC component in the wires, that will improve reliability over long lines, as now all you need is to sense the polarity of the signal, not the level. You'd also need high power reference voltage (for "1") wire going all over the board, which will be nasty polluted with uncorrelated switching noise, will sag in uncorellated way with respect to the "2" (Vcc wire) etc.

cma 2y ago

I have heard somewhere it is theoretically better because it is much closer to e (2.718... natural logarithm base). Anyone have an explanation that includes that as to why it is better?

vsuperpower2021 2y ago
No, it's better because it's closer to pi (3.14...). The reason for this is that pi can be used to measure the area of a circle and circles are the most divine shape.
ykonstant 2y ago
Wikipedia has a nice article on this: https://en.wikipedia.org/wiki/Non-integer_base_of_numeration https://en.wikipedia.org/wiki/Non-integer_base_of_numeration Edit: layer8 below has a much better link
- layer8 2y ago
  This one is more to the point I think: https://en.wikipedia.org/wiki/Optimal_radix_choice https://en.wikipedia.org/wiki/Optimal_radix_choice
  - ykonstant 2y ago
    You are absolutely right, your link is the way to go.
Qem 2y ago
http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf http://bit-player.org/wp-content/extras/bph-publications/AmS...
- cma 2y ago
  That one's good, the cost metric for figure 2 I think is what I had seen.

actinium226 2y ago

> A system using ternary logic can give one of three answers. Because of this, it requires only one query: “Is x less than, equal to, or greater than y?” So what does the answer to that query look like? I get a trit which is -1 for x < y, 0 for x == y, and 1 for x > y? It would make more sense for the query to be "Is x less than or equal to y?" so that I can get back a true/false value and jump to the next instruction as appropriate. This of course raises a lot of questions as to how to program a ternary computer.

dbdr 2y ago
The 'if' construct (or equivalently, conditional jump) is inherently tied to binary. On a ternary computer, the natural conditional would have three branches, not two.
- saulpw 2y ago
  In some ways this seems like it might be very useful. I can easily see an if-statement with true/false/error cases, or less-than/equal-to/greater-than cases.
  - davrosthedalek 2y ago
    What percentage of real-world if cases would benefit from it, in the sense that the three branches are actually different. I would say it's a small number. Yes, there are range checks which could benefit, but more often than not, the two extreme ranges are treated the same, as an error condition. My guess is that most hot conditional jumps are loop returns. That's certainly binary.

danielvaughn 2y ago

> A binary logic system can only answer “yes” or “no.” Maybe I'm missing something, but this sounds like a silly argument for ternary. A ternary system seems like it would be decidedly harder to build a computer on top of. Control flow, bit masking, and a mountain of other useful things are all predicated on boolean logic. At best it would be a waste of an extra bit (or trit), and would also introduce ambiguity and complexity at the lowest levels of the machine, where simplicity is paramount. But again, maybe I'm missing something. I'd be super interested to read about those soviet-era ternary systems the author mentioned.

titchard 2y ago
its been a while since I read some of this book and enjoying it, and I remember it refering to 3bit computing in the soviet era but it might be right up your street https://mitpress.mit.edu/9780262534666/how-not-to-network-a-nation/ https://mitpress.mit.edu/9780262534666/how-not-to-network-a-...
__tmk__ 2y ago
I would think there wouldn't be much of a difference because the smallest unit you can really work with on modern computers is the byte. And whether you use 8 bits to encode a byte (with 256 possible values) or 5 trits (with 243 possible values), shouldn't really matter?
- AnotherGoodName 2y ago
  3 fewer lanes for the same computation. FWIW 8bits is the addressable unit. Computers work with 64bits today, they actually mask off computation to work with 8bits. A ternary computer equivalent would have 31trits (the difference is exponential - many more bits only adds a few trits). That means 31 conductors for the signal and 31 adders in the alu rather than 64. The whole cpu could be smaller with everything packed closer together enabling lower power and faster clock rates in general. Of course ternary computers have more states and the voltage differences between highest and lowest has to be higher to allow differentiation and then this causes more leakage which is terrible. But the actual bits vs trits itself really does matter.

alberth 2y ago

I've always wonder if our own human intelligence is limited by the language we speak and the base number system we use. E.g. Tonal languages allows individuals to express way more than Latin based languages. Sumerians used a Base-60 numbering system, and were exceedingly advanced in mathematics. EDIT: Why the downvotes?

