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Constraint Consistency

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

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  1. Constraint Consistency Chapter 3 CSCE 990-06 Spring 2003 B.Y. Choueiry

  2. 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

  3. 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

  4. 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

  5. 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} CSCE 990-06 Spring 2003 B.Y. Choueiry

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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 CSCE 990-06 Spring 2003 B.Y. Choueiry

  12. 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

  13. 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

  14. 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

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