Basic Consistency Methods
Foundations of Constraint Processing
CSCE421/821, Spring 2009
www.cse.unl.edu/~choueiry/S09-421-821/
All questions: cse421@cse.unl.edu
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
choueiry@cse.unl.edu
Tel: +1(402)472-5444
Foundations of Constraint Processing, Spring 2009
February 20, 2009
Basic Consistency Methods
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Lecture Sources
Required reading
1. Algorithms for Constraint Satisfaction Problems, Mackworth and
Freuder AIJ'85
2. Sections 3.1, 3.2, 3.3. Chapter 3. Constraint Processing. Dechter
Recommended
1. Sections 3.4—3.10. Chapter 3. Constraint Processing. Dechter
2. Networks of Constraints: Fundamental Properties and Application
to Picture Processing, Montanari, Information Sciences 74
3. Consistency in Networks of Relations, Mackworth AIJ'77
4. Constraint Propagation with Interval Labels, Davis, AIJ'87
5. Path Consistency on Triangulated Constraint Graphs, Bliek & SamHaroud IJCAI'99
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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Consistency checking
Motivation
• CSP are solved with search (conditioning)
• Search performance is affected by:
- problem size
- amount of backtracking
• Backtracking may yield.. thrashing
Thrashing
Exploring non-promising sub-trees and rediscovering the
same inconsistencies over and over again
 Malady of backtrack search
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(Some) causes of thrashing
Search order:
V1, V2, V3, V4, V5
V1
{1, 2, 3}
{1, 2, 3}
V3<V1
V3
{ 0, 1, 2}
V3<V4
V2
V3<V2
V3>0
V3<V5
V4
V5
{1, 2}
V5<V4
{ 1, 2}
What happens in search if we:
• do not check CV3 (node inconsistency)
• do not check constraint CV3,V5 (arc inconsistency)
• do not check constraints CV3,V4, CV3,V5, and CV4,V5 (path
inconsistency)
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Consistency checking
Goal: eliminate inconsistent combinations
Algorithms (for binary constraints):
•
•
•
•
Node consistency
Arc consistency (AC-1, AC-2, AC-3, ..., AC-7, AC-3.1, etc.)
Path consistency (PC-1, PC-2, DPC, PPC, etc.)
Constraints of arbitrary arity:
– Waltz algorithm ancestor of AC's
– Generalized arc-consistency (Dechter, Section 3.5.1 )
– Relational m-consistency (Dechter, Section 8.1)
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Overview of Recommended Reading
• Davis, AIJ'77
– Focuses on the Waltz algorithm
– Studies its performance and quiescence for given:
• Constraint types (bounded diff, algebraic, etc.)
• Domains types (continuous or finite)
• Mackworth, AIJ'77
– presents NC, AC-1, AC-2, PC-1, PC-2
• Mackworth and Freuder, AIJ'85
– studies their complexity
 Mackworth and Freuder concentrate on finite domains
 More people work on finite domains than on continuous ones
 Continuous domains can be quite tough
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Constraint Propagation with Interval Labels Ernest Davis
• `Old' paper (1987): terminology slightly different
• Interval labels: continuous domains
– Section 8: sign labels (discrete)
• Concerned with Waltz algorithm (e.g.,
quiescence, completeness)
• Constraint types vs. domain types (Table 3)
• Addresses applications of reasoning about
– Physical Systems (circuit, SPAM)
– time relations (TMM)
•  Advice: read Section 3, more if you wish
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Waltz algorithm for label inference
Vi
• REFINE(C(Vi, Vk, Vm, Vn), Vi) finds a
new label for Vi consistent with C.
C
Vk
Vm
Vn
• REVISE(C (Vi, Vk, Vm, Vn)) refines the domains of Vi, Vk, Vm, Vn by
iterating over these variables until quiescence (i.e., no domains is
further refined)
• Waltz Algorithm
– revises all constraints by iterating over each constraint connected to a
variable whose domain has been revised
– Is the ancestor of arc-consistency
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WARNING
→ Completeness: (i.e., for solving the CSP) Running the Waltz
Algorithm does not
B solve the problem.
A
{ 1, 2, 3 }
A
{ 2, 3 }
{ 2, 3, 4 }
B
{ 2, 3 }
A=2  B=3 is still not a solution!
→ Quiescence: The Waltz algorithm may go into infinite loops even
if problem is solvable
x  [0, 100]
x=y
y  [0, 100]
x = 2y
→ Davis characterizes the completeness and quiescence of the Waltz
algorithm (see Table 3) in terms of
– constraint types
– domain types
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Importance of this paper
