slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Foundations of Constraint Processing CSCE421/821, Spring 2009 PowerPoint Presentation
Download Presentation
Foundations of Constraint Processing CSCE421/821, Spring 2009

Loading in 2 Seconds...

play fullscreen
1 / 35

Foundations of Constraint Processing CSCE421/821, Spring 2009 - PowerPoint PPT Presentation


  • 136 Views
  • Uploaded on

More on Constraint Consistency Properties & Algorithms. 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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Foundations of Constraint Processing CSCE421/821, Spring 2009' - etoile


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

More on Constraint Consistency

Properties & Algorithms

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

outline
Outline
  • Global properties
  • Local properties
    • Binary CSPs Dechter Sections 3.1, 3.2, 3.3, 3.4
    • Non-Binary CSPsDechter Sections 3.5.1, 8.1
  • Effects of Consistency Algorithms
    • Domain filtering
    • Constraint filtering
    • Constraint synthesis
  • Beyond finite, crisp CSPs
    • Continuous domainsDechter Sections 3.5.3
    • Weighted CSPs
global consistency properties
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
outline1
Outline
  • Global properties
  • Local properties
    • Binary CSPs
    • Non-Binary CSPs
  • Effects
    • Domain filtering
    • Constraint filtering
    • Constraint synthesis
  • Beyond finite, crisp CSPs
    • Continuous domains
    • Weighted CSPs
local properties binary csps
Local Properties: Binary CSPs
  • Classical ones
    • Arc, path, i, strong i, (i,j)-consistency
  • More recently
    • Singleton Arc Consistency
    • Inverse Consistency
    • Neighborhood Inverse Consistency
    • (Conservative) Dual consistency
  • Special Constraints
classical local consistency properties
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
classical local consistency algorithms
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
local properties binary csps1
Local Properties: Binary CSPs
  • Classical ones
    • Arc, path, i, strong i, (i,j)-consistency
  • More recently
    • Singleton Arc Consistency
    • Inverse Consistency
    • Neighborhood Inverse Consistency
    • (Conservative) Dual consistency
  • Special Constraints
singleton arc consistency sac
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
inverse consistency
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
neighborhood inverse consistency nic algorithm
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
summary binary csps
Summary: Binary CSPs
  • Arc, path, i-consistency
  • (i,j)-consistency
  • SAC
  • PIC
  • (1,m)-consistency
  • NIC
local properties binary csps2
Local Properties: Binary CSPs
  • Classical ones
    • Arc, path, i, strong i, (i,j)-consistency
  • More recently
    • Singleton Arc Consistency
    • Inverse Consistency
    • Neighborhood Inverse Consistency
    • (Conservative) Dual consistency
  • Special Constraints
ac for special constraints van hentenryck et al aij 92
AC for Special Constraints [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 (respectivelyv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

  • Monotonic

A constraint C is monotonic with respect to a domain D iff there exists a total ordering on D such that, for all vand 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

outline2
Outline
  • Global properties
  • Local properties
    • Binary CSPs
    • Non-Binary CSPs
  • Effects of Consistency Algorithms
    • Domain filtering
    • Constraint filtering
    • Constraint synthesis
  • Beyond finite, crisp CSPs
    • Continuous domains
    • Weighted CSPs
how about non binary csps
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:
    • Domain filtering techniques (a.k.a. domain reduction, domain propagation)
      • Do not change ‘topology’ of network (width/arity)
      • Do not modify constraints definitions
    • Relational m-consistency [Dechter, Chap 8]
      • Add constraints/change constraint definitions
non binary csps
Non-Binary CSPs
  • Domain filtering
    • Generalized Arc Consistency (GAC) Dechter 3.5.1
    • Singleton Generalized Arc Consistency (SGAC)
    • maxRPWC, rPIC, RPWC, etc. [Bessiere et al., 08]
  • 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)
slide18

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)
generalized arc consistency algorithm1
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)
generalized arc consistency algorithm2
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)
slide21
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
relational consistency
Relational Consistency
  • Dechter generalizes consistency Dechter 8.1.1

properties to non-binary constraints

    • 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
relational 1 consistency dechter def 8 1
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 constraint C, generate all constraints 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: Adds a huge number of new constraints
  • Complexity: polynomial in the largest scope
relational 2 consistency dechter def 8 2
Relational 2-Consistency Dechter Def 8.2
  • Property
    • For every two constraint C1and C2
    • Let s = scope(C1)  scope(C2)
    • Every consistent partial solution of length |s|-1
    • Can be extended to a consistent partial solution of length |s|
  • (Sketchy) Algorithm Dechter Equation (8.4)
    • For each constraints C1and C2, generate all constraints of arity|s|-1 by
      • Joining C1, C2, and the domain of a variable (in C1 and C2 ) and
      • Projecting the result on remaining variables
  • Effect: Adds a huge number of new constraints
  • Complexity: polynomial in the largest |s|
relational m consistency dechter def 8 3
Relational m-Consistency Dechter Def 8.3
  • Property
    • For every m constraints C1, C2 , .., Cm
    • Let s = imscope(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
relational i m consistency dechter def 8 4
Relational (i,m)-Consistency Dechter Def 8.4
  • Property
    • For every m constraints C1, C2 , .., Cm
    • Let s = imscope(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 iby
      • 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
m wise consistency
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
summary non binary csps
Summary: Non-Binary CSPs
  • Domain filtering
    • Generalized Arc Consistency (GAC)
    • Singleton Generalized Arc Consistency (SGAC)
    • maxRPWC, rPIC, RPWC, etc. [Bessiere et al., 08]
  • 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)
outline3
Outline
  • Global properties
  • Local properties
    • Binary CSPs
    • Non-Binary CSPs
  • Effects of Consistency Algorithms
    • Domain filtering
    • Constraint filtering
    • Constraint synthesis
  • Beyond finite, crisp CSPs
    • Continuous domains
    • Weighted CSPs
effects of consistency algorithms
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
solving csps by constraint synthesis freuder 78
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
outline4
Outline
  • Global properties
  • Local properties
    • Binary CSPs
    • Non-Binary CSPs
  • Effects of Consistency Algorithms
    • Domain filtering
    • Constraint filtering
    • Constraint synthesis
  • Beyond finite, crisp CSPs
    • Continuous domains
    • Weighted CSPs
box consistency on interval constraints
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
weighted csps
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
summary
Summary
  • Global properties
  • Local properties
    • Binary CSPs
    • Non-Binary CSPs
  • Effects of Consistency Algorithms
    • Domain filtering
    • Constraint filtering
    • Constraint synthesis
  • Beyond finite, crisp CSPs
    • Continuous domains
    • Weighted CSPs