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Ordering Products

P: n/a
Here might be an interesting puzzle for people who like sorting
algorithms ( and no I'm not a student anymore and the problem is not a
students 'homework' but a particular question associated with a
computer algebra system in Python I'm currently developing in my
sparetime ).

For motivation lets define some expression class first:

class Expr:
def __init__(self, name=""):
self.name = name
self.factors = [self]

def __mul__(self, other):
p = Expr()
if isinstance(other,Expr):
other_factors = other.factors
else:
other_factors = [other]
p.factors = self.factors+other_factors
return p

def __rmul__(self, other):
p = M()
p.factors = [other]+self.factors
return p

def __repr__(self):
if self.name:
return self.name
else:
return "*".join([str(x) for x in self.factors])

One can create arbitrary products of Expr objects ( and mixing numbers
into the products ):
a,b,c = Expr("a"),Expr("b"),Expr("c")
a*b a*b 7*a*8*9 7*a*8*9

The goal is to evaluate such products and/or to simplify them.

For expressions like
x = 7*a*8*9
this might be easy, because we just have to sort the factor list and
multiply the numbers.
x.factors.sort()
x a*7*8*9

-> a*504

This can be extended to arbitrary products:
x = 7*a*b*a*9
x.factors.sort()
x

a*a*b*7*9

-> (a**2)*b*63

Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?

Regards,
Kay

Jul 21 '05 #1
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15 Replies


P: n/a
Kay Schluehr wrote:
Here might be an interesting puzzle for people who like sorting
algorithms ( and no I'm not a student anymore and the problem is not a
students 'homework' but a particular question associated with a
computer algebra system in Python I'm currently developing in my
sparetime ).


<folded>
x = 7*a*b*a*9
x.factors.sort()
x


a*a*b*7*9

-> (a**2)*b*63

Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?

Regards,
Kay


Looks interesting Kay.

I think while the built in sort works as a convenience, you will need to
write your own more specialized methods, both an ordering (parser-sort),
and simplify method, and call them alternately until no further changes
are made. (You might be able to combine them in the sort process as an
optimization.)

A constrained sort would be a combination of splitting (parsing) the
list into sortable sub lists and sorting each sub list, possibly in a
different manner, then reassembling it back. And doing that possibly
recursively till no further improvements are made or can be made.
On a more general note, I think a constrained sort algorithm is a good
idea and may have more general uses as well.

Something I was thinking of is a sort where instead of giving a
function, you give it a sort key list. Then you can possibly sort
anything in any arbitrary order depending on the key list.

sort(alist, [0,1,2,3,4,5,6,7,8,9]) # Sort numbers forward
sort(alist, [9,8,7,6,5,4,3,2,1,0]) # Reverse sort
sort(alist, [1,3,5,7,9,0,2,4,6,8]) # Odd-Even sort
sort(alist, [int,str,float]) # sort types

These are just suggestions, I haven't worked out the details. It could
probably be done currently with pythons built in sort by writing a
custom compare function that takes a key list. How fine grained the key
list is is also something that would need to be worked out. Could it
handle words and whole numbers instead of letters and digits? How does
one specify which? What about complex objects?
Here's a "quick sort" function that you might be able to play with..
There are shorter versions of this, but this has a few optimizations added.

Overall it's about 10 times slower than pythons built in sort for large
lists, but that's better than expected considering it's written in
python and not C.

Cheers,
Ron

# Quick Sort
def qsort(x):
if len(x)<2:
return x # Nothing to sort.

# Is it already sorted?
j = min = max = x[0]
for i in x:
# Get min and max while checking it.
if i<min: min=i
if i>max: max=i
if i<j: # It's not sorted,
break # so stop checking and sort.
j=i
else:
return x # It's already sorted.

lt = []
eq = []
gt = []

# Guess the middle value based on min and max.
mid = (min+max)//2

# Divide into three lists.
for i in x:
if i<mid:
lt.append(i)
continue
if i>mid:
gt.append(i)
continue
eq.append(i)

# Recursively divide the lists then reassemble it
# in order as the values are returned.
return q(lt)+eq+q(gt)
Jul 21 '05 #2

P: n/a
Kay Schluehr <kay.schluehr <at> gmx.net> writes:
Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?


I don't think that sorting is the answer here.
Firts of all IMHO you have to add an
additional constraint - associativity of the operation in question
So the problem could be reduced to making the constant
parts be more associative than the non-constant parts.
which you should be able to
do with a parser. The BNF grammar could look like this:

expr ::= v_expr "*" v_expr | v_expr
v_expr ::= variable | c_expr
c_expr ::= l_expr "*" literal | l_expr
l_expr ::= literal | "(" expr ")"

The trick is to create a stronger-binding multiplication operator on constants
than on mixed
expressions.

This grammar is ambigue of course - so a LL(k) or maybe even LALR won't work.
But earley's method
implemented in spark should do the trick.
If I find the time, I'll write an short implementation
tomorrow.

Diez

Jul 21 '05 #3

P: n/a
Diez B.Roggisch wrote:
Kay Schluehr <kay.schluehr <at> gmx.net> writes:
Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?


I don't think that sorting is the answer here.
Firts of all IMHO you have to add an
additional constraint - associativity of the operation in question
So the problem could be reduced to making the constant
parts be more associative than the non-constant parts.
which you should be able to
do with a parser.


Hi Diez,

I have to admit that I don't understand what you mean with the
'constant parts' of an expression?

The associativity of __mul__ is trivially fullfilled for the dummy
class M if an additional __eq__ method is defined by comparing factor
lists because those lists are always flat:

def __eq__(self, other):
if isinstance(other,M):
return self.factors == other.factors
return False

The sorting ( or better 'grouping' which can be represented by sorting
in a special way ) of factors in question is really a matter of
(non-)commutativity. For more advanced expressions also group
properties are important:

If a,b are in a center of a group G ( i.e. they commute with any
element of G ) and G supplies an __add__ ( besides a __mul__ and is
therefore a ring ) also a+b is in the center of G and (a+b)*c = c*(a+b)
holds for any c in G.

