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How to (repeatedly) change preferences *

This paper discusses approaches for repeatedly modifying and evaluating preferences in various contexts. We explore binary relations between tuples, emphasizing the desirability of attributes like quality and timeliness. Our framework includes preference revision strategies, composition operators, and incremental evaluation methods that aim to minimize changes while preserving order properties. We present algorithms for efficiently finding optimal items according to specified preferences, addressing challenges in handling excessive results, and propose directions for future research in integrating these methods into broader relational query optimizations.

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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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