 Establishes that constraints of bounded differences
(temporal reasoning) can be efficiently solved by the
Waltz algorithm (O(n3), n number of variables)
 Early paper that attracts attention on the difficulty of
label propagation in interval labels (continuous domains).
This work has been continued by Faltings (AIJ 92) who
studied the early-quiescence problem of the Waltz
algorithm.
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Basic consistency algorithms
Examining finite, binary CSPs and their complexity
Notation:
Given variables i, j, k with values x, y, z, the predicate
Pijk(x, y, z) is true iff the 3-tuple x, y, z  Cijk
Node consistency:
Arc consistency:
Path consistency:
checking Pi(x)
checking Pij(x,y)
bit-matrix manipulation
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Worst-case asymptotic complexity
Worst-case complexity
time
space
as a function of the input parameters
Upper bound: f(n) is O(g(n)) means that f(n)  c.g(n)
f grows as g or slower
Lower bound: f(n) is  (h(n)) means that f(n)  c.h(n)
f grows as g or faster
Input parameters for a CSP:
n = number of variables
a = (max) size of a domain
dk = degree of Vk ( n-1)
e = number of edges (or constraints)  [(n-1), n(n-1)/2]
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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Node consistency (NC)
Procedure NC(i):
Di  Di  { x | Pi(x) }
Begin
for i  1 until n do NC(i)
end
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Complexity of NC
Procedure of NC(i)
Di Di  { x | Pi(x) }
Begin
for i  1 until n do NC(i)
end
• For each variable, we check a values
• We have n variables, we do n.a checks
• NC is O(a.n)
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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Arc-consistency
Adapted from Dechter
Definition: Given a constraint graph G,
• A variable Vi is arc-consistent relative to Vj iff for every value aDVi,
there exists a value bDVj | (a, b)CVi,Vj.
Vi
Vj
1
1
2
Vj
1
2
2
3
Vi
3
2
3
• The constraint CVi,Vj is arc-consistent iff
– Vi is arc-consistent relative to Vj and
– Vj is arc-consistent relative to Vi.
• A binary CSP is arc-consistent iff every constraint (or sub-graph of
size 2) is arc-consistent
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Procedure Revise
Revises the domains of a variable i
Procedure Revise(i,j):
Begin
DELETE  false
for each x  Di do
if there is no y  Dj such that Pij(x, y) then
begin
delete x from Di
DELETE  true
end
return DELETE
end
• Revise is directional
• What is the complexity of Revise?
{a2}
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Revise: example
R. Dechter
Apply the Revise procedure to the
following example
X
Y
X+Y = 10
{ 1, 2, .., 10}
{5, 6, .. 15}
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Effect of Revise
Question:
Adapted from Dechter
Vj
Vi
1
1
Vi
Vj
1
Given
2
2
2
2
• two variables Vi and Vj
3
3
3
• their domains DVi and DVj, and
• the constraint CVi,Vj,
write the effect of the Revise procedure as a sequence of operations
in relational algebra
Hint:
• Think about the domain DVi as a unary constraint CVi and
• consider the composition of this unary constraint and the binary
one..
Solution:
DVi  DVi  Vi (CVi,Vj
DVi)
This is actually equivalent to DVi  Vi (CVi,Vj
DVi)
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Arc Consistency (AC-1)
Procedure AC-1:
1 begin
2 for i  1 until n do NC(i)
3 Q  {(i, j) | (i,j)  arcs(G), i  j }
4 repeat
5
begin
6
CHANGE  false
7
for each (i, j)  Q do CHANGE  ( REVISE(i, j) or CHANGE )
8
end
9 until ¬ CHANGE
10 end
• AC-1 does not update Q, the queue of arcs
• No algorithm can have time complexity below O(ea2)
• Alert: Algorithms do not, but should, test for empty domains
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Warning
• What I don’t like about the algorithms as
specified in most papers is that they do not
check for domain wipe-out.
• In your code, always check for domain
wipe out and terminate/interrupt the
algorithm when that occurs
• In practice may save lots of #CC and
cycles
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Arc consistency
1. AC may discover the solution
Example borrowed from Dechter
V1
V1
V3
V2
V1
V2
V3
V2
V1
V3
V2
V3
V2
V1
V1
V2
V3
V1
V3
V2
V3
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Arc consistency
2. AC may discover inconsistency
Example borrowed from Dechter
X
{ 1, 2, 3 }
Z<X
X<Y
Y
{ 1, 2, 3 } Z
{ 1, 2, 3 }