It would be nice ( and much more efficient ) not to force expansion of
the product assuming distributivity of __add__ and __mul__ and
factorization after the transposition of the single factors but
recognizing immediately that a+b is in the center of G because the
center is a subgroup of G.
Regards,
Kay

Jul 21 '05 #4

P: n/a


Ron Adam wrote:
Kay Schluehr wrote:
Here might be an interesting puzzle for people who like sorting
algorithms ( and no I'm not a student anymore and the problem is not a
students 'homework' but a particular question associated with a
computer algebra system in Python I'm currently developing in my
sparetime ).
<folded>
>x = 7*a*b*a*9
>x.factors.sort()
>x


a*a*b*7*9

-> (a**2)*b*63

Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?

Regards,
Kay


Looks interesting Kay.


I think so too :) And grouping by sorting may be interesting also for
people who are not dealing with algebraic structures.
I think while the built in sort works as a convenience, you will need to
write your own more specialized methods, both an ordering (parser-sort),
and simplify method, and call them alternately until no further changes
are made. (You might be able to combine them in the sort process as an
optimization.)

A constrained sort would be a combination of splitting (parsing) the
list into sortable sub lists and sorting each sub list, possibly in a
different manner, then reassembling it back. And doing that possibly
recursively till no further improvements are made or can be made.
I think a comparison function which is passed into Pythons builtin
sort() should be sufficient to solve the problem. I guess the
comparison defines a total order on the set of elements defined by the
list to sort.
On a more general note, I think a constrained sort algorithm is a good
idea and may have more general uses as well.

Something I was thinking of is a sort where instead of giving a
function, you give it a sort key list. Then you can possibly sort
anything in any arbitrary order depending on the key list.

sort(alist, [0,1,2,3,4,5,6,7,8,9]) # Sort numbers forward
sort(alist, [9,8,7,6,5,4,3,2,1,0]) # Reverse sort
sort(alist, [1,3,5,7,9,0,2,4,6,8]) # Odd-Even sort
sort(alist, [int,str,float]) # sort types
Seems like you want to establish a total order of elements statically.
Don't believe that this is necessary.
These are just suggestions, I haven't worked out the details. It could
probably be done currently with pythons built in sort by writing a
custom compare function that takes a key list.
Exactly.
How fine grained the key
list is is also something that would need to be worked out. Could it
handle words and whole numbers instead of letters and digits? How does
one specify which? What about complex objects?


In order to handle complex objects one needs more algebra ;)

Since the class M only provides one operation I made the problem as
simple as possible ( complex expressions do not exist in M because
__mul__ is associative - this is already a reduction rule ).

Kay

Jul 21 '05 #5

P: n/a
Kay Schluehr wrote:

Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?


Hello Kay,

take this into account:
Restrictions like commutativity, associative, distributive and flexibility
laws don't belong neither to operands nor to operators themselves.
Instead these are properties of fields (set of numbers with respect to a
certain operation).
For a famous example for a somewhat "alternative" behaviour look at the
Octonions (discovered in 1843 by Graves and 1845 by Cayley), which are not
associative with respect to addition and/or multiplication.
(http://en.wikipedia.org/wiki/Octonions) or the Quarternions, which are
non-commutative (http://en.wikipedia.org/wiki/Quaternion)

Obviously, it's not correct to say: addition is associative, or, that
multiplication is. With the same right, you could say, multiplication is
not associative.
With the same reasoning, we can show that it's not easy to generalize
sorting, commutation, association or distribution mechanisms.

Maybe it would be a very fascinating goal to solve your algorithmic approach
in such a limited environment like the Quarternions.
A solution for this set of numbers, if achieved in a clean, mathematically
abstract way, should hold for most other numbers/fields too, natural and
real included.

I guess that the approach might be this way:
- define/describe the fields which shall be handled
- define/describe the rules which shall be supported
- find methods to reduce sequences of operations to simple binary or unary
operations (tokens) - this may introduce brackets and stacking mechanisms
- a weighing algorithm might be necessary to distinguish between plain
numbers and place holders (variables)
- application of the distributivity (as far as possible) might help to find
a rather flat representation and a base for reordering according to the
weights of the individual sub-expressions

Nevertheless, there are lots of commercial programs which do such sort of
symbolic mathematics, and which would badly fail when it would come to such
awkward fields like Quarternions/Octonions.
Bernhard
Jul 21 '05 #6

P: n/a
> I have to admit that I don't understand what you mean with the
'constant parts' of an expression? From what I percieved of your example it seemed to me that you wanted to evaluate the constants like 7*9 first, so that an expression like

a * 7 * 9 * b

with variables a,b is evaluated like this:

a * 63 * b

So my suggestion was simply to make the *-operator more precedent when
in between two constants. What I mean with constants here are of course
integer/float literals. The concept of a differing operator precedence
can be extended to arbitray elements when their types are known - which
should be possible when variable values are known at parsing
time.
The associativity of __mul__ is trivially fullfilled for the dummy
class M if an additional __eq__ method is defined by comparing factor
lists because those lists are always flat:
I don't care about that, as my approach deosn't use python's built-in parser
- it can't, as that wouldn't allow to re-define operator precedence.

What you do is to
simply collect the factors as list. But what you need (IMHO) is a parsing
tree (AST) that reflects your desired behaviour by introducing a different
precedence thus that the expression

a * 7 *9 * b

is not evaluated like

((a*7)*9)*b

(which is a tree, and the standard way of evaluationg due to built-in parsers
precedence rules) but as

a*(7*9)*b

which is also a tree.
The sorting ( or better 'grouping' which can be represented by sorting
in a special way ) of factors in question is really a matter of
(non-)commutativity. For more advanced expressions also group
properties are important:
No, IMHO associativity is the important thing here - if

(a * 7) * 9

yields a different solution than

a *(7*9)

your reordering can't be done - in the same way as re-arranging
factors a*b to b*a only works if the commute - or, to put in in
algebraic terms, the group is abelian.
If a,b are in a center of a group G ( i.e. they commute with any
element of G ) and G supplies an __add__ ( besides a __mul__ and is
therefore a ring ) also a+b is in the center of G and (a+b)*c = c*(a+b)
holds for any c in G.

