[SCL] two comments

[Jay Halcomb]

Gentlemen:

Re (PH):

> We could get over this by being having just a single category of
> 'constant', and let the interpretations distinguish between the
> cases, which is what I had in mind when writing the previous message.
> This would be in line with the basic 'extendable lexicon' intuition.
> This however would be a step back to the CL wild-west syntax, which I
> guess we should try to avoid (?).  Or we could just accept it, and
> remark that unambiguous context-free parsing requires that the basic
> categories can be assigned unambiguously in a context-free way (duh),
> and hence *derive* a conventional FO syntax from this syntactic
> condition.

Without directly addressing disambiguation and the parsing questions, but
with respect to having a single category of 'constants' and to extensible
lexicons -  a lexicon which includes (in some way) both logical operators
and non-logical  constants (such as relations and functions):

Might the following suggest a potentially useful, uniform (and possibly
parametizable) notation for being explicit about defining and rendering
syntactical features of both logical and non-logical constants (in both FOL
and its extensions)? This is the notation for variable-binding operators
(including the standard FOL quantifiers and Boolean operators) presented in
Kalish and Montague, _Techniques_of_Formal_Reasoning, (1st Ed. Kalish and
Montague, Ch. VIII and IX  ; 2nd Ed., Kalish, Montague and Mar, Ch. X and
XI). At least parts of this scheme might be adaptable to an SCL context.

>From Chapter VIII (P. 271-2, 1st Ed.; Ch. X, P.438-9, 2nd Ed.):

"... in symbolic languages... we distinguish two kinds of meaningful
expressions, terms and formulas. Reverting to our informal characterization
of these expressions, we may say that a term is an expression that becomes a
name once its free variables are replaced by names, and a formula is an
expression that becomes a sentence once its free variables are replaced by
names...."

"Variables are the simplest terms. Formulas, together with more complicated
terms, are formed with the aid of constants, which fall into two classes,
formula-makers and term-makers, according to the kind of expression that
they generate. Each constant may be used in combination with a certain
number of variables and previously generated terms and formulas to construct
a new term or formula. To make this procedure precise, we shall associate
with every constant a fixed degree, which will be a quadruple <I, M, N, P>
of nonnegative integers, in which I is either 0 or 1. Here I is 0 or 1
according as the constant in question is a term-maker or a formula-maker,
and M, N, and P are respectively the number of variables, the number of
terms, and the number of formulas that the constant demands. Our terms and
formulas can then be exhaustively characterized as follows.

...

...We wish to include among our constants all the special symbols introduced
in the previous chapters: '~', '=>', 'and',..., '=',... 'forall', 'exists',
'the'. These symbols are called logical constants and are all formula-makers
except for 'the' [which is a term-maker]. ... Constants other than [these]
are called non-logical."


K&M(&M) then proceed to rigorously develop this informal notation, in the
context of (possible) definitional extensions of FOL theories, illustrating
this development with various constants (logical and non-logical) for (and
theories of):

'the'
identity
set membership
convergence of sequences, and limits (in calculus)
roots (in algebra)

and so forth. This style could also be adapted to record (for translation
purposes) syntactical features of modal operators, etc. I don't know if
SCLers are interested in the broader question of treating both logical and
non-logical constants in this way, but, e.g., see also, for example:

<http://suo.ieee.org/email/msg06577.html>

wherein CM discusses variable-binding operators (like 'kappa') as such.

Returning to parsing issues: K&M adopt a parenthesis-free prefix notation
for their offical purposes, but an  associated CFG wouldn't address the kind
of ambiguity Pat is discussing, since K&M require the traditional
non-overlap between predicates and function symbols, including individuals,
Informally, of course, they use an parenthesized infix style which needs all
sorts of further (contextual) disambiguation.

But as we discussed, if the concrete syntax is infix there are still
opportunities for predicate / function ambiguity. Only when one requires
pre-declaration of predicates and functions (and plays interesting games
with lexer to parser tie-ins -- games very familiar to those who've written
parsers for C and C++) can those ambiguities be sidestepped.

The real point is that expunging all context sensitivities, no matter how
manageable they are in practice, is overkill and tends to produce languages
that humans find tedious to write and read. Since recognizer technology has
advanced beyond its early state, admitting some constructs that are not
strictly context-free is actually desirable. Parsers remain tractable, the
tools allow one to construct them readily and the resulting concrete syntax
is more aesthetically palatable to human producers and consumers of that
syntax.


Cheers,
Jay and Randall