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I'm fairly new to the syntax of relational algebra, and I'm having a hard time understanding how I could set a "at least one" clause.

Example: I have:

  • a table with books (listing the title, year published and ID),
  • a table with authors (listing their name and ID),
  • a table which lists what author wrote what book (through a tuple of the IDs mentioned before).

How could I, in relational algebra, get "All the authors that have published at least one book per year between 2008 and 2010"?

I have figured this so far. At step "b", the Natural join is used since both tables have PublicationID in common. Thus, the resulting table is |PublicationID|AuthorID|Year|. So I'm simply missing the step "c", where I don't understand how to gather a sub-set of the authors that published at least one book per year between 2008 and 2010.

$ a \leftarrow \pi_{PublicationID,Year} (Publication)$

$ b \leftarrow a \bowtie AuthorPublication $

$ c \leftarrow \sigma_{something} $

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    $\begingroup$ A query like $A.\exists\, T,Y,I,J \; \text{book}(T,Y,I),\text{author}(N,J),\text{authorbook}(J,I),{\lt}(Y,2011),{\lt}(2007,Y)$ should work to return a multiset of author names, using Chandra and Merlin's notation from their 1977 paper. $\endgroup$ – András Salamon May 22 '13 at 22:42
  • $\begingroup$ @AndrásSalamon Thanks! Unfortunately I was looking for a more "classical" notation (using projections, selections and joins). $\endgroup$ – Balaam May 23 '13 at 5:58
  • $\begingroup$ Check the definition of natural join. Are there any other attributes common to two relations ? $\endgroup$ – András Salamon May 23 '13 at 9:00
  • $\begingroup$ @AndrásSalamon Indeed, that is a good idea. A friend gave me a possible solution, I shall edit my post with it. $\endgroup$ – Balaam May 23 '13 at 10:32
  • $\begingroup$ The FOL answer is not correct as it returns authors who wrote at least one book between 2007 and 2011, not at least a book per year $\endgroup$ – Romuald May 23 '13 at 12:34
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A first hint toward a solution is to think about "what's the result of a natural join between $Author$, $Publication$ and $AuthorPublication$?" The answer is "the universal relation" $R(BookID,AuthorID,Title,Year,Name)$ describing who wrote a book and when. Note that a book without any author or an author without any book written won't appear in $R$. So an author in $R$ as written at least one book. The following query gives the authors who wrote at least a book in year 2008: $$ \pi_{Name}(Author \Join AuthorPublication \Join \sigma_{(2008 = year)}(Publication))$$

For the final answer, compute the intersection of this query with its variants: $$ \pi_{Name}(Author \Join AuthorPublication \Join \sigma_{(2008 = year)}(Publication)) \cap \pi_{Name}(Author \Join AuthorPublication \Join \sigma_{(2009 = year)}(Publication)) \cap \pi_{Name}(Author \Join AuthorPublication \Join \sigma_{(2010 = year)}(Publication))$$

You may provide other equivalent answers with outer $Author \Join$.

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  • $\begingroup$ This answer does seem correct, thanks! But for curiosity's sake, is my EDIT2 answer any good? It's another way of doing it no? Or am I totally wrong? $\endgroup$ – Balaam May 23 '13 at 15:13
  • $\begingroup$ It looks like, but it's a little bit contrived IMHO : Mix2 is (almost) a natural join written from product and selection (almost because redundant attributes are "removed" by $\Join$), moreover you use several distinct intermediate variables which are not that useful (subresult and YXXXX, but i'm ok with "mix2"). $\endgroup$ – Romuald May 26 '13 at 16:48
  • $\begingroup$ As a side remark, from the computational point of view, $\sigma_{year=X}$ should be done as "early" as possible (and $Author \Join$ should be done as "late" as possible). $\endgroup$ – Romuald May 26 '13 at 16:55
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This is an answer by the OP, which is removed from the question.

A friend gave me a tip and a possible solution appeared:

For Author(Name,AuthorID) | Publication(Title,Year,BookID) | AuthorPublication(BookID,AuthorID)

$ RenamedAP = \alpha_{(AuthorID:linkAuthorID, PublicationID:linkPubID)} (AuthorPublication) $

$Mix \leftarrow RenamedAP \times Author \times Publication $

$Mix2 \leftarrow \sigma_{(AuthorID=linkAuthorID \wedge PublicationID = linkPubID)} (Mix) $

$ Y2008 \leftarrow \sigma_{(Year=2008)} (Mix2)$

$ Y2009 \leftarrow \sigma_{(Year=2009)} (Mix2)$

$Y2010 \leftarrow \sigma_{(Year=2010)} (Mix2) $

$ Subresult \leftarrow Y2008 \cap Y2009 \cap Y2010 $

$ Result \leftarrow \pi_{Name} (Subresult) $ \

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