What examples do you have in mind? I see your point as being more obvious for full-text (enhanced by XML) than for pure XML use cases.

Preserving order is essential when a database is used to store XML documents.

Interestingly, columnar database architectures laugh at the notion of order. And just about every row-based analytic DBMS vendor has a columnar option coming. Greenplum’s is here; Oracle’s is known to me and not under NDA; others are surely coming too.

We seem to have different requirements for properties of n-tuples.

Tuples were introduced in set theory to support the concept of a function, f(a)=b. This required a notation (with a proper underlying membership condition) that distinguished (a,b) from (b,a) when a was not equal to b. This was expanded to a general definition of n-tuple where (a(1),..,a(n)) = (b(1),..,b(n)) iff a(i)=b(i) for all i. A requisite property for n-tuples is “order”. Your definition does not seem to support this property.

For your database interests your definition may be more than adequate. For my set theoretic interest in support of functional behavior, it does not work.

For a better understanding of my set theoretic needs for tuples please read the first and last pages of Axioms, and as much of others as you have time for.

Axioms: http://xsp.xegesis.org/X_axioms.pdf
Tuples: http://xsp.xegesis.org/xst_etc5a.pdf
Klasses: http://xsp.xegesis.org/N_isntk.pdf
Records(p. 6): http://xsp.xegesis.org/Spio.pdf
Tables(p. 5): http://xsp.xegesis.org/SSDAA.pdf

1) What base collection of axioms are you using: ZFC, Quine, NBG, other?

Anything compatible with elementary-school textbooks.

2) What is the set-membership of your “set of ordered pairs”: Kuratowski, Wiener, Quine, other?

Ditto

3) How do your “pairs” and “tuples” behave under intersection?

4) Using (a,b) and (x,y,z) for pairs and tuples what is the cardinality of the set { (a,b,c), ((a,b),c), (a,(b,c)), ((a,1), (b,2), (c,3)} }? In ZFC the answer is 3. In XST the answer is 4.

That’s four different sets.

5) How do you define the McCarthy rho operator that plucks the i-th element from a n-tuple? Ex. rho(4) of (a,b,c,d,e) equals d.

As per the above, I don’t define tuples to be ordered. There’s might be an “AddressLineOne” element, but there isn’t a “third” element.

By the way, my definitions may not work as well unless we say all the sets discussed are finite. I was taking that as implicit.

6) How do you define “tables” as sets? Tables (Codd uses “arrays”) are not a formal constituent of the RDM (see http://xsp.xegesis.org/Relations_70.pdf ). Tables are “expository” representations to conveniently represent “relationships”, which in turn are equivalence classes of relations. The RDM uses set theory in spirit but not in fact. Tables are not sets!

Again as per the above, a table is a set of tuples, subject to a constraint about the set of first-values of the ordered pairs that make up each tuple.

Perhaps I should define that set explicitly as the “implied-table-schema” of the tuple of something.

The foundations of the RDM suffer all the failings imbued by the Kuratowski definition of ordered pair. Thus any definition of tuple that shares the same formal support as the RDM will not work for my purposes.

I don’t see why, so long as you don’t go down the path that, for example, an ordered triple is an ordered pair of a singleton and an ordered pair.

If your definitions work for your purposes, then that is all that matters. In order to answer if they will work for my purposes, I will need answers to the following questions:

1) What base collection of axioms are you using: ZFC, Quine, NBG, other?

2) What is the set-membership of your “set of ordered pairs”: Kuratowski, Wiener, Quine, other?

3) How do your “pairs” and “tuples” behave under intersection?

4) Using (a,b) and (x,y,z) for pairs and tuples what is the cardinality of the set { (a,b,c), ((a,b),c), (a,(b,c)), ((a,1), (b,2), (c,3)} }? In ZFC the answer is 3. In XST the answer is 4.

5) How do you define the McCarthy rho operator that plucks the i-th element from a n-tuple? Ex. rho(4) of (a,b,c,d,e) equals d.

6) How do you define “tables” as sets? Tables (Codd uses “arrays”) are not a formal constituent of the RDM (see http://xsp.xegesis.org/Relations_70.pdf ). Tables are “expository” representations to conveniently represent “relationships”, which in turn are equivalence classes of relations. The RDM uses set theory in spirit but not in fact. Tables are not sets!

The foundations of the RDM suffer all the failings imbued by the Kuratowski definition of ordered pair. Thus any definition of tuple that shares the same formal support as the RDM will not work for my purposes.

A more comprehensive analysis of tuples, tables, records and sets is presented in the following:
Tuples: http://xsp.xegesis.org/xst_etc5a.pdf
Klasses: http://xsp.xegesis.org/N_isntk.pdf
Records(p. 6): http://xsp.xegesis.org/Spio.pdf
Tables(p. 5): http://xsp.xegesis.org/SSDAA.pdf

And now back to my original post.

If we define a tuple as a set of ordered pairs, a relational tuple as a set of ordered pairs in which each of the first elements is unique, and a relational table as set of relational tuples that each have the same set of first-elements — doesn’t that get the job done?

I did keep things simple by only defining an equi-join, but in another line or two I could have covered the general case of joins just as well.

The tap dance to go from the true Cartesian product (ordered pair of sets) to the pseudo-Cartesian product (union of the two sets in the ordered pair) wasn’t bad at all.

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Hmm. I’m seeing one little thing I didn’t say, which is that two different tables have to have disjoint sets of field-names. But that can be assumed true as a tautology.

I don’t understand your last posting, but on rereading you questions and applying them to the middle level component of a three tiered architecture (APP-PRO-STO each component separated by a data independence that allows communication only by set-membership exchange), then the answers to your questions are:

1. Which primitives?
No predetermine universal collection, but any collection with XST definitions will work. The only architectural concern is what allowable operations will support the best data representations and access performance given a chosen hardware configuration.

2. How agnostic?
Since all data is stored and accessed as a set, the only belief required is in the consistency of XST axioms.

In an ideal world, computer systems would be designed to give great performance for any or all of the applications they are logically capable of running, with no user optimization needed, whether at purchase time, installation time, or later.

Reality, as you know, is rather far from that ideal.

CAM

I’m not sure I understand your questions which seem to imply that an application has the ability to dictate performance directives to the underlying system data provider. This only makes sense if the system architecture subverts data independence between user environment and processing environment. As is the case when allowing index structures to be defined at the application level.

In a truly data independent architecture an application can only express desired results and a minimum data covering that supports derivation of these resells – a set specification of the desired result. The capability is latent in SQL and can be easily achieved if execution plans are eliminated at the application level.

The real value of set processing, as I see it, is at the architectural level, not at the programming level. Using set theory to program seems akin to using Dedekind cuts to balance a checkbook.

If by “agnostic as to physical primitives” you mean a belief in a Maxwell Demon that will only let informational dense data pass through the I/O interface, then, yes, any precise expression of a desired result (like an SQL statement or a SQUARE expression) can be executed as optimally as the underlying set processing engine and hardware performance potential allow.

This “answer” may produce more questions then it answers. A current paper in progress (SET-PROCESSING I/O ARCHITECTURES For Application Independent Data Access) expands on the architectural requirements for using set processing to tap the I/O performance potential of any given hardware configuration. Hopefully it will do a better job of reducing questions than in creating them.