©2007 Tarik Hadzic1 Lecture 11: Consistency Techniques 1. Arc Consistency 2. Directional Consistency 3. Generalized Arc Consistency Efficient AI Programming

©2007 Tarik Hadzic 1

Lecture 11: Consistency Techniques

1. Arc Consistency 2. Directional Consistency3. Generalized Arc Consistency

Efficient AI Programming


Today’s Program• Arc-Consistency

- AC-1 - AC-3, - AC-4

• Directional Consistency – DAC-1– Backtrack-free search

• Global Constraints– Generalized Arc Consistency– Domain Store Propagation


Bounded Constraint Inference

• Guarantee: any consistent instantiation of i-1 variable is extendible to any i-th variable

• Algorithms:– i=2: Arc-Consistency– i=3: Path-Consistency– i≥4: i-consistency

• i-consistency algorithms exponential in i. Trade-off between preprocessing and subsequent search


Arc-Consistency


Arc-Consistency

• Guarantee: any value in the domain of a single variable can be extended consistently by any other variable

• Example: network X={x,y}, Rxy: x<y, Dx=Dy={1,2,3}

1 •2 • 3 •

• 1• 2• 3

x < y

x y


Example

• Network X={x,y}, Rxy: x<y, Dx=Dy={1,2,3}

Not Arc-Consistenty

1 •2 • • 2

• 3

x < y

x

1 •2 • 3 •

• 1• 2• 3

x < y

x yArc-Consistent


Definition

• Given constraint problem <X,D,C> and constraint Cij(Sij,Rij) in C

• An arc (xi, xj) is arc-consistent iff for every ai Di there exists aj Dj such that (ai,aj) Rij

• If (xi, xj) is arc-consistent, it doesn’t imply that (xj, xi) is arc-consistent.

• A CSP <X,D,C> is arc-consistent iff all of its arcs are arc-consistent


Revise ((xi),xj)

1. deleted false

2. for each ai Di

3. if there is no aj Dj, (ai,aj) Rij

4. then delete ai from D, deleted true5. endif6. endfor 7. return deleted

• Complexity?

• O(k2) where k bounds the domain size


Revise ((xi),xj)

1 •2 •

• 1• 2• 3

x < y

x y

1 •2 • 3 •

• 2• 3

x < y

x y

1 •2 • 3 •

• 1• 2• 3

x < y

x y Revise((x),y)

Revise((y),x)


Revise ((xi),xj)

To achieve arc-consistency, apply Revise until no change in the domain of any variable in the

network

x

• • •

y• • •

z• • •

•Revise((x),z)


Revise ((xi),xj)


network

x

• •

y• • •

z• • •

•Revise((x),z)

•Revise((y),x)


Revise ((xi),xj)


network

x

• •

y • •

z• • •

•Revise((x),z)

•Revise((z),y)•Revise((y),x)


Revise ((xi),xj)


network

x

• •

y • •

z • •

•Revise((x),z)

•Revise((z),y)

•Revise((x),z)

•Revise((y),x)


Revise ((xi),xj)


network

x

•

y • •

z • •

•Revise((x),z)

•Revise((z),y)

•Revise((x),z)•Revise((y),x)

•Revise((y),x)


Revise ((xi),xj)


network

x

•

y •

z • •

•Revise((x),z)

•Revise((z),y)•Revise((x),z)•Revise((y),x)•Revise((z),y)

•Revise((y),x)


Revise ((xi),xj)


network

x

•

y •

z •

•Revise((x),z)

•Revise((z),y)•Revise((x),z)•Revise((y),x)•Revise((z),y)

•Revise((y),x)


AC-1 ()1. repeat2. for every {xi,xj} participating in a constraint3. Revise((xi),xj)4. Revise((xj),xi)5. endfor6. until no domain changed

• Complexity?

• O(enk3) where n variables, e binary constraints, k bounds the domain size


AC-3 ()

• Improvement over AC-1– No need to process all constraints

– Revise((xi),xj) is processed only when Dj is reduced

– queue maintains pairs (xi,xj) involved in a constraint where Dj is reduced

– Initially, for every constraint Cij, (xi,xj) and (xj,xi) are added to queue


AC-3 ()

1. for every constraint Cij

2. Q ← Q U {(xi,xj),(xj,xi)}3. endfor4. while Q ≠ {}

5. select and delete (xi,xj) from Q

6. Revise((xi),xj)

7. if Revise((xi),xj)=true then

8. Q ← Q U {(xk,xi), k ≠i, k≠j}9. endif10. endwhile


Example: AC-3 ()X = {x,y,z},

Dz={2,5}, Dx={2,5}, Dy={2,4};

Rzx: ”z divides x”; Ryz: “z divides y”

2,5

2,5 2,4

x

z

y

Q = {(z,x),(x,z),(z,y),(y,z)}



Dz={2,5}, Dx={2,5}, Dy={2,4};

Rzx: ”z divides x”; Rxz: “z divides y”

2,5

2,5 2,4

x

z

y

Q = {(z,x),(x,z),(z,y),(y,z)}

Revise((z),x)

No effect!