It would be nice ( and much more efficient ) not to force expansion of
the product assuming distributivity of __add__ and __mul__ and
factorization after the transposition of the single factors but
recognizing immediately that a+b is in the center of G because the
center is a subgroup of G.


Well, you don't need to expand that product - the subexpression a+b is
evaluated first. If you can sort of "cache" that evaluation's result because
the expressions involved are of a constant nature, you can do so.

The rason (a+b) is evaluated first (at least in the standard python parser,
and in my proposed special parser) is that the parentheses ensure that.

To sum things up a little: I propose not using the python built-in parser
which results in you having to overload operators and lose control
of precedence, but by introducing your own parser, that can do the
trick of re-arranging the operators based on not only the "usual" precedence
(* binds stronger than +), but by a type-based parser that can even change
precedence of the same operator between different argument types is's
applied to. That might sound complicated, but I think the grammar
I gave in my last post shows the concept pretty well.

regards,

Diez
Jul 21 '05 #7

P: n/a
Kay Schluehr wrote:

Ron Adam wrote:
Kay Schluehr wrote: On a more general note, I think a constrained sort algorithm is a good
idea and may have more general uses as well.

Something I was thinking of is a sort where instead of giving a
function, you give it a sort key list. Then you can possibly sort
anything in any arbitrary order depending on the key list.

sort(alist, [0,1,2,3,4,5,6,7,8,9]) # Sort numbers forward
sort(alist, [9,8,7,6,5,4,3,2,1,0]) # Reverse sort
sort(alist, [1,3,5,7,9,0,2,4,6,8]) # Odd-Even sort
sort(alist, [int,str,float]) # sort types

Seems like you want to establish a total order of elements statically.
Don't believe that this is necessary.


I want to establish the sort order at the beginning of the sort process
instead of using many external compares during the sort process. Using
a preprocessed sort key seems like the best way to do that. How it's
generated doesn't really matter. And of course a set of standard
defaults could be built in.
These are just suggestions, I haven't worked out the details. It could
probably be done currently with pythons built in sort by writing a
custom compare function that takes a key list.

Exactly.


The advantage of doing it as above would be the sort could be done
entirely in C and not need to call a python compare function on each
item. It would be interesting to see if and how much faster it would
be. I'm just not sure how to do it yet as it's a little more
complicated than using integer values.
How fine grained the key
list is is also something that would need to be worked out. Could it
handle words and whole numbers instead of letters and digits? How does
one specify which? What about complex objects?

In order to handle complex objects one needs more algebra ;)

Since the class M only provides one operation I made the problem as
simple as possible ( complex expressions do not exist in M because
__mul__ is associative - this is already a reduction rule ).

Kay


I'm played around with your example a little bit and think I see how it
should work... (partly guessing) You did some last minute editing so M
and Expr were intermixed.

It looks to me that what you need to do is have the expressions stored
as nested lists and those can be self sorting. That can be done when
init is called I think, and after any operation.

You should be able to add addition without too much trouble too.

a*b -> factors [a],[b] -> [a,b] You got this part.

c+d -> sums [c],[d] -> [c,d] Need a sums type for this.

Then...

a*b+c*d -> sums of factors -> [[a,b],[c,d]]

This would be sorted from inner to outer.

(a+b)*(b+c) -> factors of sums -> [[a,b],[c,d]]

Maybe you can sub class list to create the different types? Each list
needs to be associated to an operation.

The sort from inner to outer still works. Even though the lists
represent different operations.

You can sort division and minus if you turn them into sums and factors
first.

1-2 -> sums [1,-2]

3/4 -> factors [3,1/4] ? hmmm... I don't like that.

Or that might be...

3/4 -> factor [3], divisor [4] -> [3,[4]]
So you need a divisor type as a subtype of factor. (I think)
You can then combine the divisors within factors and sort from inner to
outer.

(a/b)*(c/e) -> [a,[b],c,[e]] -> [a,c,[b,e]]

Displaying these might take a little more work. The above could get
represented as...

(a*c)/(b*e)

Which I think is what you want it to do.
Just a few thoughts. ;-)
Cheers,
Ron





Jul 21 '05 #8

P: n/a
Bernhard Holzmayer schrieb:
Kay Schluehr wrote:

Now lets drop the assumption that a and b commute. More general: let be
M a set of expressions and X a subset of M where each element of X
commutes with each element of M: how can a product with factors in M be
evaluated/simplified under the condition of additional information X?

It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?

Hello Kay,

take this into account:
Restrictions like commutativity, associative, distributive and flexibility
laws don't belong neither to operands nor to operators themselves.
Instead these are properties of fields (set of numbers with respect to a
certain operation).
For a famous example for a somewhat "alternative" behaviour look at the
Octonions (discovered in 1843 by Graves and 1845 by Cayley), which are not
associative with respect to addition and/or multiplication.
(http://en.wikipedia.org/wiki/Octonions) or the Quarternions, which are
non-commutative (http://en.wikipedia.org/wiki/Quaternion)

Obviously, it's not correct to say: addition is associative, or, that
multiplication is. With the same right, you could say, multiplication is
not associative.


It was associative in the tiny example I presented. I did not mentioned
to discuss the evolving structure of the whole CAS here in detail which
would be better done in an own newsgroup once an early version is
released.

Maybe the setting of the original question should be made more precise:
associative, non-commutative multiplicative groups.

Handling non-associative algebras like Lie algebras is a completely
different matter and I'm not even sure which one is the best way to
represent operations in Python?

Maye this way?
lie = Lie() # create an arbitrary Lie algebra (lie is again a class )
A,B = lie(),lie() # create two arbitrary elements of the Lie algebra
lie[A,B] # create the commutator of the lie algebra by overloading lie[A,B] # the __getitem__ method
lie[A,B] == -lie[-A,B] True

If one wants to enforce assertions like
lie[r*A,B] == r*lie[A,B] True

for certain elements r of some group acting on lie, one must refine
creation of lie in the initial assignment statement e.g.
lie = Lie(V)


where V is some vectorspace and the elements of lie are homomorphisms
on V. V is created elsewhere. There are a lot of constraints induced by
all the objects dynamically coupled together.
With the same reasoning, we can show that it's not easy to generalize
sorting, commutation, association or distribution mechanisms.

