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SandBoxTensorProduct

Edit detail for SandBoxTensorProduct revision 4 of 9

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Editor: Bill Page Time: 2009/05/12 13:35:17 GMT-7
Note: tensor product axioms

added:
Demonstrating the axioms of the tensor product:
\begin{axiom}
x:M
y:M
u:M
p:=2*x+3*u
q:=5*x+7*y+11*u
MxM:=TensorProduct(Integer,Symbol,Symbol,M,M);
r:=product(p,q)$MxM
w:= 13*y+17*y+19*u
test( product(p+q,w)$MxM = product(p,w)$MxM + product(q,w)$MxM )
test( product(p,q+w)$MxM = product(p,q)$MxM + product(p,w)$MxM )
test( product(p,23*w)$MxM = 23*product(p,w)$MxM )
test( product(23*p,w)$MxM = 23*product(p,w)$MxM )
\end{axiom}

On Date: Thu, 14 May 2009 21:46:06 +0200 Franz Lehner wrote:

Attached is a prototype for tensor products. It is free modules over commutative rings. There seems to be some confusion with free modules.

FreeModule? is defined in poly.spad with a local category. FreeModuleCat? is defined in xpoly.spad. FreeModule1? is of category FreeModuleCat?, its Rep however is FreeModule?.

spad

)abbrev category TENSORC TensorProductCategory
TensorProductCategory(R:CommutativeRing, M : Module(R), N : Module(R)):Category == Module(R) with
    product: (M, N) -> %

)abbrev category TENSORP TensorProductProperty
TensorProductProperty(R:CommutativeRing, M : Module(R), N : Module(R), _
      MxN : TensorProductCategory(R, M, N), S : Module(R)): Category == with
    eval: (MxN, (M, N) -> S) -> S

)abbrev package TENSORD TensorProduct
TensorProduct(R:CommutativeRing, B1:SetCategory, B2:SetCategory,M1:FreeModuleCat(R,B1),M2:FreeModuleCat(R,B2)): TPcat == TPimp where 
    TPcat == Join(TensorProductCategory(R,M1,M2),FreeModuleCat(R,Product(B1,B2))) 
    TERM1 == Record(k: B1, c: R)
    TERM2 == Record(k: B2, c: R)
    B1xB2 == Product(B1,B2)
    TERM  == Record(k: B1xB2, c: R)
    TPimp == FreeModule1(R,Product(M1,M2)) add 
       -- Representation
       -- Rep == FreeModule1(R,Product(M1,M2))
       Rep == List TERM
       import Rep, TERM1, TERM2, TERM, B1xB2

       -- One should expect the following 3 functions to be inherited from FreeModule1
       -- but there are problems if these are omitted.
       outTerm(r:R, s:B1xB2):OutputForm ==
            r=1  => s::OutputForm
            r::OutputForm * s::OutputForm
       coerce(a:%):OutputForm ==
            empty? (a pretend Rep) => (0$R)::OutputForm
            reduce(_+, reverse_! [outTerm(t.c, t.k) for t in (a pretend Rep)])$List(OutputForm)
       leadingMonomial(x:%):B1xB2 == (x pretend Rep).first.k

       product(x1:M1,x2:M2):% == 
            zero? x1 or zero? x2 => return 0
            x1l:List TERM1 := ListOfTerms x1
            x2l:List TERM2 := ListOfTerms x2
            res:List Rep := [[[makeprod(s1.k,s2.k),s1.c*s2.c ] for s2 in x2l] for s1 in x1l]
            (reduce(concat,res)) pretend %

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/4415200928774662710-25px001.spad using 
      old system compiler.
   TENSORC abbreviates category TensorProductCategory 
------------------------------------------------------------------------
   initializing NRLIB TENSORC for TensorProductCategory 
   compiling into NRLIB TENSORC 

;;;     ***       |TensorProductCategory| REDEFINED
Time: 0 SEC.