Y<Z
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NC & AC
Example courtesy of M. Fromherz
In the temporal problem below
1. AC propagates bound
2. Unary constraint x>3 is imposed
3. AC propagates bounds again
x
[0, 10]
x
x  y-3
y
[0, 10]
[0, 7]
x
x  y-3
x  y-3
y
[3, 10]
[4, 7]
y
[7, 10]
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Complexity of AC-1
Procedure AC-1:
1 begin
2 for i  1 until n do NC(i)
3 Q  {(i, j) | (i,j)  arcs(G), i  j
4 repeat
5 begin
6 CHANGE  false
7
for each (i, j)  Q do CHANGE  (REVISE(i, j) or CHANGE)
8 end
9 until ¬ CHANGE
10 end
Note: Q is not modified and |Q| = 2e
• 4  9 repeats at most n·a times
• Each iteration has |Q| = 2e calls to REVISE
• Revise requires at most a2 checks of Pij
 AC-1 is O(a3 · n · e)
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AC versions
• AC-1 does not update Q, the queue of arcs
• AC-2 iterates over arcs connected to at least
one node whose domain has been modified.
Nodes are ordered.
• AC-3 same as AC-2, nodes are not ordered
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Arc consistency (AC-3)
AC-3 iterates over arcs connected to at least one
node whose domain has been modified
Procedure AC-3:
1 begin
2 for i  1 until n do NC(i)
3 Q  { (i, j) | (i, j)  arcs(G), i  j }
4 While Q is not empty do
5 begin
6 select and delete any arc (k, m) from Q
7 If Revise(k, m) then Q  Q  { (i, k) | (i, k)  arcs(G), i  k, i  m }
8 end
9 end
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Complexity of AC-3
Procedure AC-3:
1 begin
2 for i  1 until n do NC(i)
3 Q  {(i, j) | (i, j) arcs(G), i  j }
4 While Q is not empty do
5 begin
6 select and delete any arc (k,
m) from Q
7 If Revise(k, m) then Q  Q
 { (i, k) | (i, k)  arcs(G), i 
k, i  m }
8 end
9 end
•
•
First |Q| = 2e, then it grows and shrinks
48
Worst case:
– 1 element is deleted from DVk per iteration
– none of the arcs added is in Q
• Line 7: a·(dk - 1)
n
• Lines 4 - 8: 2e  k 1 a  d k  1  2e  a  (2e  n)
• Revise: a2 checks of Pij
 AC-3 is O(a2(2e + a(2e-n)))
 Connected graph (e  n – 1), AC-3 is
O(a3e)
 If AC-p, AC-3 is O(a2e)  AC-3 is (a2e)
 Complete graph: O(a3n2)
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Example: Apply AC-3
Example: Apply AC-3
Thanks to Xu Lin
V2
V1
{ 1, 2, 3, 4, 5 }
{ 1, 2, 3, 4, 5 }
V3 { 1, 2, 3, 4, 5 }
{ 1, 2, 3, 4, 5 }
V2
V1
V4
V3
{ 1, 3, 5 }
{ 1, 2, 3, 4 }
{ 1, 3, 5 }
{ 1, 2, 3, 5 }
V4
DV1 = {1, 2, 3, 4, 5}
DV2 = {1, 2, 3, 4, 5}
DV3 = {1, 2, 3, 4, 5}
DV4 = {1, 2, 3, 4, 5}
CV2,V3 = {(2, 2), (4, 5), (2, 5), (3, 5), (2, 3), (5, 1), (1, 2), (5, 3), (2, 1), (1, 1)}
CV1,V3 = {(5, 5), (2, 4), (3, 5), (3, 3), (5, 3), (4, 4), (5, 4), (3, 4), (1, 1), (3, 1)}
CV2,V4 = {(1, 2), (3, 2), (3, 1), (4, 5), (2, 3), (4, 1), (1, 1), (4, 3), (2, 2), (1, 5)}
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Applying AC-3
Thanks to Xu Lin
• Queue = {CV2, V4, CV4,V2, CV1,V3, CV2,V3, CV3, V1, CV3,V2}
• Revise(V2,V4): DV2 DV2 \ {5} = {1, 2, 3, 4}
• Queue = {CV4,V2, CV1,V3, CV2,V3, CV3, V1, CV3, V2}
• Revise(V4, V2): DV4  DV4 \ {4} = {1, 2, 3, 5}
• Queue = {CV1,V3, CV2,V3, CV3, V1, CV3, V2}
• Revise(V1, V3): DV1  {1, 2, 3, 4, 5}
• Queue = {CV2,V3, CV3, V1, CV3, V2}
• Revise(V2, V3): DV2  {1, 2, 3, 4}
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Applying AC-3 (cont.)
Thanks to Xu Lin
• Queue = {CV3, V1, CV3, V2}
• Revise(V3, V1): DV3  DV3 \ {2} = {1, 3, 4, 5}
• Queue = {CV2, V3, CV3, V2}
• Revise(V2, V3): DV2  DV2
• Queue = {CV3, V2}
• Revise(V3, V2): DV3  {1, 3, 5}
• Queue = {CV1, V3}
• Revise(V1, V3): DV1  {1, 3, 5}
END
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Main Improvements
Mohr & Henderson (AIJ 86):
• AC-3 O(a3e)  AC-4 O(a2e)
• AC-4 is optimal
• Trade repetition of consistency-check operations with heavy
bookkeeping on which and how many values support a given value
for a given variable
• Data structures it uses:
–
–
–
•
m: values that are active, not filtered
s: lists all vvp's that support a given vvp
counter: given a vvp, provides the number of support provided by a
given variable
How it proceeds:
1. Generates data structures
2. Prepares data structures
3. Iterates over constraints while updating support in data structures
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Step 1: Generate 3 data structures
• m and s have as many rows as there are vvp’s in the problem
• counter has as many rows as there are tuples in the constraints
m
(V4 , 5)
(V3 , 2)
(V3 , 3)
(V2 , 2)

