Re: [stack] language hierarchy

William Tanksley, Jr <[email protected]> wrote:
> My definition, however, is now *very* specific to syntax. Perhaps I'd
> better phrase it as: "A language's syntax is concatenative if the
> language it accepts is its own Kleene closure." Then, your definition
> rephrased to be parallel to mine would be: "A language's semantics is
> concatenative if every term denotes a function, and every
> concatenation of terms denotes an associative operation on those
> functions." (Hmm, is the word "semantics" plural?)

Okay, I just realized, thanks to
http://en.wikipedia.org/wiki/Kleene_closure, that the Kleene closure
of a language forms an associative monoid. If only I'd read
http://www.latrobe.edu.au/philosophy/phimvt/joy/j02maf.html closely
enough, I would have seen (in the first paragraph) that "The
denotation of Joy programs maps a syntactic monoid of program
concatenation to a semantic monoid of function composition."

How about I condense (using the frigid temperatures of mathematical
language) the two definitions I proposed above into a single one: "A
concatenative language consists of a mapping from a syntactic monoid
over program concatenation to a semantic monoid over an associative
operation on functions."

Now, the question is whether function composition is the only possible
function operation that maintains concatenativity. I would claim it's
not: the trivial counterexample would be the null operation, which
discards both functions. I think 'nip' is also an associative
operation (discard the result of the first function). Is there some
reason why these -- admittedly uninteresting -- operations produce a
non-concatenative language?

-Wm