Constraint Consistency

Wednesday, January 29, 2003 CSCE 990-06 Spring 2003 B.Y. Choueiry

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: Rij ij)

• 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!!


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


Section 3.4.1

• for binary CSPs,Path-consistency 3-consistency

• with ternary CSPs, ternary constraints are accounted for


Section 3.5.1

• Generalized arc-consistency– non-binary CSPs– checks value support in domain of variables– updates domains– complexity

• Relational arc-consistency– non-binary CSPs– updates relations RS-{x}


Section 3.5

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


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


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)


Historical note

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


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


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– Path-consistency tightens/adds binary constraints

• Non-binary case (non-negative integer domains, why?)– Generalized arc-consistency filters domains– Relational arc-consistency tightnes/adds constraints


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


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)


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.

Constraint Consistency

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