Maybe it would be a very fascinating goal to solve your algorithmic approach
in such a limited environment like the Quarternions.
No CAS can represent infinitely many different representations of
quaternions. But it should not be to hard to deal with an algebra that
represents admissable operations on quaternions in an abstract fashion.
A solution for this set of numbers, if achieved in a clean, mathematically
abstract way, should hold for most other numbers/fields too, natural and
real included.

I guess that the approach might be this way:
- define/describe the fields which shall be handled
- define/describe the rules which shall be supported
- find methods to reduce sequences of operations to simple binary or unary
operations (tokens) - this may introduce brackets and stacking mechanisms
- a weighing algorithm might be necessary to distinguish between plain
numbers and place holders (variables)
- application of the distributivity (as far as possible) might help to find
a rather flat representation and a base for reordering according to the
weights of the individual sub-expressions

Nevertheless, there are lots of commercial programs which do such sort of
symbolic mathematics, and which would badly fail when it would come to such
awkward fields like Quarternions/Octonions.


If you take a look on Mathematica or Maple both programs seem to
interpret pure symbols as members of an associative and commutative
algebra:

expand( (a+x)^2) -> a^2 + 2ax + x^2

This works very fast and accurate but is mathematically too restricted
for me. For doing more advanced stuff one needs to do a lot of
programming in either language shipped with the CAS for creating new
packages. But then I ask myself: why not doing the programming labor in
Python and redesign and optimize the core modules of the CAS if
necessary?

Kay

Jul 21 '05 #9

P: n/a
I see, you're sensitive for the difficulties which might arise.
That's the thing I wanted to point out.
Maybe I was looking too far forward...

My first thought was to add attributes/qualifiers to the operands to improve
the sorting.
Then I realized that these attributes/qualifiers were related to the
operators, since multiplication and division use the same operands, but
while in one case it is associative and commutative, it isn't in the other.

I agree that all this leads too far.
But one thing creeps into my mind again:

I guess you'll always need an inverse operation:
A class which can handle multiplication will certainly require an inverse
operation like division.

Bernhard
Jul 21 '05 #10

P: n/a
Kay Schluehr wrote:
Here might be an interesting puzzle for people who like sorting
algorithms ( and no I'm not a student anymore and the problem is not a
students 'homework' but a particular question associated with a
computer algebra system in Python I'm currently developing in my
sparetime ).

For motivation lets define some expression class first:

This works for (simple) expressions with mixed multiplication and addition.
class F(list):
def __init__(self,*x):
#print '\nF:',x
list.__init__(self,x)
def __add__(self, other):
return A(self,other)
def __radd__(self, other):
return A(other,self)
def __mul__(self, other):
return M(self,other)
def __rmul__(self, other):
return M(other,self)
def __repr__(self):
return str(self[0])
def __order__(self):
for i in self:
if isinstance(i,A) \
or isinstance(i,M):
i.__order__()
self.sort()

class A(F):
def __init__(self, *x):
#print '\nA:',x
list.__init__(self, x)
def __repr__(self):
self.__order__()
return "+".join([str(x) for x in self])

class M(F):
def __init__(self,*x):
#print '\nM:',x
list.__init__(self,x)
def __repr__(self):
self.__order__()
return "*".join([str(x) for x in self])
a = F('a')
b = F('b')
c = F('c')
d = F('d')

print '\n a =', a

print '\n b+a+2 =', b+a+2

print '\n c*b+d*a+2 =', c*b+d*a+2

print '\n 7*a*8*9+b =', 7*a*8*9+b

a = a

b+a+2 = 2+a+b

c*b+d*a+2 = 2+a*d+b*c

7*a*8*9+b = 9*8*7*a+b <-- reverse sorted digits?

The digits sort in reverse for some strange reason I haven't figured out
yet, but they are grouped together. And expressions of the type a*(c+b)
don't work in this example.

It probably needs some better logic to merge adjacent like groups. I
think the reverse sorting my be a side effect of the nesting that takes
place when the expressions are built.

Having the digits first might be an advantage as you can use a for loop
to add or multiply them until you get to a not digit.

Anyway, interesting stuff. ;-)

Cheers,
Ron
Jul 21 '05 #11

P: n/a
Ron Adam wrote:
Kay Schluehr wrote:
Here might be an interesting puzzle for people who like sorting
algorithms ( and no I'm not a student anymore and the problem is not a
students 'homework' but a particular question associated with a
computer algebra system in Python I'm currently developing in my
sparetime ).

For motivation lets define some expression class first:

This works for (simple) expressions with mixed multiplication and addition.
class F(list):
def __init__(self,*x):
#print '\nF:',x
list.__init__(self,x)
def __add__(self, other):
return A(self,other)
def __radd__(self, other):
return A(other,self)
def __mul__(self, other):
return M(self,other)
def __rmul__(self, other):
return M(other,self)
def __repr__(self):
return str(self[0])
def __order__(self):
for i in self:
if isinstance(i,A) \
or isinstance(i,M):
i.__order__()
self.sort()

class A(F):
def __init__(self, *x):
#print '\nA:',x
list.__init__(self, x)
def __repr__(self):
self.__order__()
return "+".join([str(x) for x in self])

class M(F):
def __init__(self,*x):
#print '\nM:',x
list.__init__(self,x)
def __repr__(self):
self.__order__()
return "*".join([str(x) for x in self])
a = F('a')
b = F('b')
c = F('c')
d = F('d')

print '\n a =', a

print '\n b+a+2 =', b+a+2

print '\n c*b+d*a+2 =', c*b+d*a+2

print '\n 7*a*8*9+b =', 7*a*8*9+b
>>>
a = a

b+a+2 = 2+a+b

c*b+d*a+2 = 2+a*d+b*c

7*a*8*9+b = 9*8*7*a+b <-- reverse sorted digits? >>>

The digits sort in reverse for some strange reason I haven't figured out
yet, but they are grouped together. And expressions of the type a*(c+b)
don't work in this example.

It probably needs some better logic to merge adjacent like groups. I
think the reverse sorting my be a side effect of the nesting that takes
place when the expressions are built.

Having the digits first might be an advantage as you can use a for loop
to add or multiply them until you get to a not digit.

