[25 January 2010; correct botched formulation, 26 January]
The XPath 1.0 puzzle introduced and discussed in earlier posts continues to occupy some of my thoughts. Consider the document which can be written
The central question is this: given an instance of the XPath 1.0 data model corresponding to this document (or to any equivalent document), how many element nodes are present in the instance?
It turns out that not only are the answers four and two both consistent (as far as I can tell) with the definition of the XPath 1.0 data model, but that other answers are possible as well. Well, another answer.
Consider the document
<!DOCTYPE a [
<!ELEMENT a ANY>
<!ELEMENT b ANY>
<!ENTITY b '<b/>' >
This is not the same serial-form XML document as the one at the top of the page, and the data-model equivalent is not necessarily the same, either, I think. But they both have a document element of type ’
a’, whose ordered list of children has length three, with each child being of type ‘
If we take token as denoting some physical object, marks on paper or some other way of writing a character type (here, pixels on screens, magnetic fields on suitable media, or optical effects on other media), which I am told is its proper meaning as it was introduced by Peirce, then here there appear to me to be three string tokens matching the element production (one ‘a’ and two ‘b’s), not four and not two, and thus three element nodes in the data model, not four and not two.
I’m not really at all sure that the right way to define XML documents (and by extension XML elements) is as strings. But if we do wish to say that an element is a string (of some kind), there seem to be at least three different approaches we can take:
- a sequence of character types (i.e. a string-type)
- a sequence of character tokens (i.e. a string-token)
- an occurrence, within some context, of a sequence of character types
Some of the issues relating to these concepts are very well laid out in the article “Types and Tokens” written by Linda Wetzel for the Stanford Encyclopedia of Philosophy. It is interesting to know that SGML and XML accomplish simply, through entity references, what some philosophers have regarded as impossible: we have more occurrences of a thing than we have tokens for it. In the second document given above, it seems clear (at least to me, but I am no philosopher) that there are two string-tokens of type “
<b/>” — one in the entity declaration and one in the document content — and three occurrences of that string-type in the document body. When entity expansion is performed by a machine, we are quite likely to be able to identify different tokens with the different occurrences of the string-type (as long as we are willing to entertain time-slices as part of our definition of string-token), but when I just look at the document and count the occurrences of the string-type, I don’t create new tokens in the physical world (unless you want to include mental acts as tokens, but I am pretty sure that that would be a bad idea).
The good news is that there is plenty of philosophical light to throw on these issues, and the issues are well understood by philosophers. Also good news is that the XPath 1.0 data model appears to be consistent with all three possible views.
The bad news, I think, is that while the philosophers understand the issues quite well, they don’t agree on what to do about them. Also bad news is that the XPath 1.0 spec is consistent with all three views, even though it clearly wants to have a single answer to questions like “How many elements are in this document?”. Also, it seems clear that the XPath 1.0 spec really wants to view elements as occurrences (or, at least, the occurrence story produces the results the XPath 1.0 spec relies on its data model producing), but the notion of occurrence appears to be regarded by philosophers as easily the most problematic of the three concepts competing for our patronage.
Fortunately, to make the XPath 1.0 data model do what its creators wanted it to do, all that is necessary is to say explicitly that no node occurs more than once (there’s that word again!) in its parent’s ordered list of children. Or, equivalently, that the cardinality of the set of child nodes is the same as the length of the ordered list of children.
[In full XPath terms, we may also think of it as forbidding any node to appear among the result of evaluating the expression “
preceding-sibling::* | following-sibling::*” with that node as the context item. But the data model does not provide a primitive definition of sibling-hood, so it can’t conveniently be formulated that way in the data model.]