MV logics and Gröbner bases

1. MV logics and Gröbner bases History of Logic in Serbia, Belgrade, 14-15. 06. 2010.

2. Motivation ,,History of Logic in Serbia” History of Logic in Serbia Recent interests: Gröbner bases Search: Logic+Gröbner bases→MVL+GB The talk • is missing the topic of the conference, • has no original results, • relies on few references, etc.

3. Boolean algebra ↔ Boolean ring

4. A polynomial model for propositional logic

5. Ideals

6. A polynomial model for propositional logic

7. Ideals This model cannot be extended to p-valued modal logics, but something can be done. We map propositional variables to polynomial variables and logic operations (connectives) to polynomial operations using interpolation formula.

8. GB

9. Additional properties • All ideals are radical ideals, i.e. each is equal to its radical. • In Boolean case, all ideals are principal.

10. Kleene’s and Łukasiewicz’s three-valued logic

11. Kleene’s three-valued logic

12. Łukasiewicz’s three-valued logic

13. Polynomial translation of the logical connectives

14. Tautological consequence

15. Connecting “To be a tautological consequence” with an ideal membership

16. The main theorem adapted to knowledge extraction in RBES

17. Backward reasoning in rule based expert systems whose underlying logic is classic Boolean logic

19. Shortcuts and RBES • RBES rule based expert systems • CAS computer algebra system

20. RBES

21. RBES

22. RBES

23. RBES

25. Abstarct 2008

26. 2009

27. Abstarct 2009

29. Reference