Anyway, interesting stuff. ;-)

Cheers,
Ron


Hi Ron,

I really don't want to discourage you in doing your own CAS but the
stuff I'm working on is already a bit more advanced than my
mono-operational multiplicative algebra ;)

Mixing operators is not really a problem, but one has to make initial
decisions ( e.g about associativity i.e. flattening the parse-tree )
and sub-algebra generation by means of inheritance:
a,b = seq(2,Expr)
type(a+b) <class '__main__.Expr'>
class X(Expr):pass
x,y = seq(2,X)
type(x+y) <class '__main__.X'>

This is not particular hard. It is harder to determine correspondence
rules between operations on different levels. On subalgebras the
operations of the parent algebra are induced. But what happens if one
mixes objects of different algebras that interoperate with each other?
It would be wise to find a unified approach to make distinctive
operations visually distinctive too. Infix operators may be
re-introduced just for convenience ( e.g. if we can assume that all
algebras supporting __mul__ that are relevant in some computation have
certain properties e.g. being associative ).
################################################## ########################

After thinking about M ( or Expr ;) a little more I come up with a
solution of the problem of central elements of an algebra ( at least
the identity element e is always central ) that commute with all other
elements.

Here is my approach:

# Define a subclass of list, that provides the same interface as list
and
# a customized sorting algorithm

import sets

class Factors(list):
def __init__(self,li):
list.__init__(self,li)
self.elems = sets.Set(li) # raw set of factors used in the
__mul__
self._center = () # storing central elements
commuting with
# with all others

def _get_center(self):
return self._center

def _set_center(self,center):
Center = sets.Set(center)
if not Center<=self.elems:
raise ValueError,"Subset required"
else:
self._center = Center

center = property(_get_center, _set_center)

def __add__(self,li):
return Factors(list.__add__(self,li))

def sort(self):
center = list(self.center)
def commutator(x,y):
if isinstance(x,(int,float,long)): # numeral literals
should
return -1 # always commute
if isinstance(y,(int,float,long)):
return 1
if x == y:
return 0
if x in center:
if y in center:
if center.index(x)<center.index(y): # induce an
aritrary
return -1 # order on
central
else: # elements by
center
return 1
else:
return -1
return 0
list.sort(self,commutator)
# Define an associative multiplicative algebra

class M(object):
def __init__(self, name=""):
self.name = name
self.factors = Factors([self]) # implement factor list as
Factors

def _get_center(self):
return self.factors.center

def _set_center(self,center):
self.factors.center = center

center = property(_get_center, _set_center)

def __mul__(self, other):
p = M()
if isinstance(other,M):
other_factors = other.factors
else:
other_factors = Factors([other])
p.factors = self.factors+other_factors
return p

def __rmul__(self,other):
p = M()
p.factors = Factors([other])+self.factors
return p

def __repr__(self):
if self.name:
return self.name
else:
return "*".join([str(x) for x in self.factors])

a,b,c,d = M("a"),M("b"),M("c"),M("d")
y = c*3*a*d*c*b*7*c*d*a
y c*3*a*d*c*b*7*c*d*a
y.center = (c,d)
y.factors.sort()
y

7*3*c*c*c*d*d*a*b*a
Regards,
Kay

Jul 21 '05 #12

P: n/a
Diez B.Roggisch wrote:
I have to admit that I don't understand what you mean with the
'constant parts' of an expression?
From what I percieved of your example it seemed to me that you wanted to

evaluate the constants like 7*9 first, so that an expression like

a * 7 * 9 * b

with variables a,b is evaluated like this:

a * 63 * b

So my suggestion was simply to make the *-operator more precedent when
in between two constants. What I mean with constants here are of course
integer/float literals. The concept of a differing operator precedence
can be extended to arbitray elements when their types are known - which
should be possible when variable values are known at parsing
time.


O.K.
The associativity of __mul__ is trivially fullfilled for the dummy
class M if an additional __eq__ method is defined by comparing factor
lists because those lists are always flat:
I don't care about that, as my approach deosn't use python's built-in parser
- it can't, as that wouldn't allow to re-define operator precedence.


Diez, I try not to care too much about global operator precedence of
builtin infix operators. The hard problems in designing a CAS beyond
Mathematica are related to a bunch of interoperating algebras all
defining their own operations. Finally only local precedences exist
that are characteristic for certain patterns of expressions with a lot
of tangled operators ( e.g. 'geometric algebra' with vector products,
wedge products, inner products, additions and subtractions ). I don't
want a system defining a syntactically extendable language with 10
custom punctuations per module that no one ( at least not me ) can
remind and which looks as awkward as regular expressions.

What you do is to
simply collect the factors as list. But what you need (IMHO) is a parsing
tree (AST) that reflects your desired behaviour by introducing a different
precedence thus that the expression

a * 7 *9 * b

is not evaluated like

((a*7)*9)*b

(which is a tree, and the standard way of evaluationg due to built-in parsers
precedence rules) but as

a*(7*9)*b

which is also a tree.


Yes, but I tend to use __mul__ just for convenience. It is reflecting
an associative and non-commutative operator whereas __add__ is a
convenient way to fix an associative and commutative operator. In an
idealized mathematical interpretation they represent nothing specific
but as language elements they shall be fixed somehow.

For more general operations one may define functional operators e.g.
r_assoc and l_assoc where following (in)equations hold:

l_assoc(a,b,c) == l_assoc(l_assoc(a,b),c)
l_assoc(a,b,c) != l_assoc(a, l_assoc(b,c))

r_assoc(a,b,c) == r_assoc(a,r_assoc(b,c))
r_assoc(a,b,c) != r_assoc(r_assoc(a,b),c)

This kind of pattern can be used to define rules about l_assoc and
r_assoc.

Nevertheless, there is no loss of generality. The system lacks
prevention from deriving some class providing __mul__ and overwrite the
implementation of __mul__ using l_assoc. People may do this on their
own risk.

Kay

Jul 21 '05 #13

P: n/a
Kay Schluehr wrote:

Hi Ron,

I really don't want to discourage you in doing your own CAS but the
stuff I'm working on is already a bit more advanced than my
mono-operational multiplicative algebra ;)
I figured it was, but you offered a puzzle:

"Here might be an interesting puzzle for people who like sorting
algorithms ..."

