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Comment by tromp | Hacker News Reader

Concatenative languages [1] have the property that every token sequence is a valid program.

For languages using single bits as tokens, every bit sequence is a valid program. One such language is Chris Barker's zot [2].

Inspired by zot, I defined a concatenative version of Binary Lambda Calculus that shares the same property [3].

[1] https://en.wikipedia.org/wiki/Concatenative_programming_lang...

[2] https://en.wikipedia.org/wiki/Iota_and_Jot#Zot

[3] https://cstheory.stackexchange.com/questions/32309/concatena...

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> Concatenative languages [1] have the property that every token sequence is a valid program.

I don't think this is correct? Concatenative languages have the property that if a and b are both valid programs, that the program a || b is valid (where || means "concatenate"). But that property doesn't imply that every sequence of tokens is valid.

For example, in Cat,

  [1 2

is not grammatically valid.

You're right. I should have qualified it as "some concatenative languages".

Is there a particular term for concatenative languages with this property?

Trivially it's all concatenative languages where each single token is valid.

Not sure if they have a name but I would be tempted to call it a free language.

So does Prolog count as a concatenative language given the standard definition of concatenative languages?

For context, a Prolog program is a set of definite clauses so the union of two Prolog programs is a Prolog program.

In fact, this property forms the basis of the concept of (syntactic) generality in Prolog: if P1, P2, P3 are three Prolog programs such that P1 is the union of P2 and P3, then we say that P1 generalises each of P2 and P3. Does this concept of generality also exist for concatenative languages?

They used an OCR program though, which probably has a bias toward attempting to close parentheses.

> making Jot a natural Gödel numbering of all algorithms.

This sounds very cool. I wish I understood both Jot and that sentence.

A Gödel numbering is simply a mapping to integers (that is easily decoded). If your programs are arbitrary binary strings, then you're basically already done, since bitstrings are in 1-1 correspondence with integers:

    empty  0  1  00  01  10  11  000  001  ...
        0  1  2   3   4   5   6    7    8  ...

> A Gödel numbering is simply a mapping to integers (that is easily decoded).

"easily" is arguable. Sure, I could multiply/factor an enormous number and count how many copies of each prime it has, but I'd much rather concatenate/split some digits.

Factoring numbers is difficult. Manipulating digits is easy, or it looks that way, but this is partially an illusion resulting from the number being given to you with the digits already known.

If you have a quantity in mind but you're not sure what its digits are, it can be a lot of work to learn.

I don't know what you mean by "not sure what its digits are".

If you mean what base to represent the number in, that's no more arbitrary than knowing the exact way Gödel numbers use primes.

Maybe it's an unfair example but: the number of people on Earth. We know the quantity more or less, the leftmost digit in base ten, then it's a lot of work to figure out the other digits.

Definitely unfair, because we're discussing this as a way to encode and decode a string of numbers into a single number, no measurements involved.

And factoring the number of people on earth would be just as hard.

You don't need to factor, if you know the nth prime you can easily find what value is at the nth position.

If you're clever about the way you write your syntax tree you can limit the primes to below 13 or so. Don't think there are many useful language constructs that consist of more than 5 or so terms (I think you can limit it to 3 if you encode lists as (list a (list b (list c nil))).

But doesn't Gödel numbering imply some sort of uniqueness, i.e. that each algorithm is only present once?

Otherwise, wouldn't any language assigning a valid meaning to any sequence of 0 and 1 be a Gödel numbering, even if only by saying that "an unrecognised sequence is a noop"?

Yes, in order to be decodable it must be an injective mapping.

No, an algorithm may be written in many ways, each of which would encode to a different Gödel number.

> To encode an entire formula, which is a sequence of symbols, Gödel used the following system. Given a sequence (x_1, x_2, x_3, ..., x_n) of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence

https://en.wikipedia.org/wiki/Gödel_numbering

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