Transcript More on Consistency Properties

More on Constraint Consistency
Properties & Algorithms
Problem Solving with Constraints
CSCE421/821, Fall 2015
www.cse.unl.edu/~choueiry/F15-421-821/
All questions to Piazza
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
Tel: +1(402)472-5444
Foundations of Constraint Processing
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Summary
1. Global properties
2. Local properties
– Binary CSPs
– Non-Binary CSPs
3. Effects of Consistency Algorithms
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
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Outline
1. Global properties
2. Local properties
– Binary CSPs
Dechter Sections 3.1, 3.2, 3.3, 3.4
– Non-Binary CSPs
Dechter Sections 3.5.1, 8.1
3. Effects of Consistency Algorithms
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
Dechter Sections 3.5.3
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Global Consistency Properties
• Minimality & Decomposability
– Originally defined for binary CSPs
– Easily extendable to non-binary CSPs
• Minimality
Dechter Definition 2.6
– Every constraint is as tight as it can be
– Minimality  n-consistency
– In DB, the relations are said to “join completely”
• Decomposability
– Every consistent partial solution can be terminated backtrack
free
– Decomposability ≡ strong n-consistency
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Graphical Illustration
Minimality
tuple
n-2 variables
Decomposability
any consistent assignment of length k
Arc consistency
n-k variables
(1,1)-consistency
vvp
(2,1)-consistency
Path consistency
tuple
(k-1,1)-consistency
k-consistency
any consistent assignment of length k-1
kth variables
(i,j)-consistency
any consistent assignment
of length i
j variables
(1,j)-consistency
vvp
j variables
vvp
deg variables
inverse consistency
NIC
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Outline
1. Global properties
2. Local properties
– Binary CSPs
– Non-Binary CSPs
3. Effects
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
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Local Properties: Binary CSPs
• Classical ones
– Based on variations of (i,j)-consistency
• More recently
– Singleton Arc Consistency
– Inverse Consistency
– Neighborhood Inverse Consistency
– (Conservative) Dual consistency
• Special Constraints
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Classical Local Consistency: Properties
• Arc consistency
– Every vvp can be extended to a partial solution of length 2
• Path consistency
– Every partial solution of length 2 can be extended to a partial
solution of length 3
• i-consistency
– Every partial solution of length (i-1) can be extended to a partial
solution of length i
• (i,j)-consistency
– Every partial solution of length i can be extended to a partial
solution of length i+j
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Classical Local Consistency: Algorithms
• Arc consistency:
– AC-1, 2, 3, …, 7, AC-2001, AC-*, …
– Effect: domain filtering
– Complexity: in n2
• Path consistency
– PC-1, 2, 3, …, 8, PC2001, PPC, …
– Effect: adds binary constraints, modifies the width of network
– Complexity: in n3
• i-consistency
– Dechter Figure 3.14 & 3.15
– Effect: adds constraints of arity i-1, modifies the arity of network
– Complexity: in ni
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Local Properties: Binary CSPs
• Classical ones
– Based on variations of (i,j)-consistency
• More recently
– Singleton Arc Consistency
– Inverse Consistency
– Neighborhood Inverse Consistency
– (Conservative) Dual consistency
• Special Constraints
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Singleton Arc Consistency (SAC)
• Property: The CSP is AC for every vvp
• (Sketchy) Algorithm
Repeat until no change occurs
Repeat for each variable
Repeat for each value in domain
Assign this value to this variable.
If the CSP is AC, keep the value.
Otherwise, remove it.
• Effect: domain filtering
• Note
– Proposed by Debruyne & Bessière, IJCAI 97
– Quite expensive, but can be quite effective
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Inverse Consistency
• Path Inverse Consistency (PIC)
– Equivalent to (1,2)-consistency
• Inverse m-consistency
– Equivalent to (1,m)-consistency
• Neighborhood Inverse Consistency (NIC)
– Every vvp participates in a solution in the CSP
induced by its neighborhood
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Neighborhood Inverse Consistency (NIC): Algorithm
Repeat until no change occurs
Repeat for each variable
Consider only the neighborhood of the variable
Repeat for each value for the variable
If the value appears in a complete solution for the
neighborhood, then keep it.
Otherwise, remove it.
