Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Constraint Consistency Chapter 3

Wednesday, January 29, 2003CSCE 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: R ij 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!!

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Section 3.4 i-consistency –A relation is i-consistent (D y, 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

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Section for binary CSPs, Path-consistency  3-consistency with ternary CSPs, ternary constraints are accounted for

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Section Generalized arc-consistency –non-binary CSPs –checks value support in domain of variables –updates domains –complexity Relational arc-consistency –non-binary CSPs –updates relations R S-{x}

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Section 3.5 No transition between 3.5 and 3.5.1, it would be good to have one

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Section 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

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Section 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)

Wednesday, January 29, 2003CSCE Spring 2003 B.Y. Choueiry Historical note The concepts of global constraint and bound consistency were developed in the context of Constraint Programming.

Wednesday, January 29, 2003CSCE 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

Wednesday, January 29, 2003CSCE 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 –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

Wednesday, January 29, 2003CSCE 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

Wednesday, January 29, 2003CSCE 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)

Wednesday, January 29, 2003CSCE 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.