Improving Recovery for Belief Bases

1. Improving RecoveryforBelief Bases Frances L. Johnson & Stuart C. Shapiro Department of Computer Science and Engineering, Center for Multisource Information Fusion and Center for Cognitive Science University at Buffalo, The State University of New York 201 Bell Hall, Buffalo, NY 14260-2000 {flj|shapiro}@cse.buffalo.edu http://www.cse.buffalo.edu/~{flj|shapiro}/

2. RecoveryInformal Definition • A property of a set of belief change operations • Given a set of beliefs B with p ∈ B • Recovery holds if • whenever p is removed from B • then added back in • everything that was originally in B is also back • (is recovered) F. L. Johnson & S. C. Shapiro

3. Two ApproachestoSets of Beliefs • Foundations uses a belief base B • Coherence uses a belief set K F. L. Johnson & S. C. Shapiro

4. Foundations Approach • Belief base B is finite set of beliefs • with independent justification • Other believed propositions • are derived from • and depend on beliefs in B F. L. Johnson & S. C. Shapiro

5. Coherence Approach • Belief set K = Cn(K) • where Cn(Φ) = {p | Φ├ p} • All beliefs in K have independent • standing • justification F. L. Johnson & S. C. Shapiro

6. Recoveryfor Belief Sets • Should hold for any reasonable set of belief change operations • Note, for every q ∈ K for every p p → q ∈ K since q├ p → q • If remove p, p → q stays So if then add p back, q is recovered F. L. Johnson & S. C. Shapiro

7. Question • For what sets of belief base belief change operations does recovery hold? F. L. Johnson & S. C. Shapiro

8. Recovery is Controversial • But, informally, if you remove something then put it back, the result should be a noop F. L. Johnson & S. C. Shapiro

9. Operations on K and B • Contraction by p: p ∉ Cn(K~p) p ∉ Cn(B~p) • Consolidation: B!├ • Expansion by p: K+p =def Cn(K  {p}) B+p =def B  {p} F. L. Johnson & S. C. Shapiro

10. Recovery Terminology • Recovery: K ⊆ (K~p)+p • Base Recovery: B ⊆ Cn((B~p)+p) • Strict Base Recovery: B ⊆ (B~p)+p • More demanding than Base Recovery • The old base is recovered in the base, itself • Not just its deductive closure F. L. Johnson & S. C. Shapiro

11. When Does Base Recovery Hold? Base Recovery: B ⊆ Cn((B~p)+p) • If B├ ¬p holds trivially • If B\{p} ├ p holds • If B\{p}├ p might not hold / F. L. Johnson & S. C. Shapiro

12. Why Does Recovery Fail? / • If B\{p}├ p,then possibly B ⊆ Cn((B~p)+p) • Example: • B = {q, qp,p}, • B~p = {qp} • (B~p)+p = {qp,p} … ├ q • Take out beliefs that imply p. • Expansion by p doesn’t put them back. • Recovery depends on coordination of contraction and expansion. / F. L. Johnson & S. C. Shapiro

13. Solution(Informal) • If don’t forget beliefs lost during contraction can bring them back later. • Two approaches: • Reconsideration • Liberation F. L. Johnson & S. C. Shapiro

14. Reconsideration†Uses Knowledge State • KS = B,B,≥  • B is the current base • Bis the union of current and all past bases • ≥ is epistemic ordering of beliefs in B † Johnson & Shapiro, AAAI-05 F. L. Johnson & S. C. Shapiro

15. !  ! !   Operations on KS • Contraction by p: KS~p =def B~p,B,≥  • Reconsideration†: KS =def B!,B,≥  • Expansion by p: KS+p =def B+p ,B+p,≥’  • Optimized-addition of KS by p: KS+ p =def (KS+ p) Cf. Semi-Revision by p : B?p=def(B+p)! F. L. Johnson & S. C. Shapiro

16. ! !   Revisit non-Recovery Example ({q, qp, p} , {q, qp, p} ,≥ ~p) + p = {qp} , {q, qp, p} ,≥ + p = {q, qp, p} , {q, qp, p} ,≥ F. L. Johnson & S. C. Shapiro

17. ! !   Shorthand Notation When B , ≥ and ≥’ are known or obvious, write KS+ p as B + p write KS~p =defB~p,B,≥  as B~p F. L. Johnson & S. C. Shapiro

18. ! !   Optimized-Recovery (OR)Holds • Optimized Base Recovery : B ⊆ Cn((B~p) + p) • Optimized Strict Base Recovery: B ⊆ (B~p) + p • OR (both versions) holds, except when ¬p ∈ Cn(B) F. L. Johnson & S. C. Shapiro

19. Liberation†Uses a Sequence •  = p1,…,pn • (, p): the maximal subsequence of  s.t.(,p) ├p • () = (, ) • K = Cn(()) • K ~ p = Cn((,p)) • K+p = Cn(K  {p}) (traditional) / † Booth, Chopra, Ghose and Meyer 2003 F. L. Johnson & S. C. Shapiro

20. Liberation Example •  = qp, q, p, r, r ¬p • K = Cn(()) = Cn({qp, q, p, r}) • K ~p = Cn((,p) = Cn({qp, r, r ¬p}) • (note liberation of r  ¬p) • (K ~ p)+p = Cn({qp, r, r ¬p}  {p})├ q • Adheres to Recovery in this example (trivially) F. L. Johnson & S. C. Shapiro

21. Liberation Recovery • Liberation Recovery (LR): K ⊆ (K ~ p) + p • Liberation fails to adhere to LR in some cases where\{p}├ p • Example: given  = qp, q, p • () = {qp, q, p} • (,p) = {qp} • (K~p)+p = Cn({qp}{p} ) ├ q / F. L. Johnson & S. C. Shapiro

22. -Recovery (R) • K = Cn(()) •  + p = p, p1,…,pn if  = p1,…,pn • -Recovery: K ⊆ K(+¬p)+p • Holds except when ¬p ∈ K F. L. Johnson & S. C. Shapiro

23. !  ConclusionsRecovery Adherence / / F. L. Johnson & S. C. Shapiro

24. References Alchourrón, C. E.; Gärdenfors, P.; and Makinson, D. 1985. On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic 50(2):510–530. Booth, R.;Chopra, S.;Ghose, A.; and Meyer,T. 2003. Belief liberation (and retraction). In Proceedings of the Ninth Conference on Theoretical Aspects of Rationality and Knowledge (TARK’03), 159-172. Hansson, S. O. 1991. Belief Base Dynamics. Ph.D. Dissertation, Uppsala University. Hansson, S. O. 1997. Semi-revision. Journal of Applied Non-Classical Logic 7:151–175. Hansson, S. O. 1999. A Textbook of Belief Dynamics, Kluwer Academic Publishers Johnson, F. L. and Shapiro, S. C. 2005. Dependency-directed reconsideration: Belief base optimization for truth maintenance systems. In Proceedings of AAAI-2005, Menlo Park, CA. AAAI Press (http://www.aaai.org/). F. L. Johnson & S. C. Shapiro