Reasoning with global constraints

1
Reasoning with global constraints

• Christian Bessiere
• LIRMM (CNRS/U. Montpellier)
• France

2
Background
3
Constraint networks

• A set XX1,,Xn of variables
• A set DD(X1),D(Xn) of domains, where D(Xi) is
  the finite set of values for Xi
• A set CC1,Ce of constraints

4
Agreements on domains

• Domains are given in extension to allow any type
  of reasoning
• ? If D(Xi) denotes the size of the encoding,
  D(Xi) D(Xi)
• Example
• D(X)1,..,100
• D(X) 100 (not log2(100))

5
Agreements on domains

• Domains are subsets of the set Z of integers
• ? We inherit the total ordering on integers
• Example
• D(X)1,..,100
• Special values min(X)1, max(X)100

6
Set variables

• It is sometimes convenient to consider a Boolean
  vector ltS1,,Sngt as a set variable S with domain
  21..n
• In this case, D(S) is represented by its required
  values lb(S) and possible values ub(S)

S can take values 1,2, 1,2,3, 1,2,4,
1,2,3,4
Gervet94
7
Constraints

• A constraint c specifies the allowed
  combinations of values on a sequence
  X(c)(Xi1,,Xiq) of variables
• Classical definition in CSPs
• c ? D(Xi1) x ... X D(Xiq)
• Example
• c ? X lt Y D(X)3, 7, 8, 10
• D(Y)1, 2, 4, 7, 9
• c(3,4), (3,7), (3,9), (7,9), (8,9)

8
Not satisfactory

• because it does not fit what is done in practice
  X lt Y is not encoded as a set of pairs!
• And even if it was, we do not want to remove all
  pairs (v,w) from c each time a value v is removed
  from D(X)

• because it does not permit to express all types
  of local consistencies (see bound consistency in
  Dechter03)

9
Constraints

• A constraint c specifies the allowed
  combinations of values on a sequence
  X(c)(Xi1,,Xiq) of variables

• c ? ZX(c)
• c allowed tuples on X(c)
• c is independent of the effective domains of its
  variables
• So, a constraint c is defined by any Boolean
  function fc with domain ZX(c)

c ? X lt Y is the set (-11,-7),, (0,4), (0,5),
(0,6),, (7,9), (7,10) of all tuples (v,w)
where vltw.
10
Non-binary versus global constraints

• Non-binary c with scope X(c), X(c)gt2
• Global constraint class G of constraints defined
  by a Boolean function fG with domain Z
• alldiff fG(t)1 iff ?v,w ? t, v ? w
• A constraint c on X(c) belongs to a class G iff
  c t ? ZX(c) fG(t)1
• Alldiff(X1,X2,X3)
  t ? Z3 t1 ?
  t2, t1 ? t2, t2 ? t3

Restriction checking if fG(t)1 is polynomial
11
Some global constraints

• Nvalue(X1..Xn,N) ? number of values taken by Xis
  is equal to N
• (2,1,3,3, 3) is allowed
• Alldiff(X1..Xn) ? Nvalue(X1..Xn,n)
• Atmostk,v(X1..Xn) ? at most k variables take
  value v
• (2,5,1,4) is allowed if k2, v2
• See Beldiceanus catalog (gt200 global constraints)

Belal05
12
Do we need global constraints?

• or at least do we need all of them when
  specifying a CP toolkit?
• Why global constraints?
• Expressiveness (semantic globality)
• ? expressing new things
• Effectiveness of propagation (operational
  globality)
• ?producing powerful effects
• Efficiency of propagation (algorithmic globality)
• ?producing the result with little wasted effort

BesVanH03
13
Part 1 Semantic Globality
14
Expressiveness

• Global constraints permit a high-level
  programming approach a whole pattern in a single
  constraint
• Allowed shifts in a nurse rostering
• ? Strech constraint
• at most 5 working days and not morning after
  night
• N N R R M M A A A A R N M M R R

• But are they all necessary?
• does each permit to express what others cant?