   finalizing NRLIB TENSORC 
   Processing TensorProductCategory for Browser database:
--->-->TensorProductCategory((product (% M N))): Not documented!!!!
--->-->TensorProductCategory(constructor): Not documented!!!!
--->-->TensorProductCategory(): Missing Description
------------------------------------------------------------------------
   TensorProductCategory is now explicitly exposed in frame initial 
   TensorProductCategory will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/TENSORC.NRLIB/code

   TENSORP abbreviates category TensorProductProperty 
------------------------------------------------------------------------
   initializing NRLIB TENSORP for TensorProductProperty 
   compiling into NRLIB TENSORP 

;;;     ***       |TensorProductProperty| REDEFINED
Time: 0 SEC.

   finalizing NRLIB TENSORP 
   Processing TensorProductProperty for Browser database:
--->-->TensorProductProperty((eval (S MxN (Mapping S M N)))): Not documented!!!!
--->-->TensorProductProperty(constructor): Not documented!!!!
--->-->TensorProductProperty(): Missing Description
------------------------------------------------------------------------
   TensorProductProperty is now explicitly exposed in frame initial 
   TensorProductProperty will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/TENSORP.NRLIB/code

   TENSORD abbreviates package TensorProduct 
------------------------------------------------------------------------
   initializing NRLIB TENSORD for TensorProduct 
   compiling into NRLIB TENSORD 
   importing List Record(k: Product(B1,B2),c: R)
   importing Record(k: B1,c: R)
   importing Record(k: B2,c: R)
   importing Record(k: Product(B1,B2),c: R)
   importing Product(B1,B2)
   compiling local outTerm : (R,Product(B1,B2)) -> OutputForm
Time: 0.03 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.07 SEC.

   compiling exported leadingMonomial : $ -> Product(B1,B2)
Time: 0 SEC.

   compiling exported product : (M1,M2) -> $
Time: 0.11 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |TensorProduct| REDEFINED

;;;     ***       |TensorProduct| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] not known that (OrderedSet) is of mode (CATEGORY domain (IF (has M1 (Finite)) (IF (has M2 (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has M1 (Monoid)) (IF (has M2 (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has M1 (AbelianMonoid)) (IF (has M2 (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has M1 (CancellationAbelianMonoid)) (IF (has M2 (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has M1 (Group)) (IF (has M2 (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has M1 (AbelianGroup)) (IF (has M2 (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has M1 (OrderedAbelianMonoidSup)) (IF (has M2 (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has M1 (OrderedSet)) (IF (has M2 (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE makeprod ($ M1 M2)) (SIGNATURE selectfirst (M1 $)) (SIGNATURE selectsecond (M2 $)))

   Cumulative Statistics for Constructor TensorProduct
      Time: 0.21 seconds

--------------non extending category----------------------
.. TensorProduct(#1,#2,#3,#4,#5) of cat 
(|Join| (|TensorProductCategory| |#1| |#4| |#5|)
        (|FreeModuleCat| |#1| (|Product| |#2| |#3|)))   has no 
(|FreeModuleCat| |#1| (|Product| |#4| |#5|))    finalizing NRLIB TENSORD 
   Processing TensorProduct for Browser database:
--->-->TensorProduct(): Missing Description
------------------------------------------------------------------------
   TensorProduct is now explicitly exposed in frame initial 
   TensorProduct will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/TENSORD.NRLIB/code

axiom

M:=FreeModule1(Integer,Symbol)

(1)

Type: Domain

axiom

N:=FreeModule1(Integer,Symbol)

(2)

Type: Domain

axiom

a1:='a1::M

(3)

Type: FreeModule1?(Integer,Symbol)

axiom

a2:='a2::M

(4)

Type: FreeModule1?(Integer,Symbol)

axiom

b1:='b1::N

(5)

Type: FreeModule1?(Integer,Symbol)

axiom

b2:='b2::N

(6)

Type: FreeModule1?(Integer,Symbol)

axiom

MxN:=TensorProduct(Integer,Symbol,Symbol,M,N);

Type: Domain

axiom

t:=product(a1+a2,b1+b2)$MxN;

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))

axiom

(7)

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))

The code seems to work partially, at least elements are created, but most of the other functions are not accessible. Eq:

axiom

leadingMonomial t

(8)

Type: Product(Symbol,Symbol)

axiom

numberOfMonomials t

(9)

Type: PositiveInteger?

