How to (repeatedly) change preferences *
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How to (repeatedly) change preferences *
Jan ChomickiUniversity at Buffalo
* FOIKS’06, AMAI
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Preference relations
Binary relations between tuples Abstract way to capture a variety of criteria: desirability,
relative value, quality, timeliness… More general than numeric scoring functions
Make Year
VW 2002
VW 1998
Kia 1998
within each make, prefer more recent cars
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Preference queries
Winnow: In a given table, find the best elements according to a given preference relation.
Make Year
VW 2002
VW 1998
Kia 1998
within each make, prefer a more recent car
Too many results…
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Query modification via preference revision
Make Year
VW 2002
VW 1998
Kia 1998
within each make, prefer a more recent car
among cars of the same production year, prefer VW
Objectives: Preference composition operators
Minimal change to preferences
Preservation of order properties
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Overview
Preference representation Order axioms Preference revision Incremental evaluation of preference
queries Related work Conclusions and future work
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Preference relationsPreference relation binary relation (possibly infinite) represented by a quantifier-free first-order formula
within each make, prefer more recent cars: (m,y) Â (m’,y’) ´ (m = m’ Æ y > y’)
Winnow operator Â(r) = { t2 r | :9 t’2 r. t’ Â t} Used to select the
best tuples
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Order axioms ORD
Strict Partial Order (SPO) = transitivity + irreflexivity Preference SQL winnow is nonempty efficient algorithms for winnow (BNL,…) incremental query evaluation
Weak Order (WO) = SPO + negative transitivity:
8x,y,z. (x¨ y Æ y ¨ z) ! x¨ z often representable with a utility function single pass winnow evaluation
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Composing preference relationsUnion
t (Â1 [ Â2) s , t Â1 s Ç t Â2 s
Prioritized composition
t (Â1 B Â2) s , t Â1 s Ç (s¨1 t Æ t Â2 s)
Transitive closure(t,s) 2 TC(Â) , t Ân s for some n > 0
Pareto composition
t (Â1 Â2) s , (s ¨2 t Æ t Â1 s )Ç (s¨1 t Æ t Â2 s)
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Preference revisions
Preference relation ÂRevising pref.relation Â0
Composition operator
Order axioms ORD
Âand Â0 satisfy ORD
ORD -revision of  with Â0
Preference relation Â’ :• minimally different from • contains Â0 • satisfies ORD
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Conflicts and SPO revisions
BA
B CA
B
A C
D
BA
B CA B CA
no SPO -revision
0-conflict
1-conflict
2-conflict
solved by B
solved by
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0-conflict
1-conflict
2-conflict
?[ B
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Is lack of conflict sufficient?
BA C D
No conflicts
However, no SPO revision!
Interval Order (IO) = SPO + 8x,y,z,w. (x  y Æ z  w) ! (x  w Ç z  y)
Â, Â0satisfy SPOno 0-conflictsÂor Â0 is IO
Â’ = TC(Â [ Â0) is an SPO [-revision
Â, Â0satisfy SPOno 1-conflicts
Â0 is IO
Â’ = TC(Â0 B Â) is an SPO B-revision
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within each make, prefer more recent cars: (m,y) Â (m’,y’) ´ (m = m’ Æ y > y’)
among cars produced in 1999, prefer VW:
(m,y) Â0 (m’,y’) ´ m = vw Æ m’ ≠ vw Æ y = y’ = 1999
(m,y) Â’ (m’,y’) ´ m=m’Æ y > y’Ç m = vw Æ m’ ≠ vwÆ y ¸ 1999 Æ y’· 1999
TC( Â0 [Â)
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vw 2001
vw 2000
vw 1999 kia 1999
kia 1998
vw 2002
Â
Â
Â
Â
Â0
…
…
Â
Â
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WO revisions and utility functions
Â, Â0 satisfy WOno 0-conflicts
Â, Â0 satisfy WOwith conflicts
Â’= Â0 B Â is a WO B-revision
u’(x)=a¢u(x)+b¢uo(x)+c
a,b > 0
Ârepresented with u(x) Â0 represented with u0(x)
Â’ may be not representable with a utility function
Â’= Â0 [Â is a WO [-revision
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rnr2
r1
Incremental evaluation: preference revision
r
Â1
r1
Â2
r2
Ân
rn
…µ µ µ
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Make Year
VW 2002
VW 1998
Kia 1998
Make Year
VW 2002
Kia 1998
Make Year
VW 2002
Â: within each make, prefer more recent cars
Â0 : among cars produced in 1999, prefer VW
 TC(Â[Â0)
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Incremental evaluation: tuple insertion
r00 r1
Â
s0
1 r22 r3 …
Â
s1
Â
s2
Â
s3…
+0 +1+2
t1t2
t3
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Preference vs. belief revision
Preference revision
First-order Revising a single, finitely
representable relation Preserving order axioms
Belief revision
Propositional Revising a theory Axiomatic properties of
BR operators
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Related work
S. O. Hansson. Changes in Preferences, Theory and Decision, 1995 preferences = sets of ground formulas preference revision ' belief revision no focus on construction of revisions, SPO/WO preservation preference contraction, domain expansion/shrinking
M.-A. Williams. Belief Revision via Database Update, IIISC, 1997 revising finite ranking with new information new ranking can be computed in a simple way
S. T. C. Wong. Preference-Based Decision Making for Cooperative
Knowledge-Based Systems. ACM TOIS, 1994 revision and contraction of finite WO preferences with single pairs t Â0 s
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Summary and future work
Summary:
Preference query modification through preference revision Preference revision using composition Closure of SPO and WO under revisions Incremental evaluation of preference queries
Future work:
Integrating with relational query evaluation and optimization General revision language Preference contraction (query result too small) Preference elicitation