Pesant04
15
Early work

• Encode non-binary constraints as binary ones
• Primal representation
• Hidden representation
• Dual representation

Montanari74, DecPea89, Janal89, Dechter90,
Rosal90
16
Early work Primal graph

• Keep the same variables/domains
• Replace non-binary constraints by their binary
  projections (inside DX(c))
• Alldiff(X1,..X4)
• sol(N)alldiff(X1..X4)

17
Early work Primal graph

• But, it does not always work
• X Y Z with
• D(X)1, 2, 3
• D(Y)2, 4, 6
• D(Z)1, 3, 5, 7

18
Early work Hidden repr.

• Keep the original variables
• Add a new variable Hc for each original
  constraint c
• D(Hc)t t ? c ? DX(c)
• Binary constraints between Hc and Xi
  ? X(c) that accept (t,v) iff tXiv

Hc
X
Y
Z
19
Early work Hidden repr.

• D(Hc)t t ? c ? DX(c) ?
  D(Hc)?O(dk)
• Exponential size of the representation

20
Early work Dual repr.

• Forget original variables
• Add a new variable Vc for each original
  constraint c
• D(Vc)t t ? c ? DX(c)
• X(c1) ? X(c2)?? ? c(Vc1,Vc2)
• (t1,t2) ? c(Vc1,Vc2) iff
  t1X(c1)?X(c2) t2X(c1)?X(c2)

YZgt2W, D(W)5,6
21
Early work Dual repr.

• D(Vc)t t ? c ? DX(c) ?
  D(Vc)?O(dk)
• Exponential size of the representation

22
Semantic globality (relative to L)

• When does a global constraint help expressing
  something new in a language L?
• ? If it cannot be polynomially decomposed with
  constraints in L

23
A sample language

• Range, Roots, and basic arithmetic primitives

Comics05
24
A sample language

• Range(X1,...,Xn,S,T) ? X(S)T, where X(i)Xi

1
v1
vj
i
n
vm
25
A sample language

• Roots(X1,...,Xn,S,T) ? SX-1(T)

26
Semantic globality (relative to L)

• When does a global constraint help expressing
  something new in a language L?
• If it cannot be polynomially decomposed with
  constraints in L
• On the same variables
• With extra variables

27
Decomposition (relative to L)

• alldiff(X1,..,Xn)

• Atmost1,1(X1,..,Xn)
• Atmost1,2(X1,..,Xn)
• Atmost1,3(X1,..,Xn)
• Atmost1,v(X1,..,Xn), ?v?D(Xi), ?i

28
Decomposition with extra variables (relative to L)

• atmostk,v(X1,..Xn) ( k2, v2 )

X1X4
1 1 2 2 1 1 2 3 1 2 2 2 2 2 2 2 4 4 5 5
Sgtk
Roots(X1,..Xn,S,v) ? S?k
atmost is decomposable in L
Sk
29
Decomposition with extra variables (relative to L)

• Nvalue(X1,..Xn,N)

N 1 2 3 4
Range(X1,..Xn,1..n,T) ? TN
Nvalue is decomposable in L
30
Decomposition relative to L

• G decomposable relative to L if
• ?c ? G, ?D on X(c), ?N(X(c)?Y,D,C) such
  that
• ?Xi?X(c), D(Xi)D(Xi)
• ?c ? C, c ? L
• sol(N)X(c) sol(X(c),D,c)
• N is poly in cD

31
Absolute semantic globality

• When does a global constraint help expressing
  something new? (independently of any language)
• ? If it cannot be polynomially decomposed with
  constraints of bounded arity
• On the same variables
• With extra variables

32
Absolute decomposition

• G k-decomposable if
• ?c ? G, ?D on X(c), ?N(X(c)?Y,D,C) such that
• ?Xi?X(c), D(Xi)D(Xi)
• ?c ? C, X(c) ? k
• sol(N)X(c) sol(X(c),D,c)
• N is poly in cD
• G is decomposable if
• ?k s.t. G is k-decomposable

33
Absolute decomposition

• Alldiff ? clique of binary inequalities
• Alldiff(X1..X4)

Alldiff is decomposable
34
Absolute decomposition

• Atmostk,v(X1..Xn)?