And asked for suggestions:

"It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?"

So I took you up on it. ;-)
BTW.. Usually when people say "I don't want to discourage...", They
really want or mean the exact oppisite.
This is a organizational problem in my opinion, so the challenge is to
organize the expressions in a way that can be easily manipulated
further. Groupings by operation is one way. As far as inheritance
goes, it's just another way to organize things. And different algebra's
and sub-algebra's are just possible properties of a group. The groups
can easily be customized to have their own behaviors or be created to
represent custom unique operations.

The sort method I'm suggesting here, with examples, is constrained by
the associated properties of the group that is being sorted. Basically,
weather or not it's and associative operation or not. So when a group
is asked to sort, it first asks all it's sub groups to sort, then it
sorts it self if it is an associative group. Ie.. from inner most group
to outer most group but only the associative ones.

Playing with it further I get the following outputs.

( The parenthesis surround a group that is associated to the operation.
This is the same idea/suggestion I first proposed, it's just been
developed a little further along.)
b+a+2 = (2+a+b) <- addition group

a*(b+45+23) = ((68+b)*a) <- addition group within multiply group

a-4-3-7+b = ((a-14)+b) <- sub group within add group

c*b-d*a+2 = (2+((b*c)-(a*d))) <- mults within subs within adds

7*a*8*9+b = ((504*a)+b)

a*(b+c) = ((b+c)*a)

c*3*a*d*c*b*7*c*d*a = (21*a*a*b*c*c*c*d*d)

d*b/c*a = (((b*d)/c)*a)

(d*b)/(c*a) = ((b*d)/(a*c))

d*b-a/e+d+c = (((b*d)-(a/e))+c+d)

a/24/2/b = (a/48/b)

c**b**(4-5) = (c**(b**-1))

(d**a)**(2*b) = ((d**a)**(2*b))

The next step is to be able to convert groups to other groups; an
exponent group to a multiply group; a subtract group to an addition
group with negative prefix's.. and so on.

That would be how expansion and simplifying is done as well as testing
equivalence of equations.

if m*c**2 == m*c*c:
print "Eureka!"

Mixing operators is not really a problem, but one has to make initial
decisions ( e.g about associativity i.e. flattening the parse-tree )
and sub-algebra generation by means of inheritance:
What do you mean by 'sub-algebra generation'?

a,b = seq(2,Expr)
type(a+b)
<class '__main__.Expr'>
class X(Expr):pass
x,y = seq(2,X)
type(x+y)


<class '__main__.X'>

This is not particular hard. It is harder to determine correspondence
rules between operations on different levels. On subalgebras the
operations of the parent algebra are induced. But what happens if one
mixes objects of different algebras that interoperate with each other?
It would be wise to find a unified approach to make distinctive
operations visually distinctive too. Infix operators may be
re-introduced just for convenience ( e.g. if we can assume that all
algebras supporting __mul__ that are relevant in some computation have
certain properties e.g. being associative ).


Different algebras would need to be able to convert themselves to some
common representation. Then they would be able to be mixed with each
other with no problem.

Or an operation on an algebra group could just accept it as a unique
term, and during an expansion process it could convert it self (and it's
members) to the parents type. That would take a little more work, but I
don't see any reason why it would be especially difficult.

Using that methodology, an equation with mixed algebra types could be
expanded as much as possible, then reduced back down again using a
chosen algebra or the one that results in the most concise representation.
################################################## ########################

After thinking about M ( or Expr ;) a little more I come up with a
solution of the problem of central elements of an algebra ( at least
the identity element e is always central ) that commute with all other
elements.
What is a "central" element? I can see it involves a set, but the
context isn't clear.

Here is my approach:

# Define a subclass of list, that provides the same interface as list
and
# a customized sorting algorithm
It's not really that different from what I suggested. And since my
example is based on your first example. It has a lot in common but the
arrangement (organization) is a bit different.

Regards,
Kay

Here's the current version... It now handles more complex equations
including exponents and perenthisized groupings. It is fairly easy to
extend which is one of the advantages of having each operation
associated to a group. It would be interesting to see what other
opperators could be added to it.

Cheers,
Ronald Adam

#Factor - a single element or variable to be opperated on.
class F(list):
Type = 'Factor'
Commutative = False
Symbol = ''
Numerals = (int,float,long)
def __init__(self,*x):
list.__init__(self,x)
def __add__(self, other):
if isinstance(self,A):
self.append(other)
return self
return A(self,other)
def __radd__(self, other):
if isinstance(self,A):
self.append(other)
return self
return A(other,self)
def __sub__(self, other):
if isinstance(self,S):
self.append(other)
return self
return S(self,other)
def __rsub__(self, other):
if isinstance(self,S):
self.append(other)
return self
return S(other,self)
def __mul__(self, other):
if isinstance(self,M):
self.append(other)
return self
return M(self,other)
def __rmul__(self, other):
if isinstance(self,M):
self.append(other)
return self
return M(other,self)
def __div__(self, other):
if isinstance(self,D):
self.append(other)
return self
return D(self,other)
def __rdiv__(self, other):
if isinstance(self,D):
self.append(other)
return self
return D(other,self)
def __pow__(self, other):
return P(self,other)
def __rpow__(self, other):
return P(other,self)
def __repr__(self):
self._order_()
if self.Type == 'Operator':
self._simplify_()
Pleft,Pright = '(',')'
else:
Pleft,Pright = '',''
return Pleft+self.Symbol.join([str(x) for x in self])+Pright
def _order_(self):
for i in self:
if hasattr(i, 'Commutative'):
if i.Commutative:
i._order_()
if self.Commutative:
self.sort()
## Operator classes durived from the Factor class.
#
# These group factors and other operator groups
# together in a group with a common opperator.