The last output shows that t was really created and the above errors are due to malfunction of the inheritance mechanism.

Questions:

The missing functions are probably due to the mixup of categories. Why does the compiler not complain?
Is there a way to extract the basis automatically from FreeModule?? Typing TensorProduct?(R,B1,B2,M,N) every time is tedious. That would need functions returning domains I guess.

Is it difficult to implement a rep/per mechanism? The current magic is rather mysterious, moreover the error messages of the compiler are somewhat cryptic. I had a hard time interpreting the message:

     >> Apparent user error:
         NoValueMode
             is an unknown mode

   before I split::

     reduce(concat,[[[makeprod(s1.k,s2.k),s1.c*s2.c ]@TERM for s2 in x2l] for s1 in x1l]) pretend %

   into two lines::

     res:List Rep := [[[makeprod(s1.k,s2.k),s1.c*s2.c ]@TERM for s2 in x2l] for s1 in x1l]
     (reduce(concat,res)) pretend %

After TensorProduct? ... == FreeModule1?(R,Product(M1,M2)) add ..., is it better to use:

     Rep == FreeModule1(R,Product(M1,M2)) 

   or rather::

     Rep == List TERM 

   as I did?

On Date: Fri, 15 May 2009 03:23:33 +0200 (CEST) Waldek Hebisch wrote:

The following seem to work OK:

axiom

)clear completely

   All user variables and function definitions have been cleared.
   All )browse facility databases have been cleared.
   Internally cached functions and constructors have been cleared.
   )clear completely is finished.

spad

)abbrev package TENSORD TensorProduct
TensorProduct(R : CommutativeRing, B1 : OrderedSet, B2 : OrderedSet, _
    M1 : FreeModuleCat(R, B1), M2 : FreeModuleCat(R, B2)): TPcat == TPimp where 
    TPcat == Join(TensorProductCategory(R,M1,M2),FreeModuleCat(R,Product(B1,B2))) 
    TERM1 == Record(k: B1, c: R)
    TERM2 == Record(k: B2, c: R)
    B1xB2 == Product(B1,B2)
    TERM  == Record(k: B1xB2, c: R)
    TPimp == FreeModule1(R, Product(B1, B2)) add 
       import Rep, TERM1, TERM2, TERM, B1xB2

       product(x1:M1,x2:M2):% ==
           zero? x1 or zero? x2 => return 0
           x1l:List TERM1 := ListOfTerms x1
           x2l:List TERM2 := ListOfTerms x2
           res : % := 0
           for s1 in x1l repeat
               for s2 in x2l repeat
                   res := res + monom(makeprod(s1.k, s2.k), s1.c*s2.c)
           res

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/9103743063781848243-25px006.spad using 
      old system compiler.
   Illegal NRLIB 
   TENSORD.NRLIB claims that its constructor name is the package 
      TensorProduct but TensorProduct is already known to be the for 
      domain TENSORD .
   TENSORD abbreviates package TensorProduct 
------------------------------------------------------------------------
   initializing NRLIB TENSORD for TensorProduct 
   compiling into NRLIB TENSORD 
   importing Rep
   importing Record(k: B1,c: R)
   importing Record(k: B2,c: R)
   importing Record(k: Product(B1,B2),c: R)
   importing Product(B1,B2)
   compiling exported product : (M1,M2) -> $
Time: 0.03 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |TensorProduct| REDEFINED