Thomashuet 2y ago
> Tonal languages allows individuals to express way more than Latin based languages. Do you have any evidence of this? I've never heard this claim before.
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vkazanov 2y ago
Do you speak a tonal language? Vietnamese is one example. Having tones attached to syllables means that words and sentences are shorter. In fact, the grammar is very logical and efficient compared to baroque European ones, especially of the slavic/baltic flavour. But. The same mechanism in Indo-European languages is used for intonation. we express sarcasm, irony, etc this way be essentially using a single sentence tone. I have some good Vietnamese friends explaining how hard it is for them to use (and hear) a sentence-wide tone. So, say, some of the fluently speaking Russian Vietnamese always sound like they are sarcastic. Otoh, I always had problems ordering rice with pork in hanoi...
- 4sak3n 2y ago
  > Vietnamese is one example. Having tones attached to syllables means that words and sentences are shorter. This does not entail that more can be expressed than other languages. Please see my other reply which goes into (admittedly only slightly) more detail.
  - vkazanov 2y ago
    Yes, sure, "expressability" is something that is hard to quantify. Otoh, there should br some connection between grammar complexity and written culture. It is my hypothesis but, say, culture with a rich novel writing tradition leads to a complication of the language grammar. A 3 page long sentence, anyone? One can see how Middle Egyptian literature made the underlying language more complex.

jerf 2y ago

"In addition to its numerical efficiency, base 3 offers computational advantages. It suggests a way to reduce the number of queries needed to answer questions with more than two possible answers. A binary logic system can only answer “yes” or “no.”" Yes... but nobody uses binary systems. We use 64 bit systems (and a number of other systems, but all larger than one bit), which has abundantly more than enough space to represent "greater than/less than/equal". The main software issue with ternary computing is that with this, the entire advantage goes up in smoke. It is quite hard to articulate an actual advantage a multi-bit system would see since we do not use 1-bit or 1-trit systems in real life. (If you've got some particular small hardware thingy that could use it, by all means go for it; it's no problem to use it in just one little place and have a conventional binary interface.) Taylodl's hardware issue with ternary circuits sounds like a reasonable one as well. If you've already minimized the voltage difference for your binary circuits to as small as it can reasonably be, the addition of ternary hardware necessarily entails doubling it the maximum voltage in the system. Is this Quanta Magazine's "Brash And Stupid Claims About Computing Week" or something? https://news.ycombinator.com/item?id=41155021 https://news.ycombinator.com/item?id=41155021 The last paragraph is outright crockery, that "trit-based security system" is well known to be from a crackpot who appears to simply not comprehend that our binary systems do not in fact explode the moment they have to represent a "3", despite repeated attempts for people to explain it to the guy. Programmers and hardware designers are not in fact just total idiots stuck in the mud about digital (vs analog) and binary (vs ternary) representations. They are the way they are for good and solid engineering reasons that aren't going anywhere, and there is basically no space here for someone to displace these things. It isn't just path dependence, these really are the best choices based on current systems.

ykonstant 2y ago
I loved the penultimate paragraph; gave me a hearty laugh and a fun rabbit hole to waste time on :)
varjag 2y ago
Ternary is one of several crackpotry schools that were established in USSR. You'd have them write books on the subjects, rant in tech magazines… there even was an SF short story about evil alien robots defeated by ternary. Another big thing was TRIZ: a theory that you can codify invention by making a rulebook and arranging the rules in different permutations. There were plenty of smaller things too, especially in academia. It would typically start with one researcher sticking to some bizarre idea, then growing his own gang of grad students and adjuncts who all feed on that. Except the theory would be borderline batshit and all the publications are in-group, referring each other, and naturally a semester or two worth of this sectarian stuff is getting dropped into the undergrad curriculum.

asimpletune 2y ago

The benefits of ternary computing are best with demonstrated. A simple example: A waiter walks up to a table of two and asks “is everybody ready to order?”. One of them responds, “I’m not sure”. Then the other says, “Now we are”. (Edit: I didn’t really ask a question, so it may not seem like a riddle. To some people, when they imagine it, this scene makes perfect sense, so it’s not much of a riddle to them. To others, it’s sort of bizarre, and so the “question” - how is this possible, or what happened - is obvious. Then you can consider the answer. In any case, this is a very simple demonstration of ternary logic, and much more complicated riddles exist that all more or less build off of the same mechanism.)

