[Mulgara-dev] Re: Bug list and versions

[Life is hard, and then you die]

On Tue, Nov 14, 2006 at 10:48:48AM +1000, Andrae Muys wrote:
> 
> On 14/11/2006, at 9:51 AM, Life is hard, and then you die wrote:
[snip]
> >you get one line for every matching $b, just as expected. So it's just
> >in the context of an OR that things don't work. I understood OR to be
> >a set-theoretic union, so it shouldn't matter if I do one query with
> >3 OR terms, or 3 queries each with only one of the OR terms.
> 
> OR is not strictly speaking a set-theoretic union, but a disjunction.
> 
> So given the following tuples:
> 
> A  $a  $b      B   $a   $b
>     1   5           2    6
>                     2    7
> 
> the actual semantics are not tabular, but logical:
> 
> A = { {a->1, b->5} }               implies (a==1) ^ (b==5) -> #t
> B = { {a->2, b->6}, {a->2, b->7} } implies ((a==2) ^ (b==6)) v  
> ((a==2) ^ (b==7)) -> #t
> 
> and consequently
> 
> A or B |- ((a==1) ^ (b==5)) v ((a==2) ^ (b==6)) v ((a==2) ^ (b==7)) - 
> > #t
> 
> which in sets-of-bindingsets notation is
>   { {a->1, b->5}, {a->2, b->6}, {a->2, b->7} }
> and in tabular format is:
> 
> A or B  $a  $b
>          1   5
>          2   6
>          2   7
> 
> Given
> 
> C = { {a->2, c->8} }
> 
> A or C |- ((a==1) ^ (b==5) v ((a==2) ^ (c==8))
>       ie. { {a->1, b->5}, {a->2, c->8} }
> which in tabular format is:
> 
> A or C  $a  $b  $c
>          1   5   -
>          2   -   8
> 
> So you are right, disjunction is isomorphic with set-theoretic union,  
> and indeed when the propositions represented by the various tuples  
> are or'ed the result is the set-theoretic union of their variable  
> bindings; but the interface to that result is via a tabular form.   
> Consequently we have these "binding-doesn't-exist"/"don't  
> care"/"variable unbound" (whatever you want to call it) values  
> turning up in the mapping from the logical proposition to the table.   
> These values are not NULL's, rather they are DONTCARE's - they  
> literally indicate that the truth value of that term in the sum-of- 
> products form is independent of that variable.

Thanks for the detailed explanation. I tend to think in in terms of
sets of tuples, hence my interpretation. The 'A or C' then becomes the
union of an inverse-projection 

    -----        -----
    pr_ab(A)  u  pr_ac(C)

(where pr_ab is the projection from {(a,b,c)} to {(a,b)}), i.e. the
DONTCARE's are all possible values. But like you say, the two views
are (or should be) isomorphic.


  Cheers,

  Ronald