Recently we implemented a ConditionSet [1] object which abstractly
represents a set of elements satisfying a particular condition taken from
another set S:
{x|Cond(x) is True for x in S}
ConditionSet gives us the ability to represent unevaluated equations and
inequalities in forms like `{x|f(x)=0; x in S}` and `{x|f(x)>0; x in S}`
but a more powerful thing about ConditionSet is that it allows us to write
the intermediate steps as set to set transformation. Some of the
transformations are:
Composition: `{x|f(g(x))=0;x in S} => {x|g(x)=y; x in S,y in {z|f(z)=0; z
in S}}`
Polynomial Solver: `{x|P(x)=0;x in S} => {x_1,x_2, ...
,x_n}.intersection(S)` where `x_i` are roots of P(x)
Invert solver: `{x|f(x)=0;x in S} => {g(0)| all g such that f(g(x)) = x}`
logcombine: `{x| log(f(x)) + log(g(x));x in S}` => `{x| log(f(x)*g(x)); x
in S}` if f(x) > 0 and g(x) > 0
=> `{x| log(f(x)) +
log(g(x));x in S}` otherwise
product solve: {x|f(x)*g(x)=0; x in S} => `{x|f(x)=0; x in S} U {x|g(x)=0;
x in S}` given f(x) and g(x) are bounded
=> {x|f(x)*g(x)=0; x in S}
Since the output type is same as input type any composition of these
transformations is also a valid transformation. And our aim is to find the
right sequence of compositions (given the atoms) which transforms the given
condition set to a set which is not a condition set i.e., FiniteSet,
Interval, Set of Integers and their Union, Intersection, Complement or
ImageSet [2]. We can assign a cost function to each set, such that, more
desirable that form of set is to us, the less the value of cost function.
This way our problem is now reduced to finding the path from the initial
ConditionSet to of the lowest valued set on a graph where the atomic
transformations forms the edges. I'm hoping this is a well-studied problem
and someone on this mailing list can redirect us to existing literature.
Another good thing about this method is that we will be able to log and
output the intermediate steps required to reach the final solution. These
step-by-step solutions should be helpful in teaching and also to verify the
final solution is correct or not.
Please lets us know what do think about this new proposed method.
[1]: https://github.com/sympy/sympy/pull/9696
[2]: ImageSet is abstract class in SymPy to represent sets like {f(x)| x in
S}
--
You received this message because you are subscribed to the Google Groups
"sympy" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sympy+unsubscr...@googlegroups.com.
To post to this group, send email to sympy@googlegroups.com.
Visit this group at http://groups.google.com/group/sympy.
To view this discussion on the web visit
https://groups.google.com/d/msgid/sympy/CADN8iupEK7g1HwwOGR6-M9uwPduJjLWc7N5uYuj-ubbQorTv%2Bg%40mail.gmail.com.
For more options, visit https://groups.google.com/d/optout.