• Atmost3,2

• Atmost 3 among 4 ? atmost 3 among 3 ? universal
  constraint

Atmost is not decomposable on the same variables
35
Absolute decomposition with extra variables

• Atmostk,v(X1..Xn)
• B0Bn, D(Bi)0,,n
• (Xiv BiBi-11)?(Xi?v Bi Bi-1), ?i
• B00, Bn ? k

X1 X2 Xi Xn-1 Xn
B0 B1 Bi-1 Bi Bn-1 Bn
atmost is 3-decomposable
36
Absolute decomposition with extra variables

• c ? (?Xi k)
• Y0Yn, D(Yi)0,,k
• Yi Yi-1Xi, ?i
• Y00, Yn k

D(Yi) exponential in (X,D,c)
?Xi k is not decomposable
37
Part 2 Complexity of Reasoning
38
Reasoning with global constraints
c is XYZ

• All CP solvers propagate constraints
• propagate usually means
• remove value v in D(Xi), Xi?X(c) if
• not ?t?c, tXiv and tXj?D(Xj) ?j
• This is the maximum amount of inconsistency
  information a constraint taken alone can project
  on a variable, aka Generalized Arc Consistency
  (GAC )

39
Questions related to GAC

• GACSupportc,D,X,v
• Does value v for X have a support on c in D?

• GACSupportX,1?

? yes

• GACSupportX,2?

? no
40
Questions related to GAC

• IsItGACc,D
• Does GACSupportc, D, X, v answer yes for each
  X ? X(c) and each value v ? D(X)?

• IsItGACD?

? no

• IsItGACD1?

? yes
41
Questions related to GAC

• NoGACWipeOutc,D
• Is there any non empty D ? D on which
  IsItGACD answers yes?

• NoGACWipeOutD?

? yes

• NoGACWipeOutD2?

? no
42
Questions related to GAC

• maxGACc,D,D
• Is it the case that IsItGACc, D answers yes
  and not ? D, D ? D ? D such that
  IsItGACD?

• maxGACD,D?

? no

• maxGACD1,D?

? no
43
Questions related to GAC

• maxGACc,D,D
• Is it the case that IsItGACc, D answers yes
  and not ? D, D ? D ? D such that
  IsItGACD?

• maxGACD,D?

? no

• maxGACD1,D?

? no

• maxGACD3,D?

? yes
44
Questions related to GAC

• GACDomainc,D
• The domain D such that maxGACD,D answers yes
• GACDomainD?

? D3
45
GAC reasoning is intractable

• GACSupport is in NP (poly certificate)
• Let G be the class of constraints defined by X1 ?
  ?, where ? is a 3Sat formula
• GACSupport(c,0,1n,X,1) solves the SAT problem
• ? GACSupport is NP-complete

46
GAC reasoning is intractable

• GACSupport is NP-complete
• IsItGAC is NP-complete
• NoGACWipeOut is NP-complete
• GACDomain is NP-hard
• maxGAC?

47
maxGAC is DP-complete

• A decision problem P is DP-complete iff there
  exists two problems Q1 (NP-c) and Q2 (coNP-c)
  such that P answers yes when Q1 answers yes
  and Q2 answers no
• Let G1 and G2 be two 3Col problems (on the nodes
  s1..sn)
• X1 X2 Xn
• 1,2,3 1,2,3 1,2,3
• 4,5,6 4,5,6 4,5,6
• c(Xi,Xj) forbids (v,w) if v?(1,2,3) and w?(4,5,6)
  or vice versa
• (si,sj) ? G1 ? c(Xi,Xj) forbids (11), (22), (33)
  on Xi,Xj
• (si,sj) ? G2 ? c(Xi,Xj) forbids (44), (55), (66)
  on Xi,Xj
• maxGAC(c,1..3n,1..6n) answers yes iff
• G1 is 3colorable and G2 is not

48
Complexity summary
Problem Complexity class
GACSupport NP-complete
IsItGAC NP-complete
NoGACWipeOut NP-complete
GACDomain NP-hard
maxGAC DP-complete
49
Questions related to GAC

• Let G be a class of constraints
• GACSupport(G)
• IsItGAC(G)
• NoGACWipeOut(G)
• maxGAC(G)
• GACDomain(G)
• are the same as before but constraints must
  belong to G

50
Dependencies between GAC questions

• Suppose GACSupport(G) is NP-complete
• NoGACWipeOut is in NP
• GACSupport(G)c,D,X,v can be solved by a call to
  NoGACWipeOut(G)c,DXv
• ? NoGACWipeOut(G) is NP-complete

51
Dependencies between GAC questions
NoGACWOut
GACSupport
IsItGAC
GACDomain
maxGAC
A NP-hard ? B NP-hard
B
A
52
A question related to GAC Entailment

• Entailment captures the idea that a constraint is
  no longer necessary in the network, that is, for
  any subdomain D of D, c is GAC
• Entailedc,D
• Is DX(c) ? c ?
• Example
• c ? (X ? Y ? Z)