tromp 2y ago
> One of them responds, “I’m not sure”. In practice this answer is hardly different from saying Yes, since it's understood that the person saying Yes is just speaking for themselves and the waiter will wait for the other person to confirm.
- gklitz 2y ago
  Isn’t that the entire point, that it means “I’m ready but I don’t know about the other person”? If the first person was not ready they would say “no” because they know that they can’t possibly both be ready. Since the first person says “I’m not sure” and not “no” the second person can infer that the first person is ready, and since the second person is ready they can answer yes for both of them.
- asimpletune 2y ago
  This point is brought up a lot, but it doesn’t account for all the details. The second person says “now we are”, immediately after the first said “I don’t know”. The “in practice” example explains the first statement, but not both of them.

mrob 2y ago

The concept of radix economy assumes that hardware complexity for each logic element is proportional to the number of logic levels. In practice this isn't true, and base 2 is best. >Ternary circuits: why R=3 is not the Optimal Radix for Computation https://arxiv.org/abs/1908.06841 https://arxiv.org/abs/1908.06841 Previous HN discussion: https://news.ycombinator.com/item?id=38979356 https://news.ycombinator.com/item?id=38979356

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crdrost 2y ago
I like this article but it does kind of seem like it gets to a point of “well we know how to do binary stuff in hardware real well, we don't know how to do ternary stuff that well and doing it with binary components doesn't work great.” Also ternary gets a bit weird in some other ways. The practical ternary systems that the Soviets invented used balanced ternary, digits {–, o, +} so that 25 for example is +o–+, 25 = 27 + 0*9 – 3 + 1. If you think about what is most complicated about addition for humans, it is that you have these carries that combine adjacent numbers: and in the binary system you can prove that you relax to a 50/50 state, the carry bit is 50% likely to be set, and this relaxation happens in average by the 3rd bit or so, I think? Whereas ternary full adders only have the carry trit set ¼ of the time (so ⅛ +, ⅛ –) and it takes a few more trits for it to get there. (One of those nice scattered uses for Markov chains in the back of my head, the relaxation goes as the inverse of the second eigenvalue because the first eigenvalue is 1 and it creates the steady state. I got my first summer research job by knowing that factoid!) So you start to wonder if there's something like speculative execution possible then—half-adder-+ is definitely too simple for this but full adder + chains all these bits together and for larger numbers maybe it's not! Similarly I think that binary proliferated in part because the multiplication story for binary is so simple, it's just a few bitshifts away. But for balanced ternary it's just inversions and tritshifts too, so it has always felt like maybe it has some real “teeth” there.
- dgoldstein0 2y ago
  In terms of implementing adders, the standard solution for binary logic adders is carry lookahead adders. Perhaps an equivalent could be built in ternary logic? https://en.m.wikipedia.org/wiki/Carry-lookahead_adder https://en.m.wikipedia.org/wiki/Carry-lookahead_adder

phkahler 2y ago

The metric they use for "efficiency" seems rather arbitrary and looks like a theoretical mathematicians toy idea. Unfortunately real computers have to be constructed from real physical devices, which have their own measures of efficiency. These don't match.

NotAnOtter 2y ago
"It's more efficient because um... the written expression is shorter" Wait until the hear about base Googol

weinzierl 2y ago

Has there ever been any real hardware that was ternary based?

EnergyAmy 2y ago
There were some good attempts at it back in the day: https://en.wikipedia.org/wiki/Setun https://en.wikipedia.org/wiki/Setun

Retr0id 2y ago

*in theory

dragontamer 2y ago

No discussion of tristate logic? Tristate logic is how computer busses have been made for about the last half century. I'd hardly call USB unpopular ...

xpe 2y ago
Good point. Claim: Three-state logic can be advantageous in a digital system with a shared bus. The third state, Hi-Z, effectively disconnects a device from the bus. It allows multiple devices to share a single line efficiently. Does the above claim overlook anything fundamental about where tristate logic shines? There are alternative ways to design buses, but my EE-fu is not particularly current (pun intended). If others want to weigh in, please go ahead... e.g. open drain, multiplexing, TDMA, etc.
- dragontamer 2y ago
  I2C remains open collector today. The issue is that open collector is slow. It's hard to make a bus function above 1MHz and it won't go much beyond that. Sufficient for your typical temperature/ADC/whatever peripheral even today but 1Mhz is not very fast. 10kOhm pull-up will pull the bus back up to 3.3V or whatever your I2C operates at. Let's say each device on the bus adds 15pF of capacitance. At 6 devices, we are at 10kOhm * 6 * 15pF * 2.2 rise time == 2,000,000 ps rise time, aka 2us rise time aka 500kHz or somewhere in that neighborhood. (2.2 * RC constant gives the 80% rise time of a resistor-capacitor system. Capacitances in parallel add together. One 10kOhm pull-up resistor). ------- TriState buffers fix that by allowing a far greater drive of current into the bus. Current is the easiest way to fight back against fan-out * capacitance. Every chip will add parasitic capacitance and it actually adds up to real issues in practice.
akira2501 2y ago
The third state there is indeterminate and a read during that period is equally likely to produce a 0 or a 1. Logic /levels/ are what matters, not bus state. You can even see this in how it's implemented. You typically have an OUT line, and an OUT_ENABLE line. This is really just 2 bits.