(V4 , 4)
s
T
T
T
T

T
(V4 , 5)
(V3 , 2)
(V3 , 3)
(V2 , 2)

(V4 , 4)
counter
NIL
NIL
NIL
NIL

NIL
(V4 , V2 ), 3 0
(V4 , V2 ), 2
0
(V4 , V2 ), 1 0
(V2 , V3 ), 1 0


(V1, V3 ), 4 0
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Step 2: Prepare data structures
Data structures: s and counter.
•
•
Checks for every constraint CVi,Vj all tuples Vi=ai, Vj=bj_)
When the tuple is allowed, then update:
– s(Vj,bj)  s(Vj,bj)  {(Vi, ai)} and
– counter(Vj,bj)  (Vj,bj) + 1
Update counter ((V2, V3), 2) to value 1
Update counter ((V3, V2), 2) to value 1
Update s-htable (V2, 2) to value ((V3, 2))
Update s-htable (V3, 2) to value ((V2, 2))
Update counter ((V2, V3), 4) to value 1
Update counter ((V3, V2), 5) to value 1
Update s-htable (V2, 4) to value ((V3, 5))
Update s-htable (V3, 5) to value ((V2, 4))
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Constraints CV2,V3 and CV3,V2
S
(V4, 5)
Counter
Nil
(V3, 2)
((V2, 1), (V2, 2))
(V3, 3)
((V2, 5), (V2, 2))
(V4, V2), 3
0
(V4, V2), 2
0
(V4, V2), 1
0
(V2, 2)
((V3,1), (V3,3), (V3, 5), (V3,2))
(V2, V3), 1
2
(V3, 1)
((V2, 1), (V2, 2), (V2, 5))
(V2, V3), 3
4
(V2, 3)
((V3, 5))
(V4, V2), 5
1
(V1, 2)
Nil
(V2, V4), 1
0
(V2, 1)
((V3, 1), (V3, 2))
(V2, V3), 4
0
(V1, 3)
Nil
(V4, V2), 4
1
(V3, 4)
Nil
(V2, V4), 2
0
(V4, 2)
Nil
(V2, V3), 5
0
(V3, 5)
((V2, 3), (V2, 2), (V2, 4))
(V2, V4), 3
2
(v4, 3)
Nil
(V2, V4), 4
0
(V1, 1)
Nil
(V2, V4), 5
0
(V2, 4)
((V3, 5))
(V3, V1), 1
0
(V2, 5)
((V3, 3), (V3, 1))
(V3, V1), 2
0
(V4, 1)
Nil
(V3, V1), 3
0
(V1, 4)
Nil
…
…
(V1, 5)
Nil
(V3, V2), 4
0
(V4, 4)
Nil
Etc…
Etc
.
Updating m
Note that (V3, V2),4  0 thus
we remove 4 from the domain of V3
and update (V3, 4)  nil in m
Updating counter
Since 4 is removed from DV3 then
for every (Vk, l) | (V3, 4)  s[(Vk, l)],
we decrement counter[(Vk, V3), l] by 1
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Summary of arc-consistency algorithms
•
AC-4 is optimal (worst-case)
•
Warning: worst-case complexity is pessimistic.
Better worst-case complexity: AC-4
Better average behavior: AC-3
[Mohr & Henderson, AIJ 86]
[Wallace, IJCAI 93]
•
AC-5: special constraints
[Van Hentenryck, Deville, Teng 92]
functional, anti-functional, and monotonic constraints
•
AC-6, AC-7: general but rely heavily on data structures for bookkeeping
[Bessière & Régin]
•
Now, back to AC-3: AC-2000, AC-2001≡AC-3.1, AC3.3, etc.
•
Non-binary constraints:
– GAC (general)
– all-different (dedicated)
[Mohr & Masini 1988]
[Régin, 94]
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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CSP parameters ‹n, a, t, d›
• n is number of variables
• a is maximum domain size
forbidden tuples
• t is constraint tightness: t 
all tuples
• d is constraint density d  (e  emin )
(emax  emin )
where e is the #constraints, emin=(n-1), and
Lately, we use constraint ratio p = e/emax
emax = n(n-1)/2
→ Constraints in random problems often generated uniform
→ Use only connected graphs (throw the unconnected ones away)
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Criteria for performance comparison
1. Bounding time and space complexity (theoretical)
– worst-case: always
– average-case: never
– best- case: sometimes
2. Counting #CC and #NV (theoretical, empirical)
3. Measuring CPU time (empirical)
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Performance comparison
Courtesy of Lin XU
AC-3, AC-7, AC-4 on n=10,a=10,t,d=0.6 displaying #CC and CPU time
AC-3
AC-4
AC-4
AC-7
AC-3
AC-7
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Wisdom (?)
Free adaptation from Bessière
• When a single constraint check is very
expensive to make, use AC-7.
• When there is a lot of propagation, use AC-4
• When little propagation and constraint checks
are cheap, use AC-3x
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AC2001
• When checking (Vi,a) against Vj
– Keep track of the Vj value that supports (Vi,a)
–Last((Vi,a),Vi)
• Next time when checking again (Vi,a)
against Vj, we start from Last((Vi,a),Vi)
and don’t have to traverse again the
domain of Vj again or list of tuples in CViVj
• Big savings..
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Performance comparison
Courtesy of Shant
AC3, AC3.1, AC4 on on n=40,a=16,t,d=0.15 displaying #CC and CPU time
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AC-what?
Instructor’s personal opinion
• Used to recommend using AC-3..
• Now, recommend using AC2001
• Do the project on AC-* if you are curious.. [Régin 03]
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AC is not enough
Example borrowed from Dechter
V1
V1
a
b
b
=
=
V2
a
a
V3
b
a
b
V2
a
b
V3
b
a
=
Arc-consistent?
Satisfiable?
 seek higher levels of consistency
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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Consistency of a path
A path (V0, V1, V2, …, Vm) of length m is consistent iff
•
•
for any value xDV0 and for any value yDVm that are consistent (i.e., PV0 Vm(x, y))
 a sequence of values z1, z2, … , zm-1 in the domains of variables V1, V2, …, Vm-1,
such that all constraints between them (along the path, not across it) are satisfied
(i.e., PV0 V1(x, z1)  PV1 V2(z1, z2)  …  PVm-1 Vm(zm-1, zm) )
V2
V1
V0
Vm-1
for all y  DVm
for all x  DV0
Vm
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Note
 The same variable can appear more than once in the path
 Every time, it may have a different value
 Constraints considered: PV0,Vm and those along the path
 All other constraints are neglected
V2
V1
V0
Vm-1
for all y  DVm
for all x  DV0
Vm
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Example: consistency of a path
 Check path length = 2, 3, 4, 5, 6, ....
All mutex constraints
V2

