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points

by thaumasiotes2 days ago |

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by adrian_b2 days ago|

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There is an essential difference between binary functions and unary functions.

With binary functions you can compose them using a very complex composition graph.

With unary functions you can compose them only linearly, so in general it is impossible to make a binary function with unary functions.

You can make binary functions from unary functions only by using at least one other binary function. For instance, you can make multiplication from squaring, but only with the help of binary addition/subtraction.

So the one function that can be used to generate the others by composition must be at least binary, in order to be able to generate functions with an arbitrary number of parameters.

This is why in mathematics there are many domains where the only required primitives are a small number of binary functions, but there is none where strictly unary functions are sufficient. (However, it may be possible to restrict the binary functions to very simple functions, e.g. making a tuple from components, for instance the CONS function of LISP I.)

by thaumasiotes2 days ago|

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What are you responding to?

by adrian_b2 days ago|

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I think that you may have replied before I saved my entire response, so I am not sure how much of it you had read before replying yourself.

I have replied to your last statement:

> "you can use the second parameter of a binary function to identify a unary function just as you can use the fourth parameter of a quaternary function to identify a trinary one."

As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions).

by thaumasiotes2 days ago|

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> As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions).

I have no idea what you're trying to say. If you can use one parameter to identify a desired function, then obviously you can use a function of arity n+1 to define as many functions of arity n as you want, and it doesn't matter what the value of n is.

For example:

selector(3, "sin") = sin 3

selector(3, "log2") = log₂ 3

This works going from arity 4 to arity 3, and it also works going from arity 2 to arity 1. Your "response" talks about going from arity 1 to arity 2, a non sequitur.

by DoctorOetker1 days ago|

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he is saying that if you reserve the second argument of a binary operator as a "function selection indicator", that you have restricted yourself to an alphabet of unary functions. This means that you could at most effectively model some unary functions, but not arbitrary expression involving +,x,-,/, ^, etc.

Unless you had hit upon a very magical binary function where certain special values of the second parameter happens to coincide with useful unary functions, without those values trampling on a useful binary mode or region of your binary function, but the search space for such a special binary function is so large that you shouldn't demand us to disprove the existence, but rather employ your non-surprisal at the EML result and challenge you to present such a binary function, so we can challenge you to demonstrate how it captures binary functions like addition,products, exponentiation with arbitrary base etc.

So, can we see your construction, or if you refuse to present one, we may conclude you have implicitly reconsidered your position and understand the theoretical elegance this EML (and presumably many other) basis brings?

by adrian_b1 days ago|

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The subject of the parent article is expressing all the "elementary functions".

This requires expressing binary functions, like addition and multiplication.

You cannot do this by using only the set of unary functions, which can indeed be generated by a function with 2 parameters, one of which selects an unary function.