• Effect: domain filtering
• Note
– Proposed by Freuder & Elfe, AAAI 96
– Very effective, very expensive
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Summary: Binary CSPs
• (i,j)-consistency
– j=1
•
•
•
•
Arc consistency: (1,1)-consistency
Path consistency: (2,1)-consistency
i-consistency: (i-1,1)-consistency
Strong i-consistency: k-consistency for all k ≤ i
– i=1, inverse consistency
• Path Inverse Consistency (PIC): (1,2)-consistency
• Global Inverse Consisntecy (GIC): (1,n-1) consistency (minimal
domains)
• Neighborhood Inverse Consistency (NIC)
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Local Properties: Binary CSPs
• More recently
–
–
–
–
Singleton Arc Consistency
Partial Path Consistency (PPC)
(Conservative) Dual consistency
Conservative Path Consistency
Debruyene ICTAI 99
• Special Constraints
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Special Constraints: AC
[Van Hentenryck et al. AIJ 92]
• Specialized AC algorithms exist for special constraints
• Functional
A constraint C is functional with respect to a domain D iff for all vD
(respectively wD) there exists at most one wD (respectively vD)
such that C(v,w)
• Anti-functional
A constraint C is anti-functional with respect to a domain D iff C is
functional with respect to D
• Mon
otonic
A constraint C is monotonic with respect to a domain D iff there
exists a total ordering on D such that, for all v and wD, C(v,w)
holds implies C(v’,w)’ holds for all values all v’ and w’D such that
v’ v and w’ w
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Outline
1. Global properties
2. Local properties
– Binary CSPs
– Non-Binary CSPs
3. Effects of Consistency Algorithms
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
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How about Non-binary CSPs?
• (Almost) all properties (& algorithms) discussed so far
were restricted to binary CSPs
• Consistency properties for non-binary CSPs are the topic
of current research
• Mainly, properties and algorithms for:
1. Domain filtering techniques (a.k.a. domain reduction, domain
propagation)
• Do not change ‘topology’ of network (width/arity)
• Do not modify constraints definitions
2. Relational m-consistency
[Dechter, Chap 8]
• Add constraints/change constraint definitions
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Non-Binary CSPs
1. Domain filtering
– Generalized Arc Consistency (GAC)
Dechter 3.5.1
– Singleton Generalized Arc Consistency (SGAC)
– maxRPWC, rPIC, RPWC, etc.
[Bessiere et al., 08]
2. Relational consistency
– (strong) Relational m-consistency
• Relational Arc-Consistency (R1C)
• Relational Path-Consistency (R2C)
– Relational (i,m)-consistency
• i = 1, Relational (1,m)-consistency is a domain filtering technique
• i=1 and m=2, Relational (1,m)-consistency is known as rPIC
– Relational (*,m)-consistency (m-wise consistency)
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Generalized Arc-Consistency: Property
• First introduced by [Mohr & Masini, ECAI 88]
• Every value in the domain of every variable has a support in every
constraint in the problem
• In every constraint, every vvp participates in a consistent tuple (can
be extended to all other variables in the scope of the constraint)
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Generalized Arc-Consistency: Algorithm1
• (Sketchy) Algorithm
– Project the constraint on each of the variables in its scope to tighten the
domain of the variable.
– As domains are filtered, filter the constraint
– Repeat the above until quiescence
• When constraint is not defined in extension, GAC may
be problematic (e.g., NP-hard in TCSP)
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Generalized Arc-Consistency: Algorithm2
• Another (Sketchy) Algorithm
– Iterate over every combination of a variable and a constraint where it
appears (Vx, Ci)
– For every value for Vx, identify a support for this value in Ci, where a
support is a tuple where all vvps in the tuple are alive
– Repeat the above until quiescence
• Does not filter the constraints
• Check GAC2001 [Bessière et al., AIJ05]
• When constraint is not defined in extension, GAC may
be problematic (e.g., NP-hard in TCSP)
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SGAC
• Idea: Similar to SAC
• (Sketchy) Algorithm
Repeat until quiescence
For each vvp
Assign the vvp; Enforce GAC on the CSP;
If CSP is GAC, keep the vvp, else remove it
• Note
– Costly in practice, but polynomial as long as GAC is polynomial
– SGAC has been empirically shown to solve every known 9x9
Sudoku puzzle
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Relational Consistency
• Dechter generalizes consistency
properties to non-binary constraints
Dechter 8.1.1
– Relational m-consistency
• Relational 1-consistency  relational arc-consistency
• Relational 2-consistency  relational path-consistency
– Relational (i,m)-consistency
• Relational (1,1)consistency  GAC
• m-wise consistency (Databases)
– Relational (*,m)-consistency
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Relational 1-Consistency
Dechter Def 8.1
• Property
–
–
–
–
•
For every constraint C
Let k be the arity of C
Every consistent partial solution of length k-1
Can be extended to a consistent partial solution of length k
(Sketchy) Algorithm
Dechter Equation (8.2), (8.3)
– For each C, generate k constraints, each of arity k-1 by
• Joining C with the domain of each variable x in scope of C and
• Projecting result on remaining variables (possibly intersecting with
existing constraints)
• Effect: For each constraint, it adds k constraints
• Complexity: polynomial in the largest scope
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Relational 2-Consistency
Dechter Def 8.2
• Property
– For every two constraints C1 and C2
–
,every consistent partial
solution over
– can be extended to