V1
{a, b, c}

V3

{a, b, c}
{a, b, c}





{a, b, c}


{a, b, c}
V7
{a, b, c}

{a, b, c}
V4
V5

V6
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Path consistency: definition
 A path of length m is path consistent
 A CSP is path consistent
Property of a CSP
Definition: A CSP is path consistent (PC) iff every path is
consistent (i.e., any length of path)
Question: should we enumerate every path of any length?
Answer: No, only length 2, thanks to [Mackworth AIJ'77]
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Making a CSP Path Consistent (PC)
Special case: Complete graph
Theorem: In a complete graph, if every path of length 2 is
consistent, the network is path consistent [Mackworth AIJ'77]
 PC-1: two operations, composition and intersection
 Proof by induction.
Special case: Triangulated graph
Theorem: In a triangulated graph, if every path of length 2 is
consistent, the network is path consistent [Bliek & Sam-Haroud ‘99]
 PPC (partially path consistent)  PC
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Tools for PC-1
Two operators
1. Constraint composition: ( • )
R13 = R12 • R23
2. Constraint intersection: (  )
R13 R13, old  R13, induced
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Path consistency (PC-1)
Achieved by composition and intersection (of binary relations
expressed as matrices) over all paths of length two.
Procedure PC-1:
1 Begin
2
Yn  R
3
repeat
4
begin
5
Y0  Yn
6
For k  1 until n do
7
For i  1 until n do
8
For j  1 until n do
9
Ylij  Yl-1ij  Yl-1ik • Yl-1kk • Yl-1kj
10
end
11 until Yn = Y0
12 Y  Yn
10 end
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Properties of PC-1
Discrete CSPs
[Montanari'74]
1. PC-1 terminates
2. PC-1 results in a path consistent CSP
• PC-1 terminates. It is complete, sound (for
finding PC network)
• PC-2: Improves PC-1 similar to how AC3
improves AC-1
Complexity of PC-1..
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Complexity of PC-1
Procedure PC-1:
1
2
3
4
5
6
7
8
9
10
11
12
10
Begin
Yn  R
repeat
begin
Y0  Yn
For k  1 until n do
For i  1 until n do
For j  1 until n do
Ylij  Yl-1ij  Yl-1ik • Yl-1kk • Yl-1kj
end
until Yn = Y0
Y  Yn
end
Line 9:
a3
Lines 6–10: n3. a3
Line 3: at most n2 relations x a2
elements
PC-1 is O(a5n5)
PC-2 is O(a5n3) and (a3n3)
PC-1, PC-2 are specified using
constraint composition
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Some improvements
• Mohr & Henderson (AIJ 86)
– PC-2 O(a5n3)  PC-3 O(a3n3)
– Open question: PC-3 optimal?
• Han & Lee (AIJ 88)
– PC-3 is incorrect
– PC-4 O(a3n3) space and time
• Singh (ICTAI 95)
– PC-5 uses ideas of AC-6 (support bookkeeping)
• Also:
– PC8: iterates over domains, not constraints
– PC2001: an improvement over PC8, not tested
[Chmeiss & Jégou 1998]
Project!
[Bessière et al. 2005]
Note: PC is seldom used in practical applications unless in presence of
special type of constraints (e.g., bounded difference)
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Path consistency as inference of binary constraints
B
B
A< B
A< B
A
A
B<C
B<C
A< C
C
C
Path consistency corresponds to inferring a new constraint
(alternatively, tightening an existing constraint) between every two
variables given the constraints that link them to a third variable
 Considers all subgraphs of 3 variables
 3-consistency
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Path consistency as inference of binary constraints
Another example:
V1
V1
a b
a b
V2


