Beyond Fuzzy: Parameterized Approximations of Heyting Algebras for Uncertain Knowledge

1. Beyond Fuzzy: Parameterized Approximations of Heyting Algebras for Uncertain Knowledge Zoran Majkic Computer Science Dept. University of Maryland College Park December 2005, Pune

2. Implication-based LogicProgramming Intuitionistic logic based oh Heyting algebras Motivation:Logic for uncertain info • Aproximation for probabilistic information • Approximation for conditional probabilities • Extension of Fuzzy-logic • Common framework for uncertain measure types • Simple probability, closed intervals, couples of closed intervals,.. • Parameterization • Approximations of intuitionistic logic: conj/disj strategies

3. Parameterization:Probabilistic strategies • Latticeoperators: Heyting algebra – positive correlation • Ignorance strategy: • Negative correlation: • Independence strategy: • Mutually-exclusive strategy:

4. Disjunctive strategy Mutually-exclusive Independence Negative correlation Ignorance Positive correlation Conjunctive strategy Positive correlation Ignorance Negative correlation Independence Mutually-exclusive Beyond Fuzzy: Parameterized Approximations of Heyting Algebras for Uncertain Knowledge TopC Confidence-level measure TopB Top Belief measure Lattice ordering join(X,Y) Y Heyting Algebra X meet(X,Y) Simple probability measure BottomB BottomB Bottom

5. Par. algebra: Unique epistemic negation: Strategies: Positive correlation Ignorance/ Negative correlation Independence correlation Mutually-exclusive correlation Disjunction: De Morgan laws Implication: residuum Simple probability:Parameterized Heyting Algebra

6. Unique epistemic negation: Strategies: Positive correlation Ignorance Negative Independence Mutually-exclusive Disjunction: De Morgan laws Implication: residuum Probability interval:belief-based logic Par. algebra:

7. Par. algebra: Unique epistemic negation: Strategies: Disjunction: De Morgan laws Implication: residuum Couple of intervals:confidence-level logic

8. Logic Programming rules: Satisfied by a valuation v when, Residuum-based parameterized implications: Deduction theorem: Modus Ponens Curry Howard isomorphism Deduction theorem: when Logic Programming:Modus Ponens and Deduction theorem

9. Probabilistic clause Satisfied by Pr if Solution: Approximation by the family of parameterized Heyting algebras Example:Approximation of conditional probabilities Problem: with Herbrand base with only 100 atoms we have the very high number of possible worlds:

10. Thank you ! Any question ?