Term "Configuration"

Massimo Dentico m.dentico@virgilio.it
Sat Apr 12 13:08:02 2003

Some  reference  material,  questions and  comments  only  to start  the
discussion. Probably other will follow.

First some basic terminology about predicative languages, excerpted from
an italian  book on  logic [1],  translated and  sligthly adapted (I try
always to be pedagogical for myself and other less math-oriented  people
on this list; my apology to other with a strong mathematical  background
for which this is trivial).

The /words/ of a  language are finite sequences,  obtained concatenating
symbols and  sequences of symbols. The  concatenation operation  will be
indicated simply by juxtaposition of symbols  from left  to right  as in
Indo-European languages.

/Well-formed/ words, called /expressions/, are words in 2 category:

- /terms/,    corresponding to names;
- /formulae/, corresponding to phrases.

The set  of /terms/  is the  smallest set  of words  with the  following
closure properties:

- each words constituted by a single variable is a /term/,
- each words constituted by a single constant is a /term/,
- if F is a functional symbol with n arguments and t1,...,tn  are terms,
  even Ft1...tn is a term, of which t1,...,tn are immediate /subterms/.

Now it is clear that the definition of /term/ is mostly syntactic.

So on  Wed, 9  Apr 2003  16:37:46 -0700  (PDT), when  you (Brian  T Rice
<water@tunes.org>) wrote:

> It occurs to me that the same configuration concept could be used  for
> both  purposes.   We  could   cover  the   collection-like  or  -based
> requirements with such concepts, and this seems to blend well with the
> idea of terms.

are you alluding to  possible interpretations (semantics) of  the /term/
concept? I think to be so.

This remind me about two  possible kind of collection (set)  definitions
(to simplify the exposition here I'm ignoring some details when I equate
collections and sets, such as order, multiple instances, etc.):

- /Intension/: a definition of a Set by mentioning a defining property.

- /Extension/: the definition  of a Set  by enumerating its  members. An
  extensional definition can  always be reduced  to an Intentional  one.

Your examples are apparently of the second kind here:

> Indeed, terms are  often thought of  as collections: Lisp  uses lists,
> people  here  have  mentioned  "tuples"  (with  whatever  meaning they
> intended), and Slate uses objects  with attributes, some of which  are
> different kinds of collections (mostly Sequences).

but I'm quite  sure that the  first is implicit  in your discourse  (for
example:  in  Prolog  a variable  plays  often  the role  of  a  set via
backtracking and unification).

> This  sounds  great,  but  the  idea  needs  to  reach  some  definite
> formality; something we can see implementing. We can talk about a  set
> of objects:  a minimum  amount of  collection information  to say that
> some objects  are part  of a  configuration, just  that "they belong".
> With higher-orderness,  we can  use this  idea to  say that attributes
> also "belong", although this kind  of reflection required for just  an
> argument list is a lot of overhead at runtime which just should exist.
> So   there   should   be  optimized   versions   (subtypes)   of  this
> implementation, specialized for specific cases.

This is not clear  to me: collection-oriented languages  use intensional
(operators and functions) and extensional (some syntax for literals  and
enumeration)  terms  in  expressions   even  if  there  are   some  more
declarative and other more operational. Note that even in this case  you
can  often  establish,  under some  constraints,  some  equivalence; for
example  between  relational  algebra,  which  is,  in  a  sense,   more
operational, and the relational calculus of domains or tuples, which are
more declarative.

Expressions  are  always relationships/constraints  between  objects and
calling  some  of these  expressions  "configurations" seems  arbitrary.
Nothing  wrong  with this,  but  what are  precisely  the distinguishing
features of these configurations wrt other expressions?


[1] Gabriele Lolli, "Introduzione alla logica formale" [Introduction  to
    formal logic], 1991, Il Mulino, Bologna.

[2] and [3] below are from  Eric W. Weisstein, "Concise Encyclopedia  of
    Mathematics CD-ROM", CD-ROM edition 1.0, May 20, 1999

[2] http://mathworld.pdox.net/math/i/i161.htm

[3] http://mathworld.pdox.net/math/e/e422.htm

Massimo Dentico