V3
a b

=
a b
a b


V2
V3
aa bb
=



a b
a b
V4
V4
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Question
Adapted from Dechter
Given three variables Vi, Vk, and Vj and the constraints CVi,Vk, CVi,Vj, and
CVk,Vj, write the effect of PC as a sequence of operations in relational
algebra.
B
B
B
A< B
A< B
A< B
A
A
A
B<C
B<C
B<C
A< C
A+ 3 > C
C
C
A+3> C
Solution: CVi,Vj  CVi,Vj  ij(CVi,Vk
C
-3 < A –C < 0
CVk,Vj)
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Constraint propagation
After Arc-consistency:
1
1
2
2
3
3
courtesy of Dechter
1
2
2
3
After Path-consistency:
( 0, 1 )

( 0, 1 )
•
( 0, 1 )

( 0, 1 )

( 0, 1 )

( 0, 1 )
Are these CSPs the same?
– Which one is more explicit?
– Are they equivalent?
•
The more propagation,
– the more explicit the constraints
– the more search is directed towards a solution
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PC can detect unsatisfiability
V1
a b



V2
V3

a b
aa bb


a b
Arc-consistent?
V4
Path-consistent?
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Warning: Does 3-consistency guarantee 2-consistency?
B
{red,
{red,blue}
blue}
A


C
{ red }
{ red }
• Question:
– Is this CSP 3-consistent?
– is it 2-consistent?
• Lesson:
– 3-consistency does not guarantee 2-consistency
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PC is not enough
All mutex constraints
V2

{a, b, c}


{a, b, c}

V1
V3

{a, b, c}
V7




{a, b, c}
V4




{a, b, c}


{a, b, c}
V5
Arc-consistent?


