From: Arjohn Kampman <arjohn.kampman@aduna-software.com>

Date: Tue, 25 Mar 2008 16:20:20 +0100

Message-ID: <47E91834.2060603@aduna-software.com>

To: Andrew Newman <andrewfnewman@gmail.com>

CC: "Seaborne, Andy" <andy.seaborne@hp.com>, Richard Newman <rnewman@twinql.com>, Lee Feigenbaum <lee@thefigtrees.net>, "public-rdf-dawg-comments@w3.org" <public-rdf-dawg-comments@w3.org>

Date: Tue, 25 Mar 2008 16:20:20 +0100

Message-ID: <47E91834.2060603@aduna-software.com>

To: Andrew Newman <andrewfnewman@gmail.com>

CC: "Seaborne, Andy" <andy.seaborne@hp.com>, Richard Newman <rnewman@twinql.com>, Lee Feigenbaum <lee@thefigtrees.net>, "public-rdf-dawg-comments@w3.org" <public-rdf-dawg-comments@w3.org>

Andrew Newman wrote: > On 21/03/2008, Arjohn Kampman <arjohn.kampman@aduna-software.com> wrote: >> Hi Andrew, others, [...] >> U and 0 are both typed relations of some type T. U contains all possible >> tuples of this type, whereas 0 contains zero tuples of this type. Note >> that this means that these relations actually (can) have attributes. DEE >> and DUM are specific types of U and 0, namely the ones with zero >> attributes. > > I think the term date uses is isomorphism - I'm sayings there is one > between sets, bags, an untyped and typed and relational algebra (page > 246). > > The reason I think there is an isomorphism for U and 0 for DEE and DUM > is taken from page 261: > "As I showed earlier, the identity element with respect to > intersection is the universal relation of the pertinent type. But > join is a generalized intersection; in particular, it doesn't require > its operands to be of the same type, and indeed they usually aren't. > As direct consequence of this fact, join has was might be called a > general identity element namely, TABLE_DEE, which is the unique > relation with no attributes and exactly one tuple (necessarily the > empty tuple). To elaborate: If A is a relation of type T and U is the > corresponding universal relation, then it's certainly true that the > join of A and U is equal to A. But the join of A and TABLE_DEE is > also equal to A, and this latter equality is guaranteed to hold no > matter what the type of A happens to be. This, we might resonably say > that join (i.e., <AND>) has both (a) a specific identity element for > each specific relation type and (b) a generic identity element, > TABLE_DEE, that's independent of relation type" > > Hopefully, you can see why I interpreted this to mean that DEE and U > and the next paragraph likewise make DUM and 0 to be isomorphic. Now > he doesn't actually say that DEE UNION A = DEE but I don't think he > needs to (given the previous definitions). I read the same paragraph and already guessed that this is where your interpretation came from. However, I still think that this interpretation is incorrect. Isomorphisms exists between the univeral and empty relations in various algebras, but I doubt that they exist between DEE/DUM and U/0, respectively. [...] >> As an empty graph pattern "{}" corresponds to DEE and a false graph >> pattern "{filter(false)}" corresponds to DUM, the SPARQL algebra seems >> to be in line with Date's definitions. >> > > I'd disagree - even if I'm wrong about the untyped relational > identities - he's goes on further in chapter 12 to define the *same* > identities for bags. I'm sorry, but I can only find definitions for U and 0, I can't find any equivalences of DEE and DUM. Can you point me to a specific page and/or paragraph? -- ArjohnReceived on Tuesday, 25 March 2008 15:21:03 UTC

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