1. Disjunctive Bottom Set and Their Computation Dr Wenjin Lu & Dr Ross King Department of Computer Science, The University of Wales, Aberystwyth {wwl,rdk}@aber.ac.uk

2. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 2 Contents Background and Motivation Disjunctive Bottom Set Computing the Disjunctive Bottom Set Conclusions and Future Work

3. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 3 Background and Motivation Inverse Entailment (IE) [Muggleton 1995] Bottom Set Generation Bottom Set Generalization Hypothesis Selection A typical implementation of IE will be consisting of

4. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 4 Background and Motivation Bottom Set [Muggleton’95, Yamamoto’97] ? Progol, Aleph...HAIL Remove ‘atomic’ Remove ‘single’

5. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 5 Disjunctive Bottom Set We expect a bottom set to be a hypothesis satisfying that It should be a minimal hypothesis, in the sense that none of its proper subset is a hypothesis; It should be a weakest hypothesis, in the sense that it is subsumed by all other hypothesis; It should be complete, in the sense that all hypotheses can be obtained from it; It should be easy to compute. Questions: Existence: Is there such a hypothesis? Uniqueness: Is the hypothesis unique?

6. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 6 Disjunctive Bottom Set Dual Transformation [Yahya 2002] Theorem [Yahya 2002]

7. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 7 Disjunctive Bottom Set Definition:

8. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 8 Disjunctive Bottom Set a b c d e f g h Example:

9. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 9 Computing Disjunctive Bottom Set Hypothesis Selection Function:

10. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 10 Computing Disjunctive Bottom Set Algorithm

11. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 11 Conclusions The disjunctive bottom set seems to meet our requirements, that is It is a minimal hypothesis, in the sense that none of its proper subset is a hypothesis; It is a weakest hypothesis, in the sense that it is subsumed by all other hypothesis; It is complete, in the sense that all hypotheses can be obtained from it; It has low space complexity, as minimal models can be computed in polynomial space complexity given some syntax restriction, such as range-restriction.

12. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 12 Future Work Bottom Set [Muggleton’95, Yamamoto’97] Progol, Aleph...HAIL Remove ‘atomic’ Remove ‘single’ dBot(B,E) ?

13. Dr Wenjin Lu, Ross King: Disjunctive Bottom Set and Their Computation 29-08-2006 AIAI06 13 Thank You!