X ? Y ? Z 1 2 1

• Entailed(D1) no

2 4 3
3 6 5
53
Entailment is intractable

• If there is a single tuple in D violating c,
  Entailed(c,D) is false
• ? there is a tuple satisfying the negation nc of
  c
• So, Entailed(c,D) ? NoGACWipeOut(nc,D)
• Entailed is coNP-complete

54
Part 3 Operational Globality
55
Do we need global constraints?

• Expressiveness
• Effectiveness of propagation
• ?producing powerful effects
• Efficiency of propagation
• ?producing the result with little wasted effort

56
Effectiveness of propagation

• Global constraints give a global view on what
  values are consistent or not
• But are they all necessary?
• do they permit to rule out values decompositions
  cannot?

57
Early work decomposability of Montanari

• A constraint c is M-representable iff
• its primal graph N(X(c),D,C) is such that
  sol(X(c),D,C)c (ie, 2-decomp on the same vars)

Alldiff is M-representable
Montanari74
58
Early work decomposability of Montanari

• A constraint c is M-decomposable iff
• ?Y?X(c), sol(X(c),D,C)Y is M-representable

(1,2,3) is accepted whereas it does not
extend to X4
Alldiff is not M-decomposable
59
Early work decomposability of Montanari

• If a constraint c is M-decomposable into
  N(X(c),D,C) then N is globally consistent
• ? AC on N is equivalent to GAC on c

60
Early work decomposability of Montanari

• This condition is too restrictive
• Eg, it is well-known that if N is tree
  structured, AC prunes all non consistent values

1 2 3
1 2 3

1 2 3
lt
1 2 3
lt
61
Early work tree structure

• If a global constraint G can be decomposed as a
  tree-structured binary network N then GAC(c) is
  equivalent to AC(N)
• Allequal(X1,..X4)
• GAC(c)AC(N)

Genal00
62
Operational globality

• When do global constraints help propagating
  something more?
• If all decompositions in L hinder propagation
• If all absolute decompositions hinder propagation

63
Operational globality (wrt L)
GAC on alldiff prunes values 2 and 3 from X1,X2
because D(X3)D(X4)2,3

• alldiff(X1,..,Xn)
• Atmost1,1(X1,..,Xn)
• Atmost1,2(X1,..,Xn)
• Atmost1,3(X1,..,Xn)

Decomposition hinders GAC
64
Operational globality (wrt L)

• atmostk,v(X1,..Xn)

k2, v2
GAC on atmost prunes value 2 from X1,X2 because
there exist 2 variables X3 and X4 with D2
Roots(X1,..Xn,S,2) ? S?2
? Roots puts 3 and 4 in lb(S)

• S?2 removes 1 and 2 from ub(S)

? Roots prunes 2 from X1, X2
Decomposition preserves GAC
65
Operational globality (wrt L)

• Nvalue(X1,..Xn,N) with N1

Range(X1,..Xn,1..n,T) ? TN
? T ? 1,2,3,4,5
and both constraints are AC
whereas all 1, 4, and 5 should be pruned from
Xis
Decomposition hinders GAC
66
Operational globality

• When do global constraints help propagating
  something more?
• If all decompositions in L hinder propagation
• If all absolute decompositions hinder propagation

67
Absolute operational globality

• Alldiff is decomposable but

1,2,3,4
2,3
?
1,2,3,4
?
2,3
?
?
?
?
1,2,3,4
2,3
2,3
1,2,3,4
alldiff
Decomposition hinders GAC
68
Absolute operational globality

• Atmost

X1 X2 X3 Xn-1 Xn
B0 B1 B2 B3 Bn-1 Bn

• B0Bn, D(Bi)0,,n
• (Xiv BiBi-11)?(Xi?v Bi Bi-1), ?i
• Bn ? k

69
Operational globality

• When do global constraints help propagating
  something more?
• If all decompositions in L hinder propagation
• If all absolute decompositions hinder propagation

• Relaxing absolute decomposition

70
Relaxing absolute decomposition

• Absolute decomposition requires bounded arity
  (this was necessary from the semantic point of
  view)
• Relaxation Let LP be the language of all
  constraints on which GAC is polynomial
• G is GAC-poly-time decomposable iff there exists
  a decomposition of G wrt LP that does not hinder
  propagation
• How to know which constraints are decomposable?

