problem solving with constraints csce421/821, fall2012 cse.unl/~ choueiry/f12-421-821
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Path Consistency & Global Consistency Properties. Problem Solving with Constraints CSCE421/821, Fall2012 www.cse.unl.edu/~ choueiry/F12-421-821 All questions: Piazza Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Tel: +1(402)472-5444. Lecture Sources. Required reading - PowerPoint PPT PresentationTRANSCRIPT
Foundations of Constraint Processing
Basic Consistency Methods 1
Problem Solving with ConstraintsCSCE421/821, Fall2012
www.cse.unl.edu/~choueiry/F12-421-821All questions: Piazza
Berthe Y. Choueiry (Shu-we-ri)Avery Hall, Room 360Tel: +1(402)472-5444
Path Consistency & Global Consistency Properties
Foundations of Constraint Processing
Basic Consistency Methods 2
Lecture SourcesRequired reading1. Algorithms for Constraint Satisfaction Problems, Mackworth and
Freuder AIJ'852. Sections 3.1, 3.2, 3.3. Chapter 3. Constraint Processing. Dechter
Recommended1. Sections 3.4—3.10. Chapter 3. Constraint Processing. Dechter2. Networks of Constraints: Fundamental Properties and Application
to Picture Processing, Montanari, Information Sciences 743. Bartak: Consistency Techniques (link)4. Path Consistency on Triangulated Constraint Graphs, Bliek & Sam-
Haroud IJCAI'99
Foundations of Constraint Processing
Basic Consistency Methods 3
Outline1. Motivation2. Path consistency and its complexity3. Global consistency properties
– Minimality– Decomposability
4. When PC guarantees global consistency
Foundations of Constraint Processing
Basic Consistency Methods 4
AC is not enough Example borrowed from Dechter
Arc-consistent?Satisfiable?
seek higher levels of consistency
V1a b
a b a bV2 V3
=
V1b a
a b b aV2 V3
=
=
Foundations of Constraint Processing
Basic Consistency Methods 5
Outline1. Motivation2. Path consistency and its complexity3. Global consistency properties
– Minimality– Decomposability
4. When PC guarantees global consistency
Foundations of Constraint Processing
Basic Consistency Methods 6
Consistency of a pathA path (V0, V1, V2, …, Vm) of length m is consistent iff • for any value xDV0 and for any value yDVm 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) )
for all x DV0 for all y DVm
V0 Vm
Vm-1
V2
V1
Foundations of Constraint Processing
Basic Consistency Methods 7
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
for all x DV0 for all y DVm
V0 Vm
Vm-1
V2
V1
Foundations of Constraint Processing
Basic Consistency Methods 8
Example: consistency of a path Check path length = 2, 3, 4, 5, 6, ....
{a, b, c}
V2
{a, b, c}
V3
{a, b, c}
V1
{a, b, c}
V4
{a, b, c}V7
{a, b, c} V5
{a, b, c}
V6
All mutex constraints
Foundations of Constraint Processing
Basic Consistency Methods 9
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 isconsistent (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]
Foundations of Constraint Processing
Basic Consistency Methods 10
Tools for PC-1Two operators
1. Constraint composition: ( • )R13 = R12 • R23
2. Constraint intersection: ( )R13 R13, old R13, induced
Foundations of Constraint Processing
Basic Consistency Methods 11
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 Begin2 Yn R3 repeat4 begin5 Y0 Yn
6 For k 1 until n do7 For i 1 until n do8 For j 1 until n do9 Yl
ij Yl-1ij Yl-1
ik • Yl-1kk • Yl-1
kj
10 end11 until Yn = Y0
12 Y Yn
10 end
Foundations of Constraint Processing
Basic Consistency Methods 12
Properties of PC-1Discrete CSPs [Montanari'74]
1. PC-1 terminates2. 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..
Foundations of Constraint Processing
13
Complexity of PC-1Procedure PC-1:1 Begin2 Yn R3 repeat4 begin5 Y0 Yn
6 For k 1 until n do7 For i 1 until n do8 For j 1 until n do9 Yl
ij Yl-1ij Yl-1
ik • Yl-1kk • Yl-1
kj
10 end11 until Yn = Y0
12 Y Yn
10 end
Line 9: a3
Lines 6–10: n3. a3
Line 3: at most n2 relations x a2 elements
PC-1 is O(a5n5)
Basic Consistency Methods
PC-2 is O(a5n3) and (a3n3)
PC-1, PC-2 are specified using constraint composition
Foundations of Constraint Processing
Basic Consistency Methods 14
Enforcing Path Consistency (PC)General case: Complete graphTheorem: 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.
General case: Triangulated graphTheorem: 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
Foundations of Constraint Processing
Basic Consistency Methods
PPC versus PC
15
Foundations of Constraint Processing
Basic Consistency Methods 16
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 [Chmeiss & Jégou 1998]– PC2001: an improvement over PC8, not tested [Bessière et al. 2005]
Note: PC is seldom used in practical applications unless in presence of special type of constraints (e.g., bounded difference)
Project!
