SymPy's default pattern matcher is not a structural one, i.e. it tries to
match according to some mathematical rules. A structural pattern matcher
would instead just match the structure of the expression-tree.
While it is necessary to have some mathematical awareness such as, give the
wild *w*:
- associativity awareness (i.e. f(a, b, c) = f(a, f(b, c)) and similar):
(a+b+c).match(a+w) ==> {w: b+c}
- commutativity awareness:
(a+b).match(w+a) ==> {w: b}
- one-identity awareness (f is one-identical if f(x) = x for any x):
x.match(Add(w, evaluate=False)) ==> {w: x}
These rules are also available in Wolfram Mathematica.
There are some results which come out as an attempt to apply some kind of
equation solving:
In [20]: (x*y).match(x/w)
Out[20]:
⎧ 1⎫
⎨w: ─⎬
⎩ y⎭
That is, *Mul(x, y)* is matched to become *Mul(x, Pow(Mul(1, Pow(y, -1)),
-1))* which is equivalent mathematically, but not structurally.
I would expect this result by calling:
In [29]: solve(Eq(x*y, x/w), w)
Out[29]:
⎡1⎤
⎢─⎥
⎣y⎦
I was wondering, what are the pattern matching rules for this awareness?
This goes beyond the three rules I pointed out hereabove. Technically,
mathematical-aware patterns may yield infinite results as long as they are
combined with equation solvers.
Wolfram Mathematica seems not to have any solver-awareness, that is, the
previous pattern does not match:
In[12]:= (x y) /. x/w_ -> {w}
Out[12]= x y
It would be useful to keep SymPy's pattern matching rules as close as
possible to those of Mathematica. I suggest to remove redundant results
relying on solvers, or maybe add a parameter to the matcher to turn it off.
What do you think?
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