Applications of Category Theory in Symbolic Computation

1. Applications of Category Theory in Symbolic Computation Alina Andreica, PhD “Babes-Bolyai” University, ICT Department E-mail: alina.andreica@euro.ubbcluj.ro

2. Topics • The Working Framework • Theorema Meta-Definitions for Verifying Simplification Properties over some Algebraic Categories • Conclusions and Future Work

3. The Working Framework • categorical approach: means of defining generic contexts of performing symbolic algorithms, which can be particularized for various domains • Similar to OO meta-classes • Buchberger – Theorema: functors = an ellegant tool for such definitions since they map domains into other domains + advantages in proving

4. Theorema Meta-Definitions for Verifying Simplification Properties over some Algebraic Categories • Practical aims: • formalize in a categorial style the verification of canonical simplifiers (Knuth-Bendix algorithm) • Prove equivalence of simplified forms defined in [And03] which reduce neutral, symmetrical elements etc. over a hierarchy of algebraic structures: semigroup, abelian semigroup, monoid, abelian monoid, group, abelian group, ring, abelian ring, field, abelian field • General aim: • construct a more comprehensive categorical view on expression simplification

5. Theorema Meta-Definitions for Verifying Simplification Properties over some Algebraic Categories • Theoretical basis: definitions of simplification, canonical form, Knuth-Bendix algorithm (Buchgerger, Loos, Algebraic Simplification, 1982) • Theorema useful tools: functors, Simplifier Prover and Predicate Prover • 1. Definitions of algebraic categories underlining the reasoning aspect

6. Theorema Definitions for Operator Properties

7. Theorema Definitions for Algebraic Categories

8. Theorema Meta-Definitions for Verifying Simplification Properties over some Algebraic Categories • 2. Constructive functorial definitions of algebraic categories • 3. Extensions with simplification functions • HierMath.m package [And03] • OForm gives the external form of an expression • FForm gives the internal form of an expression • ElimUnit simplifies an expression by reducing the neutral element • ElimSim simplifies an expression by reducing symmetrical elements

9. Constructive functorial definitions for algebraic categories

10. Constructive functorial definitions for algebraic categories

11. Constructive functorial definitions for algebraic categories

12. Constructive functorial definitions for algebraic categories

13. Functorial extensions with simplification functions

14. Functorial extensions with simplification functions The simplification function over the group category reduces symmetrical and neutral elements (using the principles described in [And03])

15. Functorial extensions with simplification functions The simplification function over the ring category reduces neutral elements in respect with the multiplicative operator, followed by the reduction of symmetrical and neutral elements in respect with the additive operator (using the principles described in [And03])

16. Functorial extensions with simplification functions The simplification function over the field category reduces symmetrical and neutral elements in respect with the multiplicative operator, followed by the reduction of symmetrical and neutral elements in respect with the additive operator (using the principles described in [And03])

17. Theorema Meta-Definitions for Verifying Simplification Properties over some Algebraic Categories • It should be verified that for any defined algebraic category D, and an arbitrary but fixed expression expr over D, exprSD[expr], where SD is the simplification function within the appropriate category • Consequent to any category / domain definition, such a definition style enables testing the equivalence of any expression over that category / domain with a simplified form of the expression

18. Theorema Meta-Definitions for Verifying Simplification Properties over some Algebraic Categories • A categorical framework for proving canonicity is given by Knuth-Bendix algorithm • The definition can be applied for various domains, by providing the simplification function and the set of critical pairs • Ex. Canonicity of Groebner basis algorithm - A functorial and categorial definition of Groebner basis is given by Buchberger (2004; see references)

19. Conclusions and Future Work • categorical approach: advantages in defining generic contexts of performing symbolic algorithms • We defined a set of algebraic categories by using Theorema functors and introducing simplification functions • Such meta-definitions can be used in order to test the equivalence of an expression over a given domain with the form obtained by applying the simplification function to it

20. Conclusions and Future Work • A general context of proving canonicity is proposed by encoding Knuth-Bendix principle into a Theorema meta-definition • Such a definition could be used to prove canonicity of Groebner basis for polynomial domains • We intend to complete the simplification properties proving and to extend this work in order to offer a more extensive approach on expression simplification