Comics04
71
Using computational complexity

• Computational complexity can help deciding the
  existence of decompositions that preserve GAC
• If NoGACWipeOut(G) is NP-complete, then there
  does not exist any GAC-poly-time decomposition
  for G

72
Example

• Our decomposition of Nvalue Range(X1,..Xn,1..n
  ,T) ? TN hinders GAC
• Nvalue(X1,..Xn,n) is polynomial to propagate (?
  alldiff)
• What about Nvalue(X1,..Xn,N), where N is a
  variable?

73
Reduction of 3Sat

• Let ? be a 3Sat formula on x1..xn with m clauses
• Xi?-i,i, ?i?1..n
• Xnj ?f,-g,h, ?cjxf?? xg? xh ? ?
• N ? n
• Nvalue(X1,,Xn,Xn1,, Xnm,N) has a solution iff
  ? is satisfiable

Nvalue cannot be GAC-poly-time decomposed
X1 Xi Xn Xn1 Xnj
Xnm
-1 1
-i i
f -g h
-n n
74
Do we need global constraints?

• Expressiveness
• Effectiveness of propagation
• ?producing powerful effects
• Efficiency of propagation
• ?producing the result with little wasted effort

75
Efficiency of propagation

• Does GAC have the same complexity on
  decomposition and on the original constraint?
• Example Range can be encoded as a Gcc
  constraint, but GAC on Gcc is more expensive than
  on Range
• ? Algorithmic globality

Regin96,Comics06
76
So what?

• If a constraint has some inherent globality
• Semantic no way to escape!
• Operational do we accept losing pruning?
• Algorithmic do we accept a slow-down?
• Otherwise
• ? write a specific algorithm (see other talks)
• ? use a generic GAC algorithm

77
Part 4 Generic Algorithms
78
GAC3

• GAC3(X,D,C)
• Q ? (Xi,c) c ? C and Xi ? X(c)
• While Q ? ? do
• Pick (Xi,c) from Q
• If Revise(Xi,c) then
• If D(Xi) ? then return False
• Update Q
• Return True

Revise(Xi,c) Change ? False For each vi ? D(Xi)
do t ? seekSupport(Xi, vi, c) If tnil then
remove vi from D(Xi) Change ? True Return
Change
Mackworth77
79
Space of tuples
XYZ
D(X)1,2,3 D(Y)1,2 D(Z)3,4,5
80
seekSupport
XYZ
seekSupport(X,1)
81
seekSupport
XYZ
seekSupport(Y,1)
1
2
3
82
GAC3
seekSupport O(dk-1) checks
Nb of calls O(kd) per change

• GAC3(X,D,C)
• Q ? (Xi,c) c ? C and Xi ? X(c)
• While Q ? ? do
• Pick (Xi,c) from Q
• If Revise(Xi,c) then
• If D(Xi) ? then return False
• Update Q
• Return True

Revise(Xi,c) Change ? False For each vi ? D(Xi)
do t ? seekSupport(Xi, vi, c) If tnil then
remove vi from D(Xi) Change ? True Return
Change
Number of checks O(k2dk1)
83
Techniques used to be efficient

• Ordering Last support

84
Ordering Last
XYZ
seekSupport(Z,4)
Last(Z,4)
Last(Z,4)
85
GAC2001
Revise2001(Xi,c) Change ? False For each vi ?
D(Xi) do if Last(Xi,vi,c) non valid then t ?
seekNextSupport(Xi,vi,c,Last) If tnil then
remove vi from D(Xi) Change ? True Else
Last(Xi,vi,c) ? t Return Change
Revise(Xi,c) Change ? False For each vi ? D(Xi)
do t ? seekSupport(Xi, vi, c) If tnil then
remove vi from D(Xi) Change ? True Return
Change
O(dk-1) per value
Number of checks O(kdk)
Besal01
86
Techniques used to be efficient

• Ordering Last support
• Multidirectionality
• What does it mean?
• If a tuple t((X,v),(Y,w),) has been found (not
  to) to support (X,v), it also (doesnt) supports
  (Y,w)

87
Using multidirectionality

1. Do not check a tuple t if t has already been
   checked before
2. Do not look for support for (Xi,v) if a tuple
   supporting it has already been checked

88
Using multidirectionality

• Do not check a tuple t if t has already been
  checked before
• Observe that if t has been checked when looking
  for support for (Xi,v), Last(Xi,v) guarantees it
  wont be checked again by seekNextSupport(Xi,v)
• what about seekNextSupport(Xj,w)?
• ? GAC2001 doesnt ensure (1)