•
(Sketchy) Algorithm Dechter Equation (8.4)
–
generate all constraints of
arity |u| -1 by
• Joining C1, C2, (and the domain of
variables u) and
• Projecting the result on
a1
ai
as
C1
s
C2
• Effect:
adds |s| new constraints
• Complexity: polynomial in |u|
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Relational m-Consistency
Dechter Def 8.3
• Property
– For m constraints C1, C2,…, Cm
–
,every consistent partial
solution over
– can be extended to
•
(Sketchy) Algorithm Dechter Equation (8.5)
–
a1
as
C1
generate all constraints
C2
of arity |u| -1 by
ai
s
•
(and the domain of variables u)
• Projecting the result on
• Effect:
adds |s| new
constraints
• Complexity: polynomial in |u|
Cm
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Relational m-Consistency
Dechter Def 8.3
• Property
–
–
–
–
•
For every m constraints C1 , C2 , .., Cm
Let s = im scope(Ci)
Every consistent partial solution of length |s| -1
Can be extended to a consistent partial solution of length |s|
(Sketchy) Algorithm
Dechter Equation (8.5)
– For each m constraints, generate all constraints of arity |s| -1 by
• Joining the m constraints and the domain of a variable (at the
intersection of their scopes) and
• Projecting the result on remaining variables
• Effect: Adds a huge number of new constraints
• Complexity: polynomial in the sum of largest 2 scopes
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Relational (i,m)-Consistency
Dechter Def 8.4
• Property
–
–
–
–
•
For every m constraints C1 , C2 , .., Cm
Let s = im scope(Ci)
Every consistent partial solution of length i
Can be extended to a consistent partial solution of length |s|
Algorithm
Dechter Fig 8.1
– For each m constraints, generate all constraints of arity i by
• Joining the m constraints and the domain of a variable (at the
intersection of their scopes) and
• Projecting the result on every combination of i variables
• Effect: Adds a huge number of new constraints, except for i=1
• Complexity: exponential in s (largest union of scope of m
constraints) in time and space
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m-wise consistency
•
Property
– For every set of m constraints,
– Every tuple in each constraint appears in a consistent solution to the m
constraints
– That is, each constraint is as tight as it can be for the set of m constraints
•
(Sketchy) Algorithm
Repeat until quiescence
Join each set of m constraints
Project it on each existing constraint to filter the constraint
•
•
Effect: Filters the constraints, w/o introducing new constraints
Note:
– Defined in DB: pairwise consistency, relations join completely
– Woodward defined R(*,m)C + new algorithms that are linear space, currently
under evaluation
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Summary: Non-Binary CSPs
1. Domain filtering
– Generalized Arc Consistency (GAC)
– Singleton Generalized Arc Consistency (SGAC)
– maxRPWC, rPIC, RPWC, etc.
[Bessiere et al., 08]
2. Relational consistency
– (strong) Relational m-consistency
• Relational Arc-Consistency (R1C)
• Relational Path-Consistency (R2C)
– Relational (i,m)-consistency
• i = 1, Relational (1,m)-consistency is a domain filtering technique
• i=1 and m=2, Relational (1,m)-consistency is known as rPIC
– Relational (*,m)-consistency (m-wise consistency)
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Outline
1. Global properties
2. Local properties
– Binary CSPs
– Non-Binary CSPs
3. Effects of Consistency Algorithms
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
Foundations of Constraint Processing
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Effects of Consistency Algorithms
• Filter the domains
– Old algorithms: AC-*, GAC-*, etc.
– New algorithms: maxRPWC, R(1,m)C, etc.
• Filter the constraints
– New algorithms: R(*,m)C
• Add new constraints to the problem
– Old algorithms: PC-2, etc.
– i-consistency (i > 2), (i,j)-C, RmC, R(i,m)C
– Example: Solving the CSPs by Constraint Synthesis
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Solving CSPs by Constraint Synthesis
[Freuder 78]
• From i=2 to i=n,
– achieve i-consistency by using (i -1)-arity
constraints to synthesize i-arity constraints,
– then use the i-ary constraints to filter
constraints of arity i-1, i-2, etc.
• Process ends
– with a unique n-ary constraint
– whose tuples are all the solutions to the CSP
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Outline
1. Global properties
2. Local properties
– Binary CSPs
– Non-Binary CSPs
3. Effects of Consistency Algorithms
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
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Box Consistency (on interval constraints)
• Domains are (continuous) intervals
– Historically also called: continuous CSPs, continuous domains
• Domains are infinite:
– We cannot enumerate consistent values/tuples
– [Davis, AIJ 87] (see recommended reading) showed that even
AC may be incomplete or not terminate
• We apply consistency (usually, arc-consistency) on the
boundaries of the interval
• Sometimes, domains are split, so that boundaries can be
further filtered
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Weighted CSPs
• Weighted CSPs
– Tuples have weights in [0,m], m: intolerable cost
– Costs are added ab=min{m,a+b}
• Soft Arc Consistency
(Cooper, de Givry, Schiex, etc.)
– VAC: Virtual Arc Consistency
– EDAC: Existential Directional Arc Consistency
– OSAC: Optimal Soft Arc Consistency
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Summary
1. Global properties
2. Local properties
– Binary CSPs
– Non-Binary CSPs
3. Effects of Consistency Algorithms
– Domain filtering
– Constraint filtering
– Constraint synthesis
4. Beyond finite, crisp CSPs
– Continuous domains
– Weighted CSPs
Foundations of Constraint Processing
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