User talk:Lord Farin/Backup/Definition:Formal Language

Definition is weird
Hm, this is not how I learnt the definition of a formal language. Currently this reads:

A formal language is a structure which contains:
 * An alphabet of symbols;
 * A set of words made up of symbols from that alphabet;
 * A formal grammar which determines which words belong to the formal language and which do not.

The third bullet point seems to me extraneous. By my reckoning, a formal language consists of: As I understand it, a grammar is a way to construct a set such as $W$ and thus to define a language; what use is it to include a grammar and a set of words in the definition of a formal language? — Timwi (talk) 22:36, 18 December 2012 (UTC)
 * a set $S$ (the alphabet) and
 * a set of words $W \subseteq \left\{ { \left({ x_1, x_2, ..., x_n }\right) : \forall i, x_i \in S}\right\}$


 * I think this is done from an algorithmic perspective. Both the words and the formal grammar can be more easily realized in a computational context than the arbitrary set $W$ you propose. It seems to explain how in general formal languages are constructed (giving a grammar of WFFs); as such I think the current page is adequate and links well to most sources on the subject (though more general than most texts bother to define things). --Lord_Farin (talk) 22:44, 18 December 2012 (UTC)

Further broadening of perspective
Ben-Ari in his work that I have started covering defines a formula to be a labelled (rooted) tree (with ordering of child nodes). Given the conceptual simplicity that results (a subformula is a subtree) and in general the appeal of graphics representing things (often much more intuitive to process, e.g. commutative diagrams) I don't see a reason to restrict to words in specifying a formal language - it could equally well be a graphical language. Mind you, I'm not going into the mess of Definition:Logical Graph, that's pushing it. It's just a nice generalisation that Ben-Ari provides a source for. It will also come in handy to describe (eventually) proofs as labelled trees satisfying restrictions.

Explicitly, I'd like to (wherever such is appropriate) replace instances of "word" with the more general "expression" (as Definition:Expression isn't anything meaningful atm). If we are to pursue the aim of covering all (published or otherwise verifiably practised) approaches to a concept then I think this step will be inevitable. Thoughts? --Lord_Farin (talk) 11:30, 16 February 2013 (UTC)