Currently, IS doesn't understand semantic-equality of
bags (lists, matrices, sets) at all:
is(equal([1],[1])) => true OK
is(equal([0],[1])) => Unknown NO
The first case only works because the two expressions
are syntactically equal (is/= or ?like).
The problem is that meqp uses compare, which
compares x and y by comparing x-y to (scalar) 0,
obviously irrelevant here. The solution is to explicitly
add bags to meqp and mnqp. See code attached. With
this code, we have:
is(equal([1],[1])) => true
is(equal([0],[1])) => false
is(equal([a,0],[a,1])) => false
is(equal([a,b],[a,c])) => unknown
HOWEVER: a) this code ignores doscmxplus, so lists are
never considered equal to matrices; b) the assume
database is not consulted in cases like equal(x,[a,b]),
because compar thinks this is the same as equal([x-a,x-
b],0).
I considered also adding ordering inequalities (< etc.),
but I don't think it's worth it right now given the limited
capabilities of compar. Also, it's not clear whether
lexicographic ([a,b]<[c,d] iff (a<c or (a=c and b<d)) or
dominating order (a<c and b<d -- non-trichotomic) is
more useful.
This was originally reported by Robert Dodier in email.
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These are some design notes I posted to the Maxima mailing
list on this subject.
Robert Dodier said:
> is(equal([1],[[1]]))) => UNKNOWN.
> I would expect that different structures are always not
> equal, unless they're equivalent (i.e., you could substitute
> one for the other in any expression and get the same
result).
> So either TRUE or FALSE seems possible here, but not
UNKNOWN.
I had written code to assume that objects of differing
dimensions were different (in particular, that non-scalars can
never be equal to scalars), but ran into a whole bunch of
inconsistencies in Maxima semantics. These are not
accidental inconsistencies, they are not bugs; they are in
fact design features to make the use of Maxima more
convenient. But as is often the case, convenience conflicts
with consistency.
Clearly is(1=[1]) is false; remember, = performs structural
(syntactic) comparison. However, it is not at all clear
whether is(equal(1,[1])) is false. (And whether it is False or
True depends not only on various flag settings, but also the
context of use.) Consider:
Lists, 1xn, and nx1 matrices are coerced into each other in
various ways and with Scalarmatrixp=True (the default), a
1x1 matrix is converted to a scalar:
[1,2] . matrix([3],[4]) => 11
[1,2] . matrix([3,4]) => 11
matrix([1,2]).matrix([3,4]) => 11
matrix([1],[2]).matrix([3],[4]) => 11
So perhaps 11 == [11] and [1,2] == matrix([1,2]) == matrix
([1],[2]) ? On the other hand, matrix([11]) by itself does not
automatically simplify to 11 and 11-[11]=>[0], not 0.
Of course, outside the context of vector/matrix operations,
lists, 1xn matrices, and nx1 matrices are completely distinct:
after all,
member(a,[a,b]) => true and member(a,matrix([a,b])) =>
false.
Now consider symbolic expressions. You might think that
a==b must be false if nonscalar(a) and scalar(b). But that's
not true. If you declare(vec,nonscalar), then nonscalar
(vec.vec) => True, though in fact the .-product of two
vectors is a scalar (nonscalar is a *syntactic* check for the
presence of nonscalar objects within the expression tree). So
it is wrong to assume that is(equal(vec1.vec2,0)) is false
because nonscalar(vec1.vec2)=True and nonscalar(0)=False.
This is not theoretical. I think it perfectly reasonable that a
user might check whether q and r are perpendicular by
checking is(equal(q.r,0)), where q and r are sometimes
concrete vectors -- in which case you might well get a valid
True or False --, sometimes symbols (in which case the
answer should be Unknown, not False).
There is another simple case that has nothing to do with
vector/matrix semantics. To do bag comparisons correctly,
you need to solve
conjunctions: [x,x] =? [1,2] is equivalent to (x==1 and x==2),
which is clearly false, but Is doesn't know it. Similarly for
[x,1] =? [1,x+1].
Another limitation on the power of is/equal/bag: the compar
subsystem is not currently terribly useful for vectors etc. For
example, assume(equal(q,[a,b])), is(equal(q,[a,b+1])) =>
Unknown.
I think there are several possible conclusions for all this:
1) Make Maxima more rigorous. Distinguish between lists,
vectors, and 1xn matrices everywhere. Distinguish between
scalars, 1-vectors, and 1x1 matrices everywhere. In that
case, the correct results for is/equal are much clearer. But
pragmatism is one of the central characteristics of Maxima;
there are certainly other systems which emphasize rigor, with
its advantages and disadvantages.
2) Decide that is/equal/bag always refers to .-product
semantics (not list semantics and not + or * semantics): if
two objects act the same in the context of a .-product, then
they are the same.
2ay) The result of is/equal depends on the various switches
controlling vector/matrix operations. So if Scalarmatrixp
and Listarith are true, then 1 == [1] == matrix([1]).
2by) Assume default settings for the various switches.
2xa) Consider an n-list, an n-row, and an n-column equal
(with
appropriate switches set), even though they don't act
equally in all contexts.
2xb) Consider an n-list and an n-row equal, but not an n-
column.
3) Decide that is/equal/bag is true iff there is NO context in
which the objects can be distinguished. This emphasizes the
programming-type aspect of Maxima over the mathematical
aspect. But 'equal' is supposed to be about mathematical
equality. After all, is/equal considers 1/2, 0.5, and 0.5b0 to
be equal, even though 1/2 is an exact number and 0.5 and
0.5b0 are approximate.
3a) Assume that non-scalar *identifiers* (but not non-scalar
expressions) are never equal to scalars.
3b) Do not take into account the nonscalar property of
identifiers.
4) Maintain the current behavior, which gives Unknown for all
the difficult cases.
What I had started out trying to program was 2aa, but it was
too messy (because when I started, I was also trying to take
into account the behavior of + and *). Now that I've written
out this design note, it seems like the two reasonable
alternatives are 3a and 2ax (not sure whether 2aa or 2ab is
better). But I suspect that in actual fact, I have already
spent too much time on this, that no one but Robert will ever
care, and that 4 is just fine....
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Fixed by r1.16 src/compar.lisp (by Barton Willis).
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