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This probably has a very simple answer of some sort, but I'm not a mathematician. For the hyperoperation sequence: $$G(n,a,b)$$ ...there are obvious defined values for positive integer values of $n$ $$G(1,a,b)=a+b$$ $$G(2,a,b)=a \cdot b$$ $$G(3,a,b)=a^b$$

My question is, how to account for non-integer values of $n$, such as: $$G(2.5,a,b)=???$$

MphLee
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marcus erronius
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  • Well... do whatever you like! Why would you expect there to be such a thing in the first place? If I ask you to what is the meaning of "a 3.7-bit string", what would you say? – user21820 Jul 05 '15 at 09:58
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    I would assume a 3.7 bit string would refer to a symbol that takes 3.7 bits in a data stream. I think that makes perfect sense from an information theory standpoint... As to why I would expect it: mathematicians define things for non-integer values all the time, even when it makes no logical sense based on the classic (read: simple) definition. It is only when you get into more complex definitions that using non-integer arguments make sense, and I wondered if that was the case for the hyperoperation sequence. – marcus erronius Jul 05 '15 at 19:52
  • There is no such symbol that takes 3.7 bits in a **real-world** data stream. Either you are abusing terminology or you don't really know information theory. In any case, one has to **define** it and show **why** such a definition is meaningful or useful, so it is meaningless to ask **what the definition is** without having any reason for it. – user21820 Jul 06 '15 at 05:00
  • As for mathematicians defining such stuff, yes of course, but if there is no underlying reasons then it's purely arbitrary and hence you cannot ask them "to account for" their definitions. So the better question is whether people have invented interpolations of hyperoperation and what special properties do different interpolations have, because it is not going to have only one answer. In fact if I remember correctly I had read somewhere that there is no obvious natural extension... That is why I said it is completely up to you. The question you should ask is: **What properties do you want?** – user21820 Jul 06 '15 at 05:06
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    Not sure what you mean by "real-world"... Unless I am totally off base (entirely possible, as my education on the subject is incomplete) in an arithmetically compressed data stream, it is normal to have symbols take up fractional bits in a data stream. Of course, you always end up with an integer number of bits, but a data stream could easily contain ten copies of the same symbol, with each one taking 3.7 bits, and the data stream is 37 bits long. In fact, one very common symbol might easily take up less than one bit in a data stream. – marcus erronius Jul 07 '15 at 06:52
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    As for what properties I want, I was simply asking if there was an interpolation that is accepted as the norm. As I said, I'm not a mathematician, which is why I didn't know the correct terminology, or even how to tag the question properly :-) But I was hoping to learn something useful, and now I have. If you want to write up that there's "no obvious natural extension", I'll accept that as the answer. – marcus erronius Jul 07 '15 at 07:01
  • (On the aside about fractional bits, it's technically incorrect to talk about a symbol from the original stream "taking up space" in the compressed stream. The number you have is clearly not the number of bits in the symbol, nor bits in a string. Simply put, there is no such thing as a 3.7-bit string, but one can say the average bit entropy of a certain symbol is 3.7.) As for your question, you might want to take a look at http://math.stackexchange.com/q/238970. I don't like writing an answer that basically says "there is no natural answer" if I have nothing else to say. – user21820 Jul 07 '15 at 09:27
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    Here two possible duplicates: http://math.stackexchange.com/questions/1227761/example-x-y-and-z-values-for-x-uparrow-alpha-y-z-where-alpha-in-bbb/1241979#_=_ http://math.stackexchange.com/questions/1269643/continuum-between-addition-multiplication-and-exponentiation?lq=1 – MphLee Jan 01 '16 at 09:29

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