Disjunctive Bottom Set and Their Computation

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. Background and Motivation Disjunctive Bottom Set Computing the Disjunctive Bottom Set Conclusions and Future Work Contents

3. Background and Motivation Inverse Entailment (IE) [Muggleton 1995] A typical implementation of IE will be consisting of • Bottom Set Generation • Bottom Set Generalization • Hypothesis Selection

4. Background and Motivation ? Remove ‘single’ Bottom Set [Muggleton’95, Yamamoto’97] Remove ‘atomic’ Progol, Aleph... HAIL

5. 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. Disjunctive Bottom Set Questions: • Existence: Is there such a hypothesis? • Uniqueness: Is the hypothesis unique?

6. Disjunctive Bottom Set Dual Transformation [Yahya 2002] Theorem [Yahya 2002]

7. Disjunctive Bottom Set Definition:

8. a b d c g e f h Disjunctive Bottom Set Example:

9. Computing Disjunctive Bottom Set Hypothesis Selection Function:

10. Computing Disjunctive Bottom Set Algorithm

11. 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. Conclusions

12. Future Work Remove ‘single’ dBot(B,E) Bottom Set [Muggleton’95, Yamamoto’97] ? Remove ‘atomic’ Progol, Aleph... HAIL

13. Thank You!