Foundations of Constraint Processing
Basic Consistency Methods 17
Path consistency as inference of binary constraints
Path consistency corresponds to inferring a new constraint(alternatively, tightening an existing constraint) between every twovariables given the constraints that link them to a third variable
Considers all subgraphs of 3 variables 3-consistency
B
C
A
B < C
A < B
B
C
AA < B
A < C
B < C
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Basic Consistency Methods 18
Path consistency as inference of binary constraints
Another example:
V4
V3
a b
V4
V2
V1
a b
V3
a b
a b
V2
a b
V1
a b
a b
=
=a b a b
Foundations of Constraint Processing
Basic Consistency Methods 19
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
C
A
B < C
A < B
A + 3 > C
B
C
AA < B
A + 3 > C
A < C
B < C
B
C
AA < B
-3 < A –C < 0
B < C
Solution: CVi,Vj CVi,Vj ij(CVi,Vk CVk,Vj)
Foundations of Constraint Processing
Basic Consistency Methods 20
Constraint propagation courtesy of Dechter
After Arc-consistency:
After Path-consistency:
• 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
1
2
3
1
2
3
1
2 2
3
( 0, 1 )
( 0, 1 )( 0, 1 )
( 0, 1 )
( 0, 1 )( 0, 1 )
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Basic Consistency Methods 21
Arc-consistent?
Path-consistent?
PC can detect unsatisfiability
V3
a b
V4
a bV2
V1 a b
a b
a b
Foundations of Constraint Processing
Basic Consistency Methods 22
• Question:– Is this CSP 3-consistent? – is it 2-consistent?
• Lesson: – 3-consistency does not guarantee 2-consistency
Warning: Does 3-consistency guarantee 2-consistency?
{red, blue}
A
B
{ red }
{red, blue}
{ red }
C
Foundations of Constraint Processing
Basic Consistency Methods 23
PC is not enough
Arc-consistent?Path-consistent?Satisfiable? we should seek (even) higher levels of consistency k-consistency, k = 1, 2, 3, …. …following lecture
{a, b, c}
V2
{a, b, c}
V3
{a, b, c}
V1
{a, b, c}
V4
{a, b, c}V7
{a, b, c} V5
{a, b, c}
V6
All mutex constraints
Foundations of Constraint Processing
Basic Consistency Methods 24
Outline1. Motivation2. Path consistency and its complexity3. Global consistency properties
– Minimality– Decomposability
4. When PC guarantees global consistency
Foundations of Constraint Processing
Basic Consistency Methods 25
MinimalityPC tightens the binary constraintsThe 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, but it
need not be
Foundations of Constraint Processing
Basic Consistency Methods 26
Minimal CSPMinimal 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 constraints to the direct constraint CVi, Vj [Montanari'74]
• The binary constraints are explicit as possible. [Montanari'74]
Foundations of Constraint Processing
Basic Consistency Methods 27
Strong n-consistent
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 solutionDecomposability: any consistent combination of values for
any k variables is extendable to a solution
Decomposable Minimal Path Consistent
n-consistent Solvable
Foundations of Constraint Processing
Basic Consistency Methods 28
Relations to (theory of) DB
CSP DatabaseMinimal • |C|-wise consistent
• The relations join completelyDecomposable ?
Foundations of Constraint Processing
Basic Consistency Methods 29
Outline1. Motivation2. Path consistency and its complexity3. Global consistency properties
– Minimality– Decomposability
4. When PC guarantees global consistency
Foundations of Constraint Processing
Basic Consistency Methods 30
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)
Foundations of Constraint Processing
Basic Consistency Methods
PPC versus PC
31
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PC: Special Case• 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
Foundations of Constraint Processing
Basic Consistency Methods 33
Distributivity propertyIn PC-1, two operations:
RAB • (RBC R'BC) = (RAB • RBC) (RAB • R’BC)
When ( • ) distributes over ( ), then [Montanari'74]
• PC-1 guarantees that CSP is minimal and decomposable• The outer loop of PC-1 can be removed
Intersection,
Composition, •
RAB R’BCRBCA
B
C
Foundations of Constraint Processing
Basic Consistency Methods
Condition does not always holdConstraint composition does not always distribute over constraint intersection•R12= R23= R’23=
• ⋅( ∩ ) = =⋅
•( ) ∩ ( )= ∩ = ⋅ ⋅
34
1 10 0
0 01 0
1 00 0
1 10 0
0 00 0
0 00 0
1 10 0
0 01 0
1 00 0
1 10 0
0 01 0
1 10 0
1 00 0
1 00 0
1 00 0
1 00 0
Foundations of Constraint Processing
Basic Consistency Methods
Temporal Reasoning constraints of bounded difference
35
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
Foundations of Constraint Processing
Basic Consistency Methods
Example: Temporal Reasoning
36
Composition of intervals + :R’13 = R12 + R23 = [4, 12]R01 + R13 = [2,5] + [3, 5] = [5, 10]R01 + R'13 = [2,5] + [4, 12] = [6, 17]
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
R’13
R13 =[3,5]
R23=[1,8]R12=[3,4]
V1 V3
V2
V0 R01=[2,5]
Foundations of Constraint Processing
Basic Consistency Methods 37
Composition Distributes over • PC-1 generalizes Floyd-Warshall algorithm (all-
pairs shortest path), where – composition is ‘scalar addition’ and– intersection is ‘scalar minimal’
• PC-1 generalizes Warshall algorithm (transitive closure)– Composition is logical OR– Intersection is logical AND
Foundations of Constraint Processing
Basic Consistency Methods 38
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
Foundations of Constraint Processing
Basic Consistency Methods 39
Summary1. Alert: Do not confuse a consistency property with the algorithms for
reinforcing it2. 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 at4. 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