The document discusses: 1) The import and export of formal mathematical content between the Hets system and OMDoc format. 2) How concepts such as sorts, constants, functions, and theories are translated between the CASL specification language used in Hets and the OMDoc format. 3) Challenges in the translation like dealing with overloading and subsorting are addressed.

1. The OMDoc Import/Export of Hets
Ewaryst Schulz
DFKI Bremen, Germany
http://www.informatik.uni-bremen.de/~ewaryst
ewaryst.schulz@dfki.de
Conferences on Intelligent Computer Mathematics 2010
Content Math Training Camp
Paris, France
7th July 2010
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

2. The Hets System
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

3. The Hets System
Other Systems
OMDoc
OMDoc-based Services
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

4. Hets Resources
This Document:
http://www.informatik.uni-bremen.de/~ewaryst/CMTC2010.pdf
Hets:
http://www.informatik.uni-bremen.de/agbkb/forschung/
formal_methods/CoFI/hets/
Hets Library:
https://svn-agbkb.informatik.uni-bremen.de/Hets-lib/trunk/
Hets OMDoc Content Dictionaries:
https://svn-agbkb.informatik.uni-bremen.de/Hets-OMDoc/
trunk/ContentDictionaries/
CASL:
http://www.informatik.uni-bremen.de/cofi/wiki/
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

5. CASL Specification
library Basic/Algebra I
spec Monoid =
sort Elem
ops e : Elem;
∗ : Elem × Elem → Elem, assoc, unit e
spec CommutativeMonoid = Monoid
then op ∗ : Elem × Elem → Elem, comm
spec Group = Monoid
then ∀ x : Elem • ∃ x’ : Elem • x’ ∗ x = e %(inv Group)%
...
Source: https://svn-agbkb.informatik.uni-bremen.de/Hets-lib/trunk/Basic/Algebra_I.casl
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

6. Development Graph
Development Graph of Algebra Library
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

7. OMDoc Translation
<omdoc v e r s i o n=” 1 . 6 ” name=” B a s i c / A l g e b r a I ”>
<t h e o r y name=” Monoid ” meta=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc ? c a s l ”>
<c o n s t a n t name=” Elem ” r o l e=” t y p e ”> y p e>
<t
<OMOBJ >
< OMS b a s e=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc”
module=” c a s l ” name=” s o r t ” />
</OMOBJ </ t y p e> c o n s t a n t>
> </
<c o n s t a n t name=” ∗ ” r o l e=” o b j e c t ”>
<t y p e>
<OMOBJ >
<OMA >
< OMS b a s e=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc”
module=” c a s l ” name=” f u n t y p e ” />
< OMS name=” Elem ” />
< OMS name=” Elem ” />
< OMS name=” Elem ” />
</OMA >
</OMOBJ </ t y p e> c o n s t a n t>
> </
...
</ t h e o r y>
<t h e o r y name=” CommutativeMonoid ” meta=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc ? c a s l ”>
<s t r u c t u r e name=” g n i m p 0 ” from=” ? Monoid ”>
<open name=” Elem ” a s=” Elem ” />
...
</omdoc>
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

8. Same Name Same Thing Principle
spec Monoid =
sort Elem
ops e : Elem;
∗ : Elem × Elem → Elem, assoc, unit e
spec Commutative =
sort Elem
op ∗ : Elem × Elem → Elem, comm
spec CommutativeMonoid = Monoid and Commutative
Elem from Monoid and from Commutative are identified!
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

9. Same Name Same Thing Principle
spec Monoid =
sort Elem
ops e : Elem;
∗ : Elem × Elem → Elem, assoc, unit e
spec Commutative =
sort Elem
op ∗ : Elem × Elem → Elem, comm
spec CommutativeMonoid = Monoid and Commutative
Elem from Monoid and from Commutative are identified!
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

10. Same Name Same Thing Principle
cont.
Corresponding OMDoc fragment
<t h e o r y name=” CommutativeMonoid ” meta=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc ? c a s l ”>
<s t r u c t u r e name=” g n i m p 0 ” from=” ? Monoid ”>
<open name=” Elem ” a s=” Elem ” />
...
</ s t r u c t u r e>
<s t r u c t u r e name=” g n i m p 1 ” from=” ? Commutative ”>
<c o n a s s name=” Elem ”>
<OMOBJ >
<OMS name=” Elem ” />
</OMOBJ >
</ c o n a s s>
...
</ s t r u c t u r e>
</ t h e o r y>
name in open and conass interpreted in source-context of structure
as, OMOBJ interpreted in current context
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

11. Subsorts and Overloading
spec Int =
sorts Nat < Int; Elem
ops 0 : Nat;
+ : Int × Int → Int;
+ : Nat × Nat → Nat;
+ : Elem × Elem → Elem;
∗ : Nat × Int → Int;
∗ : Int × Nat → Int
vars x, y : Elem; n, m : Nat
•x +y =y +x %(commE)%
•n+m=m+n %(commN)%
•n∗m=m∗n %(commMult)%
end
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

12. Subsorts and Overloading cont.
Corresponding OMDoc fragment
<t h e o r y name=” I n t ” meta=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc ? c a s l ”>
...
<c o n s t a n t name=” + ” r o l e=” o b j e c t ”>
<t y p e>
<OMOBJ xmlns:om=” h t t p : //www . openmath . o r g /OpenMath”> . . .
<OMA >
< OMS b a s e=” h t t p : // c d s . omdoc . o r g / l o g i c s / c a s l / c a s l . omdoc”
module=” c a s l ” name=” f u n t y p e ” />
<OMS name=” Elem ” />
<OMS name=” Elem ” />
<OMS name=” Elem ” />
</OMA </OMOBJ </ t y p e> c o n s t a n t>
> > </
<c o n s t a n t name=”%()% o v e r 1 : + ” r o l e=” o b j e c t ”>
<t y p e> . . .</ t y p e> c o n s t a n t>
</
<n o t a t i o n f o r=”??%()% o v e r 1 : + ” r o l e=” c o n s t a n t ”>
<t e x t v a l u e=” + ” />
</ n o t a t i o n>
...
</ t h e o r y>
Encoding of overloaded names
notation stores the original name
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence

13. What else?
If you have further questions such as
How can I use Hets for my project?
How can I integrate my logic in Hets?
Should I use XSLT to translate an OMDoc from logic A to logic B?
How could I design an OMDoc interface for my tool?
I can probably answer them...
The OMDoc Import/Export of Hets German Research Center
Ewaryst Schulz for Artificial Intelligence