Constraint Consistency

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# Constraint Consistency - PowerPoint PPT Presentation

Constraint Consistency. Chapter 3. Section 3.3. Definition 3.3.2: Path Consistency, Two variables relative to a third non-binary, binary Three variables A network (note: R ij i j) Revise-3 updates binary constraints, not domains

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### Constraint Consistency

Chapter 3

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.3
• Definition 3.3.2: Path Consistency,
• Two variables relative to a third
• non-binary, binary
• Three variables
• A network (note: Rij ij)
• Revise-3 updates binary constraints, not domains
• PC-1, PC-3 (like AC-1, AC-3) update binary constraints, not domains
• This is not the PC-3 algorithm of Mackworth!!

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.4
• i-consistency
• A relation is i-consistent (Dy, y not specified in S!!)
• A network is i-consistent (i not specified distinct )
• Algorithms: Revise-i, i-consistency-1
• Should variables be distinct?
• Note: complexity

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.4.1
• for binary CSPs,

Path-consistency  3-consistency

• with ternary CSPs, ternary constraints are accounted for

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.5.1
• Generalized arc-consistency
• non-binary CSPs
• checks value support in domain of variables
• complexity
• Relational arc-consistency
• non-binary CSPs

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.5
• No transition between 3.5 and 3.5.1, it would be good to have one

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.5.2
• Global constraints:
• non-binary constraints dictated by practical applications
• scope is parametrized
• Relational description is unrealistic, defined intentionally (error: implicit)
• Specialized algorithms ensure generalized arc-consistency
• Examples: alldifferent, sum, global cardinality (generalization of alldifferent), cumulative, cycle

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.5.3
• Bounds consistency, large ordered domains, not necessarily continuous
• Bind domains by intervals
• Ensure that interval endpoints are AC
• Weaker notion of consistency, cost effective
• Mechanism: tighten endpoints until AC.
• Example: alldifferent in O(nlogn)

CSCE 990-06 Spring 2003 B.Y. Choueiry

Historical note
• The concepts of global constraint and bound consistency were developed in the context of Constraint Programming.

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.6
• Constraints with specific semantics (non-random): e.g., numeric/algebraic, boolean
• Implications on
• Arc-consistency
• Path-consistency
• Generalized arc-consistency
• Relational arc-consistency

CSCE 990-06 Spring 2003 B.Y. Choueiry

3.6 Algebraic constraints
• Too general term, in fact linear inequalities
• Constraint composition is linear elimination
• Binary case: constraints of bounded difference
• Arc-consistency filters domains
• Non-binary case (non-negative integer domains, why?)
• Generalized arc-consistency filters domains

CSCE 990-06 Spring 2003 B.Y. Choueiry

3.6 Boolean Constraints
• Domain filtering: unit clause
• Binary clauses
• Constraint composition is the resolution rule
• Arc-consistency achieved adding unit clause (unary constraint)
• Path consistency achieved adding a binary clause
• Non-binary clauses
• Generalized arc-consistency won’t yield new unit clauses
• Relational arc-consistency adds new clauses by unit resolution tractability of unit propagation algorithm

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.7
• Arc-consistency, path-consistency are sometimes guaranteed to solve the CSP
• Restricted classes
• Topologic restrictions: tree-structured
• Arc-consistency guarantees solvability
• Domains restrictions: bi-values domains, CNF theories with clause length 1 or 2
• Path-consistency guarantees solvability
• Constraint semantic: Horn Clauses
• Unit propagation/resolution (relational-arc consistency) guarantees solvability (see tractability of Horn Theories in CSE 876)

CSCE 990-06 Spring 2003 B.Y. Choueiry

Section 3.8
• Notice how non-binary constraints are depicted in Figures 3.17, 3.18: contours instead of box nodes. This is inherited from DB literature.

CSCE 990-06 Spring 2003 B.Y. Choueiry