{a, b, c}
Path-consistent?
V6
Satisfiable?
 we should seek (even) higher levels of consistency
 k-consistency, k = 1, 2, 3, ….
…following lecture
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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Minimality
PC tightens the binary constraints
The tightest possible binary constraints yield the minimal network
Minimal network
a.k.a. central problem
Given two values for two variables, if they are consistent, then they
appear in at least one solution.
Note:
• Minimal  path consistent
• The definition of minimal CSP is concerned with binary CSPs
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Minimal CSP
Minimal network
a.k.a. central problem
Given two values for two variables, if they are consistent, then they
appear in at least one solution.
Informally
• In a minimal CSP the remainder of the CSP does not add any
further constraint to the direct constraint CVi, Vj between the two
variables Vi and Vj
[Mackworth AIJ'77]
• A minimal CSP is perfectly explicit: as far as the pair Vi and Vj is
concerned, the rest of the network does not add any further
constraint to the direct constraint CVi, Vj
[Montanari'74]
• The binary constraints are explicit as possible.
[Montanari'74]
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Decomposability
• Any combination of values for k variables that satisfy the constraints
between them can be extended to a solution.
• Decomposability generalizes minimality
Minimality: any consistent combination of values for
any 2 variables is extendable to a solution
Decomposability: any consistent combination of values for
any k variables is extendable to a solution
Decomposable
 Minimal 
Path Consistent
Strong n-consistent  n-consistent  Solvable
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Relations to (theory of) DB
CSP
Database
Minimal
• |C|-wise consistent
• The relations join completely
Decomposable
?
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Outline
1.
2.
3.
4.
Motivation and background
Node consistency and its complexity
Arc consistency and its complexity
Criteria for performance comparison and CSP
parameters.
5. Path consistency and its complexity
6. Global consistency properties
– Minimality
– Decomposability
7. When PC guarantees global consistency
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PC approximates..
In general:
• Decomposability  minimality  path consistent
• PC is used to approximate minimality (which is the central problem)
When is the approximation the real thing?
Special cases:
• When composition distributes over intersection,
[Montanari'74]
PC-1 on the completed graph guarantees minimality and
decomposability
• When constraints are convex
[Bliek & Sam-Haroud 99]
PPC on the triangulated graph guarantees minimality and
decomposability (and the existing edges are as tight as possible)
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PC: algorithms and property
CSP0
Complete
the graph
PC-1
Result:
•
CSP0 is not path consistent or
•
PC-1(CSP0) is path consistent
1. It is tight, but could be tighter
2. The graph is complete
CSP0
Triangulate
the graph
PPC
Watch for PPC
Result:
•
CSP0 is not path consistent or
•
PPC(CSP0) is path consistent
1. It is generally less tight than PC-1(CSP0)
2. The graph is triangulated
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PC algorithms: special cases
CSPa
Complete
the graph
PC-1/2/4
With a(bc)=(ab)(bc)
Result:
•
CSPa is not path consistent or
•
PC-1(CSPa) is
[Montanari 74]
1. Path consistent, minimal & decomposable
