1. cse@buffalo Improving Recovery for Belief 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. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro2 Recovery Informal 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)

3. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro3 Two Approaches to Sets of Beliefs Foundations uses a belief base B Coherence uses a belief set K

4. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro4 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

5. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro5 Coherence Approach Belief set K = Cn(K) –where Cn(Φ) = {p | Φ├ p} All beliefs in K have independent –standing –justification

6. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro6 Recovery for 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

7. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro7 Question For what sets of belief base belief change operations does recovery hold?

8. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro8 Recovery is Controversial But, informally, if you remove something then put it back, the result should be a noop

9. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro9 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}

10. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro10 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

11. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro11 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 /

12. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro12 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. / /

13. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro13 Solution (Informal) If don’t forget beliefs lost during contraction can bring them back later. Two approaches: –Reconsideration –Liberation

14. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro14 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  Reconsideration † Uses Knowledge State † Johnson & Shapiro, AAAI-05

15. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro15 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)! !  !  ! 

16. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro16 (  {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}, ≥  Revisit non-Recovery Example !  ! 

17. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro17 When B , ≥ and ≥ ’ are known or obvious, write KS + p as B + p write KS~p = def  B~p, B , ≥  as B~p Shorthand Notation !  ! 

18. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro18 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) Optimized-Recovery (OR) Holds !  ! 

19. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro19  = p 1,…,p n  ( , p) : the maximal subsequence of  s.t.  ( ,p) ├ p  (  ) =  ( ,  ) K = Cn(  (  )) K ~  p = Cn(  ( ,p)) K+p = Cn(K  {p}) (traditional) Liberation † Uses a Sequence † Booth, Chopra, Ghose and Meyer 2003 /

20. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro20  = 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) Liberation Example

21. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro21 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 Liberation Recovery /

22. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro22  -Recovery (  R) K  = Cn (  (  ))  + p = p, p 1,…,p n if  = p 1,…,p n  -Recovery: K  ⊆ K (  +¬p)+p Holds except when ¬p ∈ K 

23. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro23 Conclusions Recovery Adherence TRLROR Case B⊆ Cn((B~p)+p) K⊆( K~p)+pB ⊆ (B~p) + p Recency K  ⊆ K (  +¬p)+p ≥ = ≥ ’ p  B  1 p  Cn( B) B\{p} ├ p YES optimal YES possibly incon. YES optimal YES optimal 2 p  Cn( B) B\{p} ├ p NO consistent NO possibly incon. YES optimal YES optimal 3 p  Cn( B) B+p ├  YES optimal YES optimal YES optimal NA 4 p  Cn( B) B+p ├  YES inconsistent YES inconsistent NO optimal YES optimal / !  /

24. cse@buffalo NRAC-05F. L. Johnson & S. C. Shapiro24 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/).