m3kw9 2y ago

So it’s memory efficient but you’d need a new type of “transistor” to support this

IWeldMelons 2y ago

Iam feeling extremely uncomfortable seeing people in this thread being absolutely unfamiliar wrt basic electronics and basic CS fundamentals. Ternary system has very limited Energy efficiency benefit compared to binary - roughly 1.5 more efficient and a lot more difficult to trasmit over differential lines. Today the latter is a big concern.

BuyMyBitcoins 2y ago
I would like to become more familiar with such things, but my CS education was lacking in this regard. It was almost entirely geared towards programming, and none of these things come up in my career. I suspect this is widespread.

joshu 2y ago

except we have no space efficient transistor equivalent. what would that even look like?

aaron695 2y ago

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glii 2y ago

Base-4 (in the form of PAM-4) is already used for high speed data transmission. But IMO given how "analog" digital signals become at high speed, using anything but binary for compute logic seems like a fruitless battle. https://blog.samtec.com/post/understanding-nrz-and-pam4-signaling/ https://blog.samtec.com/post/understanding-nrz-and-pam4-sign...

kortex 2y ago

Binary is a local maximum in terms of entropy vs complexity. Going from one state per element to two states per element is an increase from 0 to 1 bits: an ∞% increase. Going from two to three states increases it to 1.585 bits per element, or a 59% increase. Cheeky math aside, getting signalling logic to work with ternary, especially when clock rates go so high the signal starts to look less like square wave pulses and almost more like a modulated sine (due to slew), you now need much more sophisticated decoders.

creer 2y ago
Yes and / but you now have "long distance" cross-chip busses where driving and receiving from that internal network is part of the cost of doing business.

daft_pink 2y ago

Soon scientists will decide that base 10 beats base 3 and those ancient Egyptians were right the whole time.

dangirsh 2y ago

Base e is optimal under a certain metric, and 3 is closest to e. https://en.m.wikipedia.org/wiki/Non-integer_base_of_numeration https://en.m.wikipedia.org/wiki/Non-integer_base_of_numerati...

paulsmith 2y ago

> To see why, consider an important metric that tallies up how much room a system will need to store data. You start with the base of the number system, which is called the radix, and multiply it by the number of digits needed to represent some large number in that radix. For example, the number 100,000 in base 10 requires six digits. Its “radix economy” is therefore 10 × 6 = 60. In base 2, the same number requires 17 digits, so its radix economy is 2 × 17 = 34. And in base 3, it requires 11 digits, so its radix economy is 3 × 11 = 33. For large numbers, base 3 has a lower radix economy than any other integer base. I thought that was interesting so I made (well, Claude 3.5 Sonnet made) a little visualization, plotting the radix efficiency of different bases against a range of numbers: https://paulsmith.github.io/radix-efficiency/radix_effciency.html https://paulsmith.github.io/radix-efficiency/radix_effciency...

Manabu-eo 2y ago
Base 4 is surprisingly competitive, but of course never better than base 2. Base 5 is the highest base that could stand at the pareto frontier, but just once and then never more.

shoggouth 2y ago

Thought I would link to the Wikipedia page for ternary computers[0]. [0]: https://en.wikipedia.org/wiki/Ternary_computer https://en.wikipedia.org/wiki/Ternary_computer

throwawaymaths 2y ago

Not in the article but (pulling this out of my ass here) I wonder if ternary logic is ever so slightly more quantum resistant since modeling ternary systems using a "natural" spin-1 (-1, 0, 1) system is way less accessible vs spin-1/2 systems which "naturally" model binary operations