89
GAC-schema
seekNextSupport(Z,4)
Last(Z,4)
BesRég97
90
Using multidirectionality

• Gddf
• Do not look for support for (Xi,v) if a tuple
  supporting it has already been checked ?
  Last is not enough
• GAC-schema ensures (2) thanks to its fine-grained
  data structure
• St set of values supported by t
• SCXi,v set of supporting tuples containing
  (Xi,v)

91
GAC-schema
seekNextSupport(Xi,v)
Xi
Xj
seekNextSupport(Xj,w)?
v2 w
v w
t
92
GAC-schema

• Ordering Last support
• Multidirectionality

Number of checks O(dk)
the best complexity (like GAC4 which reads all
tuples in advance)
93
GAC-schema

• We always supposed that a constraint check is
  polynomial
• This is true when c is defined by a function fc
• But what happens with Table constraints? (ie, a
  list of allowed tuples)

94
Table constraints

• Use GAC4 ? heavy updates of his huge lists
• Use GAC-schematable
• It looks for valid tuples in the table T(c)
  instead of allowed tuples in D(X1) x x D(Xk)
• ? It can explore an exponential number of tuples
  in the table which are no longer valid
• Use GACallowed
• It jumps from T(c) to DX(c) and from DX(c) to T(c)

LhoReg05,LecSym06
95
Alternatives binary encodings

• AC on hidden representation
• AC on dual representation

96
AC on hidden representation

• AC(hidden) ? GAC(c)
• Specialized AC-hidden algorithm
• It requires updating the (exponential) hidden
  domain (? GAC4)

SamSte04
97
AC on dual representation

• AC(dual) ? Pairwise consistency

WipeOut(GAC) ? WipeOut(PW)
Beeal83,Janal89
98
Part 5 Other levels of consistency
99
Weaker levels of consistency

• Bound consistency was first introduced for
  arithmetic constraints on real variables
• Several adaptations to finite domain variables
• The most common (and most interesting) is BC(Z),
  which gets rid of domain size
• The others BC(D), RangeC explore domains,
  BC(R) is trivial on most constraints

Apt03,Dechter03,Choal05,Bessiere05
100
BC(Z)

• Definition
• A tuple t on c is a bound support iff t?c
  and
  min(Xi) ? tXi ? max(Xi), ?Xi ? X(c)
• Xi is bound consistent on c iff ?
  bound supports t, t such that tXimin(Xi) and
  tXimax(Xi)

101
Bound consistency(Z)
Y X2
X -1 0 1 2 3
Y 0 3 4
BC(Z)
GAC
102
Bound consistency(Z)

• Decomposability analysis for BC(Z)?
• It depends on domains representation
• When reasoning with BC, it is often the case that
  we accept domains represented as intervals
• ? It changes the size of the encoding!

103
Effect of the size
D(Yi) polynomial in (X,D,c) if domains
are encoded as intervals

• c ? (?Xi k)
• Y0Yn, D(Yi)0k
• Yi XiYi-1, ?i
• Y00, Yn k

? ?Xi k is decomposable on intervals

• In addition, BC(Z) on the decomposition is
  equivalent to BC(Z) on ?Xi k
• So, ?Xi k is BC-poly-time decomposable
• (remember it was not GAC-poly-time decomposable)

104
Effect of the size (2)
Y
1 2n
X

• BC(Z) can require an exponential number of checks
  even if the domain is small
• Let c(X,Y) be the constraint that accepts (v,w)
  iff v0 or (w mod 2n)base_2 is a model of 3Sat
  formula ?
• If D(X)0,1 and D(Y)1, 2n, checking BC(Z) on
  (X,1) is equivalent to solving SAT(?)
• ? BC(Z) is NP-hard whereas GAC takes 3
  checks!

0 1
?
01101001
105
Stronger levels of consistency
Pairwise consistency

• Reasoning on several constraints at a time
• Often requires deleting tuples in constraints
  (pairwise, relational)
• ? Exponential size and makes propagation difficult

DecvanB95
106
Stronger levels of consistency

• We can adapt inverse consistencies of the binary
  case
• Relational PIC, P(air)W(ise)IC
• WipeOut(PWIC) iff WipeOut(GAC(PW))
• Efficient algorithm for PWIC beats GAC quite
  often

FreElf96,SteWal06
107
The end