I'm terrible at algebra, but I'll do my best. (Extensive quoting follows, because I'll have trouble following this discussion otherwise.) Larry Evans wrote:
The problem involves my attempt to model the lookahead problem as an algebra problem whose solution involved the following steps.
1) For a grammar expression, e_gram, (a "source" algebra expression) perform a homomorphism, gram2X. to a corresponding target algebra, X, expression, e_X, where X is one of {empty,first,follow}.
For example,
gram2empty(inp | epsilon) = not_empty & yes_empty
where inp is any element in the terminals of the grammar and epsilon is, of course, the epsilon expression representing the empty string.
2) Perform a sequence of reductions, based on "laws" of the target algebra, until a "constant" in the target algebra is reached.
For example, some of the "laws" of the empty algebra are:
not_empty & e_empty -reduce-> not_empty e_empty & not_empty -reduce-> not_empty
where e_empty means any expression in the empty algebra. Based on this, and the example expression in 1):
not_empty & yes_empty -reduce-> not_empty
And not_empty is a "constant" in the empty algebra; do, the sequence of reductions is complete.
I get this. Is this part purely compile time, or are some of these reductions dependent on runtime information? If it's purely compile time, this example is fairly close to what you've just described: http://tinyurl.com/253esm (http://boost-sandbox.sourceforge.net/libs/proto/doc/html/boost_proto/users_g...)
Now, IIUC, a homomorphism maps functions with arity, n, in the source to functions with arity n in the target. Thus | in the gram algebra was mapped to & in empty algebra. If n = 0, this means constants in the source are mapped to cconstants in the target. For example, in the above example, inp_0, a constant in gram algebra was mapped to not_empty, a constant in the empty algebra. However, for gram2first, things aren't that simple. The constants in the first algebra are *sets* of constants in the gram algebra. More concretely:
1) homomorphism: gram2first
1.1) constants:
gram2first(inp_0) = {inp_0} gram2first(inp_1) = {inp_1} ... gram2first(inp_n) = {inp_n}
where n is the number of terminal symbols in the grammar.
1.2) binary |:
gram2first(|) = \/ where \/ is set union operator or function.
gram2first(e_gram_0 | e_gram_1) = gram2first(e_gram_0) \/ gram2first(e_gram_0)
However, {inp_0} and {inp_1} and ... {inp_n} are not really a constants in the first grammar. Instead they are really composite terms or expressions in a multisorted set algebra with the single constant, empty_set, and binary function, insert:
insert(empty_set,inp_0) insert(empty_set,inp_1) ... insert(empty_set,inp_n)
and this is what lead me to look for something like the hierarchical algebra, which, I guess, is similar to the "hierarchical abstract type" in (6) here:
The type, P, in the above tinyurl corresponds to the multisorted set algebra mentioned above, and insert(empty_set,inp_0) would be an example of the "primitive term" mentioned in the tinyurl.
Ooof, I followed that link and I think I hurt myself. But I perhaps get the gist, which is that "first" for a bunch of alternates is a set union of the "first" sets of each alternate. "Input" here is what? A character? A string? A pattern? I suppose in the abstract it could be anything. I don't know what a "mustisorted set algebra" is, but it sounds like complication. Seems fairly simple .. you have a transform that builds a set by folding together the "first" sets of the alternates, and you recurse. Terminals yield a "first" set with one element. proto::transform::fold_tree would be handy for this, and you pass around the set you're building as a visitor parameter to the transform.
In gram_lookahead.zip, the morphisms
cannot proceed down the expression tree after a node with arity==0 is reached.
Huh, there is nothing "down" after a node with arity 0. That's all you get. :-)
The above definition of homomorphism mapped constants (functions with arith=0) to constants. However, in the case of gram2first, the mapping was from constants to some term in the primitive subalgebra of the target algebra (I don't even know if that's been defined somewhere, but I'd be very surprised if there were'nt some reference somewhere defining the idea). So instead of mapping constants to constants, in proto I'd be mapping arity-0 terms to arity-0 terms; however, the contents of those target arity-0 terms would be the primitive subalgebra terms; hence, there would be a "down" to get at the subalgebra term (in this case, probably implemented as mpl::set<GrammarTerminalNumeral,TermNum>).
I think now you lost me. I was thinking the "first" set would be a runtime data structure. But I guess I'm wrong. So let's back up. What does the "first" set represent?
However, with the first and follow algebra, the terminal
values are not single values like true or false, but instead multiple values like {ident,value,lpar} (e.g. the first attribute of factor nonterminal in arith_expression grammar).
OK, so your terminal is a 3-tuple. It's still a terminal.
Yes, but also a composite term in the primitive subalgebra. Your probably wondering why I'm so fixated on using the algebra concepts and I'm beginning to wonder myself ;) . However, my original intent was to try and be more formal and possibly gain some insight about how to do it other ways.
I encourage you to formalize what you're doing with proto. If proto obeys some algebra, or can be used to implement a useful one, I think that would be very interesting. I might even want to write a paper about it with you. :-)
So, the reduction of values
in this grammar must continue through the value contained in the proto terminal (which is an instance of mpl::set on grammar terminals, like ident, value, or lpar).
I don't think you need proto to "recurse" into your terminals. I just think you need to write a custom transform that does something appropriate with your terminals.
OK, you're probably right and the attempt at formality is not worth the effort.
See above.
This lead me to search for something in
algebra literature to handle this, and that lead to something called "heirarchical albebras" where the arity-0 expressions are actually values in some other algebra (if I'm interpresting things correctly). This sounds like what I was looking for. I would be interesting if the transforms in proto actually were more like heirarchical algebra morphisms. Just something to think about (probably in the distant future ;) ).
Well, that's one approach. In fact, you can approximate that in proto today by wrapping a proto expression in a template, effectively hiding its proto-ness from proto, which would then treat it like any other terminal. You would be responsible for writing the transforms and/or contexts that knew about these special terminals and recursed into them.
OK, now you've inspired me to continue trying.
Please do.
Thanks for the feedback!
My pleasure. -- Eric Niebler Boost Consulting www.boost-consulting.com