Foundations of Constraint Processing All questions to Piazza Lookahead Schemas Foundations of Constraint Processing CSCE421/821, Spring 2019 www.cse.unl.edu/~choueiry/S19-421-821/ All questions to Piazza Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Tel: +1(402)472-5444

Outline: Lookahead Schemas Forward checking (FC) Directional Arc Consistency (DAC) Maintaining Arc Consistency (a.k.a. full arc-consistency)

Looking ahead: Rationale As decisions are made (conditioning) Revise the domain of future variables to propagate the effects of decisions (instantiations) i.e., eliminate inconsistent choices in future sub-problem Domain annihilation of a future variable avoids expansion of useless portions of the tree

Looking ahead: Techniques Partial forward-checking (FC) directional arc-consistency (DAC) Full Maintaining arc-consistency (MAC) a.k.a. Real Full Lookahead (RFL) Initialize the queue with the set of (Vf,Vc) Revise(Vf,Vc), Vf future variable, Vc current variable

Revise(Vi,CVi,Vj)= Revise(Vi,Vj) Revise the domain of Vi Revising the domain of Vi given a constraint CVi,Vj on Vi (i.e., Vi  scope(C)) General notation: Revise(Vi,CVi,Vj) In a binary CSP: Revise(Vi,CVi,Vj)= Revise(Vi,Vj)

Revise(Vi, Vj) NOTE: only DVi may be updated revised  nil  x  Dvi  y  DVj If Check((Vi,x),(Vj,y)) Then break revised  t DVi  DVi \ {x} Return revised Simpler, equivalent code but not as obvious as the previous one

Domain filtering in lookahead Vc current variable Vf future variable {Vf} all future variables Revise(Vf, Vc) FC(Vc):  Vf  {Vf} connected to Vc Revise(Vf,Vc) If DVf ={} then break false

Directional Arc Consistency Choose an ordering d, stick to it After instantiating a variable at level i, do the following For k from i to (n-1) in the ordering d Do If FC(Vk)= false then break false

Real Full Lookahead (RFL) AC-3 Set Q  {(Vf,Vc) | Vc is the current variable, Vf ∈ Neigh(Vf )} Run AC-3(Q) AC-1 does too many checks First revise all the tuples in {(Vf,Vc)} Then, call AC-1 ({(Vi,Vj), Vi,Vj are future variables}})

Look-ahead techniques: FC, DAC, RFL assumes a fixed variable ordering d RFL: does more pruning (search may visit fewer nodes) at the cost of more consistency checks FC(Vc) FC(Vc); While not failure: For the next Vf in the ordering d, FC(Vf) FC(Vc); AC-3({Vf}) FC(Vc); Repeat until quiescence or failure  Vf1,Vf2  {Vf}, Revise(Vf1,Vf2)

Terminology overload alert: FC FC is used to denote one 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

(BT Search +) RFL vs. FC Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES-20-2000], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003] Results: (sketchy) Low tightness High tightness Low density (sparse) FC RFL High density (dense) Note: Results depend on Variable ordering (static vs. dynamic) Problem difficulty (positive relative to crossover point)

Alert: Terminology Distinguish between MAC assumes binary branching Real Full Lookahead (RFL) Maintaining Arc Consistency (MAC) MAC assumes binary branching