Problem Solving with Constraints Lookahead Schemas 1 Foundations of Constraint Processing CSCE496/896, Fall All questions to Piazza Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Tel: +1(402) Lookahead Schemas

Problem Solving with Constraints Lookahead Schemas 2 Outline Looking ahead Schemas –Forward checking (FC) –Directional Arc Consistency (DAC) –Maintaining Arc Consistency (a.k.a. full arc- consistency)

Problem Solving with Constraints Lookahead Schemas 3 Looking ahead Rationale: –As decisions are made (conditioning), Revise the domain of future variables to propagate the effects of decisions i.e., eliminate inconsistent choices in future sub-problem –Domain annihilation of a future variable avoids expansion of useless portions of the tree Techniques –Partial: forward-checking (FC), directional arc- consistency (DAC) –Full: Maintaining arc-consistency (MAC) –Use: Revise (V f, V c ), V f future variable, V c current variable

Problem Solving with Constraints Lookahead Schemas 4 Revise the domain of V i Revising the domain of V i given a constraint C Vi,Vj on V i (i.e., V i  Scope (C)) General notation: Revise (V i,C Vi,Vj ) In a binary CSP: Revise (V i,C Vi,Vj )= Revise (V i, V j )

Problem Solving with Constraints Lookahead Schemas 5 Revise (V i, V j ) NOTE: only D Vi may be updated Revise (V i, V j ) 1.revised  nil 2.  x  D Vi 3. found  nil 4.  y  D Vj 5. If Check ((V i,x),(V j,y)) Then Begin 6. found  t 7. Break 8. End 9. If found=nil Then Begin 10. revised  t 11. D Vi  D Vi \ {x} 12. End 13.Return (revised)

Problem Solving with Constraints Lookahead Schemas 6 Revise (V i, V j ) 1.revised  nil 2.  x  D vi 3.  y  D Vj 4. If Check ((V i,x),(V j,y)) Then Break () 5. revised  t 6. D Vi  D Vi \ {x} 7.Return (revised) Simpler, equivalent code but not as obvious as the previous one

Problem Solving with Constraints Lookahead Schemas 7 Domain filtering in lookahead V c current variable V f future variable {V f } all future variables Revise (V f, V c ) FC (V c ): 1.  V f  {V f } connected to V c 2. Revise (V f,V c ) 3. If D Vf ={} then return(nil)

Problem Solving with Constraints Lookahead Schemas Directional Arc Consistency Choose an ordering d, stick to it After instantiating a variable at level i, do the following 1.For k from i to (n-1) in the ordering d Do 2. If FC (V k )=nil then Return (nil) 8

Problem Solving with Constraints Lookahead Schemas Maintaining Arc Consistency First, FC (V c ), If it does not fail, then, form a queue with all constraints (V i,V j ) and (V j,V i ) between future variables, and run AC AC-1 1.Q  {(V i,V j ),(V j,V i ), …, (V k,V m ), (V m,V K )} 2.Change  true 3.While Change Do 4. Change  false 5. For all (V a,V b ) in Q Do 6. If Revise (V a,V b ) 7. Then If Dom (V a ) =  8. Then Return (nil) 9. Else Change  true 9

Problem Solving with Constraints Lookahead Schemas 10 FC: DAC: –assumes a fixed variable ordering d MAC: –does more pruning (search may visit fewer nodes) at the cost of more consistency checks Look-ahead techniques: FC, DAC, MAC FC (V c ); While not failure: For the next V f in the ordering d, FC (V f ) FC (V c ); AC ({V f }) FC (V c ) FC (V c ); Repeat until quiescence or failure  V f1,V f2  {V f }, Revise (V f1,V f2 )

Problem Solving with Constraints Lookahead Schemas 11 Terminology overload alert: FC FC is used to denote any of the following: –a partial look-ahead schema –a specific chronological backtrack search algorithm that uses the partial look-ahead schema Meaning is inferred from context Not a healthy situation, but a fact of reality Advice: state upfront the meaning of your terms and stick to them throughout your paper

Problem Solving with Constraints Lookahead Schemas 12 (BT Search +) MAC vs. FC Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES ], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003] Results: (sketchy) Low tightnessHigh tightness Low density (sparse) FCMAC High density (dense) FC Note: Results depend on Variable ordering (static vs. dynamic) Problem difficulty (positive relative to crossover point)