<p>Modular counting gates are probably the closest thing in complexity theory to what you're asking about. Modular gates sum their inputs and compare against 0 mod <span class="math-container">$p$</span>. Many authors consider these gates as taking in values in the range <span class="math-container">$[0,p-1]$</span> since you can hook multiple wires between pairs of gates. <a href="https://link.springer.com/article/10.1007/s00037-010-0287-z" rel="nofollow noreferrer">This paper</a> provides a good summary of results in the area up until its publication date (2010) and a result on probabilistically emulating the AND function with just modular gates in constant depth.</p>
<p>As far as it relates to your original question, much more focus is given to composite moduli of two distinct prime factors (e.g. 6) than to prime moduli such as 3, since the computational power of a composite modulus with distinct factors is greater and much less well understood, as the linked paper describes.</p>

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2021-04-26T20:37:22.220Z
2021-04-27T17:48:00.677Z
2021-04-27T17:48:00.677Z
Jake
<p>I had the experience several years ago of working with a team that had developed a ternary computing system.  It ran out of funding and was abandoned but I felt it was ahead of its time.  Currently, what is the state of this (ternary computing) research and development?  Is there a place online that one could suggest to look for more information?</p>

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2021-04-26T19:18:26.257Z
I think that for current models of computation it doesn't matter if the computer is binary, ternary, or $k$-ary for any constant $k$.
Laakeri
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2021-04-26T19:26:48.070Z
@Laakeri True.  But do the models take into account cost and complexity, say, binary versus ternary?
tale852150
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2021-04-26T19:31:26.080Z
It affects the complexity only by a constant factor, and constant factors are very rarely/never analyzed.
Laakeri
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2021-04-26T19:33:41.933Z
[The Linear Speedup Theorem](https://en.wikipedia.org/wiki/Linear_speedup_theorem) (effectively) says that the alphabet size for TMs only matters up to constant factors. This is shown via some symbol carpooling argument. The same basically holds for most other models as well, up to possible caveats about needing to read the whole input. In the concrete example of 2 vs 3 symbol RAM machines, it's clear that a 4-symbol machine can directly simulate a 3-symbol machine, but 2 symbol machine can simulate a 4-symbol machine by considering pairs of symbols at a time. E.g. 00=0, 01=1, 10=2, 11=3.
Yonatan N
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2021-04-26T19:37:37.220Z
There are modular gates, which sum their inputs and compare against 0 mod p. You could consider these as taking in values [0,p-1]. But I'm not sure if that's what you're looking for.
Jake
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2021-04-26T19:50:39.503Z
@Jake is there a paper you can point to that I can review?
tale852150
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2021-04-26T20:26:09.257Z
@tale852150 This paper provides a good summary of results up until its publication date (2010), and has an interesting result on probabilistically emulating the AND function with just modular gates in constant depth: https://link.springer.com/article/10.1007/s00037-010-0287-z
Jake
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2021-04-26T20:27:49.163Z
@Jake please put this as an answer
tale852150
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2021-04-26T20:49:51.973Z
Did you search the web? There are plenty of interesting resources, the wikipedia page to begin with.
Matthieu Latapy
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2021-04-27T11:16:40.580Z
@MatthieuLatapy yes I did.  And therein lies the problem.  I was hoping to reduce the volume by asking for some sage advice from people on this site.  Thank you.
tale852150
<computability>
Where, if any, is there currently any research being done on the subject of ternary computers?
tale852150
# Where, if any, is there currently any research being done on the subject of ternary computers?

(Asked by: username_0 on 2021-04-26)

I had the experience several years ago of working with a team that had developed a ternary computing system. It ran out of funding and was abandoned but I felt it was ahead of its time. Currently, what is the state of this (ternary computing) research and development? Is there a place online that one could suggest to look for more information?

**Tags:** computability
> **Comments:**
> **username_1**: I think that for current models of computation it doesn't matter if the computer is binary, ternary, or $k$-ary for any constant $k$.  
> **username_0**: @Laakeri True.  But do the models take into account cost and complexity, say, binary versus ternary?  
> **username_1**: It affects the complexity only by a constant factor, and constant factors are very rarely/never analyzed.  
> **username_2**: [The Linear Speedup Theorem](https://en.wikipedia.org/wiki/Linear_speedup_theorem) (effectively) says that the alphabet size for TMs only matters up to constant factors. This is shown via some symbol carpooling argument. The same basically holds for most other models as well, up to possible caveats about needing to read the whole input. In the concrete example of 2 vs 3 symbol RAM machines, it's clear that a 4-symbol machine can directly simulate a 3-symbol machine, but 2 symbol machine can simulate a 4-symbol machine by considering pairs of symbols at a time. E.g. 00=0, 01=1, 10=2, 11=3.  
> **username_3**: There are modular gates, which sum their inputs and compare against 0 mod p. You could consider these as taking in values [0,p-1]. But I'm not sure if that's what you're looking for.  
> **username_0**: @Jake is there a paper you can point to that I can review?  
> **username_3**: @tale852150 This paper provides a good summary of results up until its publication date (2010), and has an interesting result on probabilistically emulating the AND function with just modular gates in constant depth: https://link.springer.com/article/10.1007/s00037-010-0287-z  
> **username_0**: @Jake please put this as an answer  
> **username_4**: Did you search the web? There are plenty of interesting resources, the wikipedia page to begin with.  
> **username_0**: @MatthieuLatapy yes I did.  And therein lies the problem.  I was hoping to reduce the volume by asking for some sage advice from people on this site.  Thank you.  

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## 1 answers

### Answer by username_3 on 2021-04-26. Score: 0 | Accepted answer

Modular counting gates are probably the closest thing in complexity theory to what you're asking about. Modular gates sum their inputs and compare against 0 mod $p$. Many authors consider these gates as taking in values in the range $[0,p-1]$ since you can hook multiple wires between pairs of gates. [This paper](https://link.springer.com/article/10.1007/s00037-010-0287-z) provides a good summary of results in the area up until its publication date (2010) and a result on probabilistically emulating the AND function with just modular gates in constant depth.

As far as it relates to your original question, much more focus is given to composite moduli of two distinct prime factors (e.g. 6) than to prime moduli such as 3, since the computational power of a composite modulus with distinct factors is greater and much less well understood, as the linked paper describes.


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