in reply to Re: decomposing binary matrices in thread decomposing binary matrices
Thanks. First, I should note that there must be at least as many values as variables, since each variable must take a distinct value within the set of possible values.
Second, variables that are part of an nelement submatrix need not have n bits set. The sparsest counterexample is: my @grid = (
# 0 1 2 3 4 5
[ 0, 0, 0, 1, 1, 0, ],
[ 0, 0, 0, 1, 0, 1, ],
[ 0, 0, 0, 0, 1, 1, ],
[ 1, 1, 0, 0, 0, 0, ],
[ 1, 0, 1, 0, 0, 0, ],
[ 0, 1, 1, 0, 0, 0, ],
);
.. which can decompose into two 3element submatrices.
The least sparse version of that is: my @grid = (
# 0 1 2 3 4 5
[ 0, 0, 0, 1, 1, 1, ],
[ 0, 0, 0, 1, 1, 1, ],
[ 0, 0, 0, 1, 1, 0, ],
[ 1, 1, 1, 1, 1, 1, ],
[ 1, 1, 1, 1, 1, 1, ],
[ 1, 1, 0, 1, 1, 1, ],
);
.. which can decompose the same way.
Update: swapped 2 bits in the last row of the sparse matrix, so it actually represents what I'm saying
Hugo
Re^3: decomposing binary matrices by BrowserUk (Pope) on Feb 16, 2007 at 15:55 UTC 
Hm. Sorry to have wasted your time. I'd picked up on this bit of the OP
... since A and E are restricted to only two values between them, they must consume those two values;
... and hung my hat on it, but that obviously doesn't apply in the same way to the two examples above.
Question: Would this example also decompose into the (same?) two groups as the above?
my @grid = (
[ 0, 1, 0, 1, 0, 0, ],
[ 0, 0, 0, 1, 1, 0, ],
[ 0, 1, 0, 0, 1, 0, ],
[ 1, 0, 0, 0, 0, 1, ],
[ 1, 0, 1, 0, 0, 0, ],
[ 1, 0, 0, 0, 0, 1, ],
);
Examine what is said, not who speaks  Silence betokens consent  Love the truth but pardon error.
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.
 [reply] [d/l] 

Yes: labelling the columns A..F and the rows 1..6, we know variables {B, D, E} must consume values {1, 2, 3} between them, and likewise {A, C, F} must consume {4, 5, 6}. In this example, however, the latter can be further decomposed: C can only be 5, so {A, F] are left with {4, 6}.
Hugo
 [reply] 
