MV logics and Gröbner bases

History of Logic in Serbia,

Belgrade, 14-15. 06. 2010.

,,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.

Motivation

Boolean algebra ↔ Boolean ring

A polynomial model for propositional logic

Ideals

A polynomial model for propositional logic

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.

Ideals

GB

All ideals are radical ideals, i.e. each is equal to its radical.

In Boolean case, all ideals are principal.

Additional properties

Kleene’s and Łukasiewicz’s three-valued logic

Kleene’s three-valued logic

Łukasiewicz’s three-valued logic

Polynomial translation of the logical connectives

Tautological consequence

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

The main theorem adapted to knowledge extraction in RBES

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

RBES rule based expert systems CAS computer algebra system

Shortcuts and RBES

RBES

RBES

RBES

RBES

Abstarct 2008

2009

Abstarct 2009

Reference