kerkeslager 2y ago

This article reads as something trying to make the case for ternary without knowing anything about it. Being able to store 0-8 numbers in 2 trits instead of 0-7 numbers in 3 bits is not a value added. The comparison (</=/>) is the only real advantage they mentioned, but the way that might provide an advantage is if it's a single trit register within a primarily binary system. I.e. your (binary) comparison instruction (comparing two binary numbers) drops the result in the single-trit register, and then your (binary) jump instructions jump to a (binary) location based on that trit register. Point is, there's no benefit to having the entire system be ternary just for one ternary value. This is a semantic benefit, because the operation really does have 3 possible outcomes, but there are potential energy efficiency downsides to this. A bigger benefit might be representing unknowns or don't-cares--notably numbers in this system are still binary, but some bits are unknown or don't matter. In this case, you can actually make some energy efficiency gains, especially in the context of don't cares, because you can simply jam whatever voltage is most convenient into that "bit"--it doesn't even have to match a voltage range. But I'm not entirely sure that it's accurate to call those systems "ternary".

creer 2y ago

Is you want more insight in what happens these days see also crazy modulation schemes like XPIC (cross-polarization interference cancelling) which is something like polarization-division multiplexing, and countless phase and amplitude transmission modulation schemes (MLT-3, up to 1024-QAM). Yes, "computation" is not done in that medium (except, really, for the complex demodulation process) but still in that field. Everyone is pushing hard to use what physical substrate and ingenuity they have on hand. Binary computing if anything has a massive advantage in the accumulation of technology underlying it. Hard to beat. But still worth considering it now and then as it might be bringing you an advantage in one function or another. Currently within a chip, obviously storage density and transmission density - which doesn't mean you shouldn't keep it in mind for other functions.

alan-crowe 2y ago

A basic exercise in mathematical logic involves looking for minimal sets of connectives. You can base logic on just NAND. But you get NOT by joining two inputs together, so you might refuse to count that and say it is NOT and AND. Another possibility for binary logic is implication, =>, and NOT. Because (NOT A) => B is equivalent to A OR B and obviously (OR and NOT) work just as well as (AND and NOT). What though are minimal sets for ternary? The familiar 74181 has control inputs to let the circuit summon any of the 16 operations on two bits. (A two bit operation has four possible inputs, and two possible outputs for each input so 2 ^ 4 = 16 possible operations). A similar ALU part for ternary would have two trits as input so nine possible inputs. Each would have three possible values, so 3 ^ 9 = 19683. It would need nine control inputs, each a trit. But 19683 is so much more than 16. How many gates do you need to be able to combine them to produce all possible (trit,trit)->trit gates? Here are three gates that get you a long way mathematically. First a swap gate -1 -> 0 0 -> -1 1 -> 1 Then a permute gate -1 -> 0 0 -> 1 1 -> -1 This lets your generate all permutations Then you need a "spot" gate that spots one input, say (0 0) and output 1, and 0 otherwise. Since you have all permutations you have all spot gates. And inverse spot gates. Does that solve the problem? It is late and I'm tired. And I can see that the mathematically minimal set is impractical. https://commons.wikimedia.org/wiki/File:Balanced_ternary_operation_tables.svg https://commons.wikimedia.org/wiki/File:Balanced_ternary_ope... has six operations. There are three for doing arithmetic, the units from multiplication, the units from addition, and the carry from addition (it is part of the attraction of balanced ternary that there is no carry out from multiplying digits). There are also three for doing logic, based on -1 is false, +1 is true, and basically degenerating into binary logic. Is that how this is done? The Instruction Set Architecture (ISA) has two kinds of words, numbers in balanced ternary, and bit-rows in degenerate binary. The hardware is built with a pragmatic mixture of binary and ternary logic gates, with the ternary gates heavily used in the Arithmetic Unit, barely at all in the Logic unit, and only opportunistically in the rest of the hardware, that is mostly binary gates as usual? I have a bizarre mad science interest in this because I see how to build AND gates with mechanical linkages, http://alan.sdf-eu.org/linkage-logic/now-with-labels.html http://alan.sdf-eu.org/linkage-logic/now-with-labels.html and wonder if the same technique works for ternary gates. But what are the operation tables for ternary gates? I should attempt the specialized ones for arithmetic first.

NotAnOtter 2y ago

> Why didn’t ternary computing catch on? The primary reason was convention .. No This article is honestly worthless, it doesn't address the actual challenges & strengths of moving to ternary or a different base system. Binary is superior mostly because the components are way more reliable and therefor can be made much much smaller for the same price. Trust me if Nvidia or Snapdragon could make a chip or even a whole prebuild computer working in base 3 that was 5% more efficient for the same price they ABSOLUTLY would and would get rich doing it.

mmooss 2y ago

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DrNosferatu 2y ago

Like DNA? (CTG)