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How to (repeatedly) change preferences *

How to (repeatedly) change preferences *. Jan Chomicki University at Buffalo. * FOIKS’06, AMAI. 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.

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How to (repeatedly) change preferences *

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  1. How to (repeatedly) change preferences * Jan Chomicki University at Buffalo * FOIKS’06, AMAI

  2. 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 within each make, prefer more recent cars

  3. Preference queries • Winnow: In a given table, find the bestelements according to a given preference relation. within each make, prefer a more recent car Too many results…

  4. Query modification via preference revision within each make, prefer a more recent car among cars of the same production year, prefer VW • Objectives: Preference compositionoperators Minimal change to preferences Preservation of order properties

  5. Overview • Preference representation • Order axioms • Preference revision • Incremental evaluation of preference queries • Related work • Conclusions and future work

  6. Preference relations Preference 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

  7. 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

  8. Composing preference relations Union t (Â1[Â2) s , t Â1 s Ç t Â2 s Prioritized composition t (Â1BÂ2) s , t Â1 s Ç (s¨1 t Æ t Â2 s) Pareto composition t (Â1­Â2) s , (s ¨2 t Æ 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

  9. Preference revisions • ORD -revision of  with Â0 • Preference relation Â’ : • minimally different from  • contains Â0   • satisfies ORD Preference relation  Revising pref.relation Â0 Composition operator  Order axioms ORD  and Â0 satisfy ORD

  10. Conflicts and SPO revisions solved by B 0-conflict A B A B solved by ­ 1-conflict A B C A B C A B C B 2-conflict no SPO -revision A C D

  11. 0-conflict 1-conflict 2-conflict ? ­ [ B

  12. Is lack of conflict sufficient? No conflicts A B C D However, no SPO revision! Interval Order (IO) = SPO + 8x,y,z,w. (x  y Æ z  w) ! (x  w Ç z  y) Â, Â0satisfy SPO no 0-conflicts  or Â0 is IO Â’ = TC(Â[Â0) is an SPO [-revision Â, Â0satisfy SPO no 1-conflicts Â0 is IO Â’ = TC(Â0BÂ) is an SPO B-revision

  13. 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 TC( Â0 [ Â ) (m,y) Â’ (m’,y’) ´ m=m’Æ y > y’Ç m = vw Æ m’ ≠ vwÆ y ¸ 1999 Æ y’· 1999

  14. Â Â … Â Â0 Â Â Â …

  15. WO revisions and utility functions Â, Â0 satisfy WO no 0-conflicts u’(x)=a¢u(x)+b¢uo(x)+c a,b > 0 Â’ = Â0 [  is a WO [-revision Â’ may be not representable with a utility function Â’ = Â0B is a WO B-revision Â, Â0 satisfy WO with conflicts  represented with u(x) Â0 represented with u0(x)

  16. Incremental evaluation: preference revision r µ Â1 µ µ Ân … Â2       r1 rn r1 rn r2 r2

  17. Â : within each make, prefer more recent cars Â0 : among cars produced in 1999, prefer VW Â TC(Â[Â0)

  18. +0 +1 +2 Incremental evaluation: tuple insertion t3 r3 t2 +2 r2 t1 +1 … r1 r0 +0 Â Â Â Â s0 s1 s2 s3 …

  19. Preference revision First-order Revising a single, finitely representable relation Preserving order axioms Belief revision Propositional Revising a theory Axiomatic properties of BR operators Preference vs. belief revision

  20. 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

  21. 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

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