# Add
class A(F):
Type = 'Operator'
Commutative = True
Symbol = '+'
def _simplify_(self):
#Was sorted first so all numerals shuold
#be to the left.
while ( len(self)>1
and isinstance(self[0],self.Numerals)
and isinstance(self[1],self.Numerals) ):
self[0:2]=[self[0]+self[1]]

# Subtract
class S(F):
Type = 'Operator'
Commutative = False
Symbol = '-'
def _simplify_(self):
while ( isinstance(self[0],self.Numerals)
and isinstance(self[1],self.Numerals) ):
self[0:2] = [self[0]-self[1]]
i = 1
while i<len(self)-1:
if ( isinstance(self[i],self.Numerals)
and isinstance(self[i+1],self.Numerals) ):
self[i:i+2]=[self[i]+self[i+1]]
else:
i += 1

# Multipy
class M(F):
Type = 'Operator'
Commutative = True
Symbol = '*'
def _simplify_(self):
# Was sorted first so all ints should
# be to the left.
while ( len(self)>1
and isinstance(self[0],self.Numerals)
and isinstance(self[1],self.Numerals) ):
self[0:2]=[self[0]*self[1]]

# Divide
class D(F):
Type = 'Operator'
Commutative = False
Symbol = '/'
def _simplify_(self):
while ( isinstance(self[0],self.Numerals)
and isinstance(self[1],self.Numerals) ):
self[0:2] = [self[0]/self[1]]
i = 1
while i<len(self)-1:
if ( isinstance(self[i],self.Numerals)
and isinstance(self[i+1],self.Numerals) ):
self[i:i+2]=[self[i]*self[i+1]]
else:
i += 1

# Power
class P(F):
Type = 'Operator'
Commutative = False
Symbol = '**'
def _simplify_(self):
# Todo
pass

a = F('a')
b = F('b')
c = F('c')
d = F('d')
e = F('e')

print '\n b+a+2 =', b+a+2

print '\n a*(b+45+23) = ', a*(b+45+23)

print '\n a-4-3-7+b = ', a-4-3-7+b

print '\n c*b-d*a+2 =', c*b-d*a+2

print '\n 7*a*8*9+b =', 7*a*8*9+b

print '\n a*(b+c) =', a*(b+c)

print '\n c*3*a*d*c*b*7*c*d*a =', c*3*a*d*c*b*7*c*d*a

print '\n d*b/c*a =', d*b/c*a

print '\n (d*b)/(c*a) =', (d*b)/(c*a)

print '\n d*b-a/e+d+c =', d*b-a/e+d+c

print '\n a/24/2/b =', a/24/2/b

print '\n c**b**(4-5) =', c**b**(4-5)

print '\n (d**a)**(2*b) =', (d**a)**(2*b)



Jul 21 '05 #14

P: n/a
Ron Adam wrote:
Kay Schluehr wrote:

Hi Ron,

I really don't want to discourage you in doing your own CAS but the
stuff I'm working on is already a bit more advanced than my
mono-operational multiplicative algebra ;)
I figured it was, but you offered a puzzle:

"Here might be an interesting puzzle for people who like sorting
algorithms ..."

And asked for suggestions:

"It would be interesting to examine some sorting algorithms on factor
lists with constrained item transpositions. Any suggestions?"

So I took you up on it. ;-)
BTW.. Usually when people say "I don't want to discourage...", They
really want or mean the exact oppisite.


Yes, but taken some renitence into account they will provoke the
opposite. Old game theoretic wisdoms ;)
This is a organizational problem in my opinion, so the challenge is to
organize the expressions in a way that can be easily manipulated
further. Groupings by operation is one way. As far as inheritance
goes, it's just another way to organize things. And different algebra's
and sub-algebra's are just possible properties of a group. The groups
can easily be customized to have their own behaviors or be created to
represent custom unique operations.

The sort method I'm suggesting here, with examples, is constrained by
the associated properties of the group that is being sorted. Basically,
weather or not it's and associative operation or not. So when a group
is asked to sort, it first asks all it's sub groups to sort, then it
sorts it self if it is an associative group. Ie.. from inner most group
to outer most group but only the associative ones.
But you seem to fix behaviour together with an operation i.e. declaring
that __mul__ is commutative. But in a general case you might have
elements that commute, others that anti-commute ( i.e. a*b = -b*a ) and
again others where no special rule is provided i.e. they simply don't
commute.

But much worse than this the definition of the operations __add__,
__mul__ etc. use names of subclasses A,D explicitely(!) what means that
the framework can't be extended by inheritance of A,D,M etc. This is
not only bad OO style but customizing operations ( i.e. making __mul__
right associative ) for certain classes is prevented this way. One
really has to assume a global behaviour fixed once as a class
attribute.

Playing with it further I get the following outputs.

( The parenthesis surround a group that is associated to the operation.
This is the same idea/suggestion I first proposed, it's just been
developed a little further along.)
b+a+2 = (2+a+b) <- addition group

a*(b+45+23) = ((68+b)*a) <- addition group within multiply group

a-4-3-7+b = ((a-14)+b) <- sub group within add group

c*b-d*a+2 = (2+((b*c)-(a*d))) <- mults within subs within adds

7*a*8*9+b = ((504*a)+b)

a*(b+c) = ((b+c)*a)

c*3*a*d*c*b*7*c*d*a = (21*a*a*b*c*c*c*d*d)
I still don't see how you distinguish between factors that might
commute and others that don't. I don't want a and b commute but c and d
with all other elements.

d*b/c*a = (((b*d)/c)*a)

(d*b)/(c*a) = ((b*d)/(a*c))

d*b-a/e+d+c = (((b*d)-(a/e))+c+d)

a/24/2/b = (a/48/b)

c**b**(4-5) = (c**(b**-1))

(d**a)**(2*b) = ((d**a)**(2*b))
If you have fun with those identities you might like to find
simplifications for those expressions too:

a*0 -> 0
a*1 -> a
1/a/b -> b/a
a+b+a -> 2*a+b
a/a -> 1
a**1 -> a

etc.
The next step is to be able to convert groups to other groups; an
exponent group to a multiply group; a subtract group to an addition
group with negative prefix's.. and so on.

That would be how expansion and simplifying is done as well as testing
equivalence of equations.

if m*c**2 == m*c*c:
print "Eureka!"

Mixing operators is not really a problem, but one has to make initial
decisions ( e.g about associativity i.e. flattening the parse-tree )
and sub-algebra generation by means of inheritance:
What do you mean by 'sub-algebra generation'?