;;;     ***       |TensorProduct| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] not known that (OrderedSet) is of mode (CATEGORY domain (IF (has B1 (Finite)) (IF (has B2 (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has B1 (Monoid)) (IF (has B2 (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has B1 (AbelianMonoid)) (IF (has B2 (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has B1 (CancellationAbelianMonoid)) (IF (has B2 (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has B1 (Group)) (IF (has B2 (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has B1 (AbelianGroup)) (IF (has B2 (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has B1 (OrderedAbelianMonoidSup)) (IF (has B2 (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has B1 (OrderedSet)) (IF (has B2 (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE makeprod ($ B1 B2)) (SIGNATURE selectfirst (B1 $)) (SIGNATURE selectsecond (B2 $)))

   Cumulative Statistics for Constructor TensorProduct
      Time: 0.03 seconds

--------------non extending category----------------------
.. TensorProduct(#1,#2,#3,#4,#5) of cat 
(|Join| (|TensorProductCategory| |#1| |#4| |#5|)
        (|FreeModuleCat| |#1| (|Product| |#2| |#3|)))    has no  ?*? : (Product(#2,#3),#1) -> % 
   finalizing NRLIB TENSORD 
   Processing TensorProduct for Browser database:
--->/var/zope2/var/LatexWiki/4415200928774662710-25px001.spad-->TensorProduct(): Missing Description
------------------------------------------------------------------------
   TensorProduct is now explicitly exposed in frame initial 
   TensorProduct will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/TENSORD.NRLIB/code

Note that both B1 and B2 is required to be OrderedSet?, and I use just FreeModule1?(R, Product(B1, B2))

Note that the version I wrote does not use pretend. My version may be quite slow -- if this is a problem than one can build a list using cons and add pretend. However, list must be created in correct order, otherwise functions from FreeMonoid? would work incorrectly.

axiom

M:=FreeModule1(Integer,Symbol)

(10)

Type: Domain

axiom

N:=FreeModule1(Integer,Symbol)

(11)

Type: Domain

axiom

a1:='a1::M

(12)

Type: FreeModule1?(Integer,Symbol)

axiom

a2:='a2::M

(13)

Type: FreeModule1?(Integer,Symbol)

axiom

b1:='b1::N

(14)

Type: FreeModule1?(Integer,Symbol)

axiom

b2:='b2::N

(15)

Type: FreeModule1?(Integer,Symbol)

axiom

MxN:=TensorProduct(Integer,Symbol,Symbol,M,N);

Type: Domain

axiom

t:=product(2*a1+3*a2,5*b1+7*b2)$MxN;

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))

axiom

(16)

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))

axiom

leadingMonomial t

(17)

Type: Product(Symbol,Symbol)

axiom

numberOfMonomials t

(18)

Type: PositiveInteger?

Demonstrating the axioms of the tensor product:

axiom

x:M

Type: Void

axiom

y:M

Type: Void

axiom

u:M

Type: Void

axiom

p:=2*x+3*u

(19)

Type: FreeModule1?(Integer,Symbol)

axiom

q:=5*x+7*y+11*u

(20)

Type: FreeModule1?(Integer,Symbol)

axiom

MxM:=TensorProduct(Integer,Symbol,Symbol,M,M);

Type: Domain

axiom

r:=product(p,q)$MxM

(21)

Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))

axiom

w:= 13*y+17*y+19*u

(22)

Type: FreeModule1?(Integer,Symbol)

axiom

test( product(p+q,w)$MxM = product(p,w)$MxM + product(q,w)$MxM )

(23)

Type: Boolean

axiom

test( product(p,q+w)$MxM = product(p,q)$MxM + product(p,w)$MxM )

(24)

Type: Boolean

axiom

test( product(p,23*w)$MxM = 23*product(p,w)$MxM )

(25)

Type: Boolean

axiom

test( product(23*p,w)$MxM = 23*product(p,w)$MxM )

(26)

Type: Boolean