Dz={2,5}, Dx={2,5}, Dy={2,4};


2,5

2,5 2,4

x

z

y

Q = {(x,z),(z,y),(y,z)}

Revise((x),z)

No effect!



Dz={2,5}, Dx={2,5}, Dy={2,4};

Rzx: ”z divides x”; Ryz: “z divides y”

2

2,5 2,4

x

z

y

Q = {(y,z),(x,z)}

Revise((y),z)

No effect!



Dz={2,5}, Dx={2,5}, Dy={2,4};


2

2,5 2,4

x

z

y

Q = {(x,z)}

Revise((x),z)

Delete 5 from Dx

No constraints added to queue!



Dz={2,5}, Dx={2,5}, Dy={2,4};


2

2 2,4

x

z

y

Q = {}

AC-3 terminates!


AC-4 ()• Optimal performance O(ek2)• Does not use Revise• For every value a Di maintains counter(xi,

xj,a) : number of values in Dj consistent with xi=a

2,5

2,5 2,4

x

z

y

• counter(z,y,2) =

• counter(z,y,5) =

• counter(z,x,2) =

• counter(z,x,5) =

2

0!

1

1

No counters between x and y


AC-4 ()• If counter(xi,xj,a)=0 then (xi,a) is unsupported• List Q: maintains unsupported (xi,a) pairs• S(xj,a) : all values in other variable domains supported

by (xj,a)• In each step AC-4:

– Picks and removes unsupported value from Q – Updates counters of potentially affected values


AC-4 ()


AC-4 ()


Directional Arc Consistency


Motivation

• Determining amount of inference for backtrack-free guarantee

• Restrict inference relative to a given ordering

• Arc consistency unnecessary when solution generated along a fixed ordering


Directional Arc-Consistency A network is directional arc-consistent relative to order d=(x1,…,xn) iff every arc (xi,xj) is arc-consistent whenever i < j in the ordering

DAC(<X,D,C>)

• for i=n to 1 by -1

• for each j<i such that Cij C

• Revise((xj),xi)


Example

<X,D,C>, X={x1,x2,x3,x4}

D1={red,white,black}, D2={green,white,black},

D3={red,white,blue}, D4={white,blue,black}

R12 : x1 = x2 R13: x1 = x3 R34: x3 = x4

d = (x1,x2,x3,x4)x4

x3

x2

x1


Example

x4

x3

x2

x1

D1={red,white,black}

D2={green,white,black}

D3={red,white,blue}

D4={white,blue,black}

R12: x1 = x2

R13: x1 = x3

R34: x3 = x4


ExampleD1={red,white,black}


D3={red,white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

Revise((x3),x4)




D3={red,white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

Delete red from D3!




D3={white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

Revise((x1),x3)


Example

D1={red,white,black}


D3={white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

Delete red, black from D1!


ExampleD1={white}


D3={white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

Revise((x1),x2)


ExampleD1={white}


D3={white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

No effect!


ExampleD1={white}


D3={white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

x4

x3

x2

x1

DAC terminates!


ExampleD1={white}


D3={white,blue}


R12: x1 = x2

R13: x1 = x3

R34: x3 = x4

Backtrack-free assignment:

1. x1 = white

2. x2 = white

3. x3 = white

4. x4 = white


Directional Arc-Consistency

DAC(<X,D,C>)

• for i=n to 1 by -1

• for each j<i such that Cij C

• Revise((xj),xi)

Complexity: O(ek2)


Global Constraints and Generalized Arc-Consistency


Definition

• Given constraint problem <X,D,C> and (n-ary) constraint C’C, C’ = (R,S)

• Variable x is (generalized) arc-consistent relative to constraint (R,S) iff for every a Dx there exists a tuple in the domains of variables in S, t R such that t[x] = a


Example

• Dx = [0,15], Dy = [0,15], Dz = [0,15]

• x+y+z 15, z ≥ 13

• Dx ← [0,2], Dy ← [0,2]

• Alldiff(x1,…,xn)


Modern Solver Architecture

D1, D2, …, Dn

C1

C2

C3

1. Each Global Constraint Prunes Domains (highest level of pruning is GAC)

2. Constraint Store ”notifies” affected global constraints

Global Constraints

Constraint Store (Domain Store)

Documents

©2007 Tarik Hadzic1 Lecture 11: Consistency Techniques 1. Arc Consistency 2. Directional Consistency 3. Generalized Arc Consistency Efficient AI Programming