Partially what I described in the subsequent example: the target of the
addition of two elements x,y of X is again in X. This is not obvious if
one takes an arbitrary nonempty subset X of Expr.
>a,b = seq(2,Expr)
>type(a+b)


<class '__main__.Expr'>
>class X(Expr):pass
>x,y = seq(2,X)
>type(x+y)


<class '__main__.X'>

This is not particular hard. It is harder to determine correspondence
rules between operations on different levels. On subalgebras the
operations of the parent algebra are induced. But what happens if one
mixes objects of different algebras that interoperate with each other?
It would be wise to find a unified approach to make distinctive
operations visually distinctive too. Infix operators may be
re-introduced just for convenience ( e.g. if we can assume that all
algebras supporting __mul__ that are relevant in some computation have
certain properties e.g. being associative ).


Different algebras would need to be able to convert themselves to some
common representation. Then they would be able to be mixed with each
other with no problem.


Well, it is a problem not only of representation. You might have three
algebras A,B,C each providing a different multiplication operator and
also interoperation capabilities:

A*B = B*A may hold but (A,*) is not associative and neither A nor B
interoperates with C i.e. an operation C*A or C*B is not defined.

Or an operation on an algebra group could just accept it as a unique
term, and during an expansion process it could convert it self (and it's
members) to the parents type. That would take a little more work, but I
don't see any reason why it would be especially difficult.

Using that methodology, an equation with mixed algebra types could be
expanded as much as possible, then reduced back down again using a
chosen algebra or the one that results in the most concise representation.
Maybe you should simply do that.
################################################## ########################

After thinking about M ( or Expr ;) a little more I come up with a
solution of the problem of central elements of an algebra ( at least
the identity element e is always central ) that commute with all other
elements.


What is a "central" element? I can see it involves a set, but the
context isn't clear.


"Central" elements are exactly those that commute with all other
elements. In In abelian groups they constitute the groups itself. In
non-abelian groups they are subgroups ( the center always exist and is
contains at least the unit element ). Since each group has a center one
can make general assertions without considering elements individually.
It is a common pattern of reasoning to abstract from concrete elements
and rely on properties of classes of elements.

Kay

Jul 21 '05 #15

P: n/a
Kay Schluehr wrote:
Ron Adam wrote:
Kay Schluehr wrote:
BTW.. Usually when people say "I don't want to discourage...", They
really want or mean the exact oppisite.
Yes, but taken some renitence into account they will provoke the
opposite. Old game theoretic wisdoms ;)


True.. but I think it's not predictable which response you will get
from an individual you aren't familiar with. I prefer positive
reinforcement over negative provocation myself. :-)

But you seem to fix behaviour together with an operation i.e.
declaring that __mul__ is commutative. But in a general case you
might have elements that commute, others that anti-commute ( i.e. a*b
= -b*a ) and again others where no special rule is provided i.e. they
simply don't commute.

But much worse than this the definition of the operations __add__,
__mul__ etc. use names of subclasses A,D explicitely(!) what means
that the framework can't be extended by inheritance of A,D,M etc.
This is not only bad OO style but customizing operations ( i.e.
making __mul__ right associative ) for certain classes is prevented
this way. One really has to assume a global behaviour fixed once as a
class attribute.
I don't know if it's bad OO style because I chose a flatter model.
Your original question wasn't "what would be the best class structure to
use where different algebra's may be used". It was how can sorting be
done to an expression with constraints. And you gave an example which
set __mul__ as associative as well.

So this is a different problem. No use trying to point that what I did
doesn't fit this new problem, it wasn't suppose to. ;-)

I'm not sure what the best class structure would be. With the current
example, I would need to copy and edit F and it's associated sub
class's to create a second algebra type, F2, A2, M2.. etc. Not the best
solution to this additional problem which is what you are pointing out I
believe.

So... We have factors (objects), groups (expressions), and algebras
(rules), that need to be organized into a class structure that can
be extended easily.

Does that describe this new problem adequately? I'm not sure what the
best, or possible good solutions would be at the moment. I'll have to
think about it a bit.

c*3*a*d*c*b*7*c*d*a = (21*a*a*b*c*c*c*d*d)

I still don't see how you distinguish between factors that might
commute and others that don't. I don't want a and b commute but c and
d with all other elements.


In my example factors don't commute. They are just units, however
factors within a group unit may commute because a group is allowed to
commute factors if the operation the group is associated to is commutable.

If you have fun with those identities you might like to find
simplifications for those expressions too:

a*0 -> 0 a*1 -> a 1/a/b -> b/a a+b+a -> 2*a+b a/a -> 1 a**1 ->
a

etc.


Already did a few of those. Some of these involve changing a group into
a different group which was a bit of a challenge since an instance can't
magically change itself into another type of instance, so the parent
group has to request the sub-group to return a simplified or expanded
instance, then the parent can replace the group with the new returned
instance.

a*a*a -> a**3 change from a M group to a P group.
a*0 -> 0 change from a M group to an integer.
a*1 -> a change from a M group to a F unit.
a+b+a -> 2*a+b change a A subgroup to a M group.
a/a -> change a D group to an integer.
a**1 -> change a P group to a M group to a F unit.

Some of those would be done in the simplify method of the group. I've
added an expand method and gotten it to work on some things also.

a*b**3 -> a*b*b*b
c*4 -> c+c+c+c

What do you mean by 'sub-algebra generation'?


Partially what I described in the subsequent example: the target of
the addition of two elements x,y of X is again in X. This is not
obvious if one takes an arbitrary nonempty subset X of Expr.


Would that be similar to the simultaneous equation below?

z = x+y <- term x+y is z
x = a*z+b <- z is in term x
x = a(x+y)+b <- x is again in x (?)

I think this would be...
x, y = F('x'), F('y')
z = x+y
x = a*z+b
x

(((x+y)*a)+b)

This wouldn't actually solve for x since it doesn't take into account
the left side of the = in the equation. And it would need an eval
method to actually evaluated it. eval(str(expr)) does work if all the
factors are given values first.
Cheers,
Ron

Jul 21 '05 #16

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