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}/
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)
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Two Approaches
to
Sets of Beliefs
• Foundations uses a belief base B
• Coherence uses a belief set K
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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
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Coherence Approach
• Belief set K = Cn(K)
– where Cn(Φ) = {p | Φ├ p}
• All beliefs in K have independent
– standing
– justification
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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
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Question
• For what sets
of belief base belief change operations
does recovery hold?
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Recovery is Controversial
• But, informally,
if you remove something
then put it back,
the result should be a noop
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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}
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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
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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
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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.
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Solution
(Informal)
• If don’t forget beliefs lost during contraction
can bring them back later.
• Two approaches:
– Reconsideration
– Liberation
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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
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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)!
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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} , ≥
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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
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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)
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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,
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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)
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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
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-Recovery (R)
• K = Cn(())
•  + p = p, p1,…,pn
• -Recovery:
if  = p1,…,pn
K ⊆ K(+¬p)+p
• Holds except when ¬p ∈ K
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Conclusions
Recovery Adherence
TR
LR
OR
B⊆Cn((B~p)+p)
K⊆(K~p)+p
Recency
Recency
B ⊆ (B~p) +! p
≥ = ≥’
Recency
K ⊆ K(+¬p)+p p  B
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
YES
inconsistent
YES
inconsistent
NO
optimal
YES
optimal
Case
1 p  Cn(B)
B\{p} ├
/p
4 p  Cn(B)
B+p ├ 
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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/).
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