Foundations of Artiﬁcial Intelligence 26. Constraint Satisfaction Problems: Path Consistency Malte Helmert

(1)

Foundations of Artificial Intelligence

26. Constraint Satisfaction Problems: Path Consistency

Malte Helmert

University of Basel

April 19, 2021

(2)

Constraint Satisfaction Problems: Overview

Chapter overview: constraint satisfaction problems:

22.–23. Introduction 24.–26. Basic Algorithms

24. Backtracking 25. Arc Consistency 26. Path Consistency 27.–28. Problem Structure

(3)

Beyond Arc Consistency Path Consistency Summary

Beyond Arc Consistency

(4)

Beyond Arc Consistency: Path Consistency

idea of arc consistency:

For every assignment to a variable u

there must be a suitable assignment to every other variablev. If not: remove values ofu for which

no suitable “partner” assignment to v exists.

tighter unary constraintonu

This idea can be extended to three variables (path consistency):

For every joint assignment to variables u,v

there must be a suitable assignment to every third variablew. If not: remove pairs of values of u andv for which

no suitable “partner” assignment to w exists.

tighter binary constraintonu andv German: Pfadkonsistenz

(5)

Beyond Arc Consistency: i -Consistency

general concept ofi-consistency fori ≥2:

For every joint assignment to variables v₁, . . . ,vi−1

there must be a suitable assignment to every i-th variablev_i. If not: remove value tuples of v₁, . . . ,v_i−1 for which

no suitable “partner” assignment for v_i exists.

tighter (i−1)-ary constraintonv₁, . . . ,vi−1

2-consistency = arc consistency 3-consistency = path consistency (*) We do not consider generali-consistency further as larger values thani = 3 are rarely used

and we restrict ourselves to binary constraints in this course.

(*)usual definitions of 3-consistency vs. path consistency differ

(*)

when ternary constraints are allowed

(6)

Path Consistency

(7)

Path Consistency: Definition

Definition (path consistent)

LetC=hV,dom,(Ruv)i be a constraint network.

(a) Two different variablesu,v ∈V are path consistent with respect to a third variable w ∈V if

for all values du ∈dom(u),dv ∈dom(v) withhd_u,dvi ∈Ruv

there is a value dw ∈dom(w) with hd_u,dwi ∈Ruw and hd_v,d_wi ∈R_vw.

(b) The constraint network C ispath consistent if for any three variablesu,v,w,

the variables u andv are path consistent with respect tow.

(8)

Path Consistency: Remarks

remarks:

Even if the constraintRuv is trivial, path consistency can infer nontrivial constraints between u andv. name “path consistency”:

pathu →w →v leads to new information on u →v

w

u v

(9)

Path Consistency: Example

red blue

v₁

red blue

v2

red blue

v₃

6= 6=

6=

arc consistent, but not path consistent

(10)

Processing Variable Triples: revise-3

analogous torevisefor arc consistency:

functionrevise-3(C,u,v,w):

hV,dom,(Ruv)i:=C for eachhd_u,dvi ∈Ruv:

if there is nodw ∈dom(w) with hd_u,d_wi ∈R_uw andhd_v,d_wi ∈R_vw:

remove hd_u,d_vi fromR_uv

input: constraint networkC and three variables u, v,w of C effect: u,v path consistent with respect tow.

All violating pairs are removed fromR_uv.

time complexity: O(k³) wherek is maximal domain size

(11)

Enforcing Path Consistency: PC-2

analogous toAC-3for arc consistency:

functionPC-2(C):

hV,dom,(Ruv)i:=C queue:=∅

for eachset of two variables{u,v}:

for each w ∈V \ {u,v}:

insert hu,v,wi into queue whilequeue6=∅:

remove any elementhu,v,wi fromqueue revise-3(C,u,v,w)

if R_uv changed in the call to revise-3:

for each w⁰∈V \ {u,v}:

insert hw⁰,u,vi into queue insert hw⁰,v,ui into queue

(12)

PC-2: Discussion

The comments for AC-3 hold analogously.

PC-2 enforces path consistency

proof idea: invariant of the whileloop:

if hu,v,wi∈/queue, then u,v path consistent with respect to w

time complexity O(n³k⁵) forn variables and maximal domain sizek (Why?)

(13)

Summary

(14)

Summary

generalization of

arc consistency(considers pairsof variables) to path consistency(considers triplesof variables) andi-consistency (considersi-tuplesof variables) arc consistency tightens unary constraints

path consistency tightensbinary constraints i-consistency tightens(i−1)-aryconstraints higher levels of consistency more powerful but more expensive than arc consistency