2. Graph is complete
CSPb
Triangulate
the graph
PPC
Constraint are convex
Result:
•
CSPb is not path consistent or
•
PPC(CSPb) is
[B&S-H 99]
1. Path consistent, minimal, decomposable
2. Existing edges are as tight as PC-1(CSPb)
3. Graph is triangulated
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Special conditions: examples
• Distributivity property
– Outer loop in PC-1 (PC-3) can be ignored
• Exploiting special conditions in temporal
reasoning:
– Temporal constraints in the Simple Temporal Problem
(STP): composition & intersection
– Composition distributes over intersection
• PC-1 is a generalization of the Floyd-Warshall algorithm (all
pairs shortest path)
– Convex constraints
• PPC
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Distributivity property
Intersection, 
In PC-1, two operations:
Composition, •
B
RAB
A
RBC
R’BC
C
RAB • (RBC  R'BC) = (RAB • RBC)  (RAB • R’BC)
When ( • ) distributes over (  ), then
[Montanari'74]
1. PC-1 guarantees that CSP is minimal and decomposable
2. The outer loop of PC-1 can be removed
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Temporal Reasoning
constraints of bounded difference
Variables: X, Y, Z, etc.
Constraints: a  Y-X  b, i.e. Y-X = [a, b] = I
Composition: I 1 • I2 = [a1, b1] • [a2, b2] = [a1+ a2, b1+b2]
Interpretation:
– intervals indicate distances
– composition is triangle inequality.
Intersection: I1  I2 = [max(a1, a2), min(b1, b2)]
Distributivity: I1 • (I2  I3) = (I1 • I2)  (I1 • I3)
Proof: left as an exercise
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Example: Temporal Reasoning
V2
Composition of intervals + :
R’13 = R12 + R23 = [4, 12]
V0
R01 + R13 = [2,5] + [3, 5] = [5, 10]
R01 + R'13 = [2,5] + [4, 12] = [6, 17]
R12=[3,4]
R01=[2,5] V1
R23=[1,8]
R’13
V3
R13 =[3,5]
Intersection of intervals: R13  R'13 = [4, 12]  [3, 5] = [4, 5]
R01 + (R13  R'13) = (R01 + R13)  (R01 + R'13)
R01 + (R13  R'13) = [2, 5] + [4, 5] = [6, 10]
(R01 + R13)  (R01 + R'13) = [5, 10]  [6,17] = [6, 10]
Here, path consistency guarantees minimality and decomposability
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PC-1 on the STP
• PC-1 generalizes Floyd-Warshall
algorithm (all-pairs shortest path), where
– composition is ‘scalar addition’ and
– intersection is ‘scalar minimal’
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Convex constraints: temporal reasoning (again!)
Thanks to Xu Lin (2002)
• Constraints of bounded difference are convex
• We triangulate the graph (good heuristics exist)
• Apply PPC: restrict propagations in PC to triangles of the
graph (and not in the complete graph)
• According to [Bliek & Sam-Haroud 99] PPC becomes
equivalent to PC, thus it guarantees minimality and
decomposability
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Summary
1. Alert: Do not confuse a consistency property with the algorithms for
reinforcing it
2. Local consistency methods
–
–
–
–
–
Remove inconsistent values (node, arc consistency)
Remove Inconsistent tuples (path consistency)
Get us closer to the solution
Reduce the ‘size’ of the problem & thrashing during search
Are ‘cheap’ (i.e., polynomial time)
3. Global consistency properties are the goal we aim at
4. Sometimes (special constraints, graphs, etc) local consistency guarantees
global consistency
–
E.g., Distributivity property in PC, row-convex constraints, special networks
5. Sometimes enforcing local consistency can be made cheaper than in the
general case
–
E.g., functional constraints for AC, triangulated graphs for PC
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