Constraint Consistency

1
Constraint Consistency

• Chapter 3

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!!

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

4
Section 3.4.1

• for binary CSPs,
• Path-consistency ? 3-consistency
• with ternary CSPs, ternary constraints are
  accounted for

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

6
Section 3.5

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

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

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)

9
Historical note

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

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

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

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 wont yield new unit
  clauses
• Relational arc-consistency adds new clauses by
  unit resolution tractability of unit propagation
  algorithm

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)

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.