Constraint Handling Rules (CHR): Rule-Based Constraint Solving and Deduction

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Constraint Handling Rules (CHR)Rule-Based
Constraint Solving and Deduction

• Jacques Robin

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Outline

• Constraint Handling Rules (CHR)
• Key ideas
• Introductory example
• CHR constraint solver over real variables
• CHR with disjunction (CHR?)
• CHR constraint solver over finite domain
  variables
• General purpose rule-based reasoning with CHR?
• A taxonomy of rule-based languages
• Production rules and ECA rules in CHR?
• Conditional term rewrite rules in CHR?
• Prolog and CLP rules in CHR?

• Deduction with CHR?
• Propositional deduction as Boolean constraint
  solving in CHR?
• First-order Horn Logic forward chaining with
  CHR?
• First-order Horn Logic backward chaining with
  CHR?
• First-order logic refutation and resolution based
  entailment with CHR?
• Description logic reasoning with CHR?

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Constraint Handling Rules (CHR)Key Ideas

• Originally a logical rule-based language to
  declaratively program specialized constraint
  solvers on top of a host programming language
  (Prolog, Haskell, Java)
• Since evolved in a general purpose first-order
  knowledge representation language and
  Turing-complete programming language
• Fact base contains both extensional and
  intentional knowledge in the form of a
  conjunction of constraints
• Rule base integrates and generalizes
• Event-Condition-Action rules (themselves
  generalizing production rules) for constraint
  propagation
• Conditional rewrite rules for constraint
  simplification
• Relies on forward chaining and rule
  Left-Hand-Side (LHS) matching
• Extended variant CHRV adds backtracking search
  and thus generalizes Prolog rules as well

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CHR by ExampleRule Base Defining ? in Terms of

• reflexivity_at_ X ? Y ltgt X Y true.
• asymmetry_at_ X ? Y, Y ? X ltgt XY.
• Constraint simplification (or rewriting) rules
• Syntax ltruleNamegt_at_ ltsimplifiedHeadgt ltgt
  ltguardgt ltbodygt
• Logically ?X?vars(head ? guard)
• ltguardgt ? (ltheadgt ?
  ?Y?vars(body - (head ? guard)) ltbodygt)
• Operationally substitute in constraint store
  (knowledge base) constraints that match
• the rule simplified head by those in rule body
  with their variables instantiated from
• the match
• transitivity_at_ X ? Y , Y ? Z gt X ? Z.
• Constraint propagation (or production) rule (in
  this case, unguarded)
• Syntax ltruleNamegt_at_ ltpropagatedHeadgt gt guard
  ltbodygt
• Logically ?X?vars(head ? guard)
• ltguardgt ? (ltheadgt ?
  ?Y?vars(body - (head ? guard)) ltbodygt)
• Operationally if constraint store (knowledge
  base) contains constraints that match
• the rule propagated head then add those in rule
  body to the store with their variables
• instantiated from the match

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CHR by Example Rule Base Defining ? in Terms of

• idempotence_at_ X ? Y \ X ? Y ltgt true.
• Constraint simpagation rule (in this case,
  unguarded)
• Syntax ltruleNamegt_at_ ltpropagatedHeadgt \
  ltsimplifiedHeadgt ltgt guard ltbodygt
• Logically ?X?vars((head ? guard) ltguardgt ?
  (ltpropagatedheadgt ? ltsimplifiedHeadgt
  ? ?Y?vars(body - (head ? guard)) ltbodygt ?
  ltpropagatedheadgt)
• Operationally if constraint store (knowledge
  base) contains constraints that match
• the rule simplified head and the rule
  propagated head, then substitute in the store
  those matching the simplified head by the rule
  body with their variables instantiated from the
  match
• query1 A ? B, C ? A, B ? C, A 2 Initial
  constraint store a constraint conjunction
• answer1 A 2, B 2, C 2, Final
  constraint store initial constraint store
  simplified through repeated rule application
  until no rule neither simplifies nor propagates
  any new constraint
• query2 A ? B, B ? C, C ? A
• answer2 A B, B C

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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
A ? B, C ? A, B ? C A 2
Matching Equations ? Guard
Built-In Constraint Store
Rule-Defined Constraint Store
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Condition for firing a rule
• Rule head matches active constraint in RDCS
• Generates set of equations between variables and
  constants from the head and the constraint
  (inserted to MEG)
• Every other head from the rule matches against
  some other (partner) constraint in the RDCS
• Generates another set of equations (inserted to
  MEG)
• Rule r fires iff?X1,...,Xi ? vars(MEG ? BICS -
  r) BICS ? ?Y1,...,Yj ? vars(r) MEG

Rule RDCS BICS MEG
r? A ? B, C ? A, B ? C A 2 X' A, Y' B, X' Y'
Active Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
r? A ? B, C ? A, B ? C A 2 X' A Y' B
Normalizing Simplification
Active Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true. ?(?A,B A 2 ?
  ?X',Y' X' A Y' B), eg, B 3 ? 2 A
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
r? A ? B, C ? A, B ? C A 2 X' A Y' B
Active Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Rule firing order depends on 3 heuristics, with
  the following priority
• Rule-defined constraint ordering to become active
• Rule ordering to try matching and entailment
  check with active constraint
• Rule-defined constraint ordering to become
  partner constraints
• Engine first tries matching and entailment check
• All rules with current active constraint, before
  trying any rule with the next constraint in the
  RDCS
• For all elements of the RDCS as partner for the
  first multi-headed rule that matches the active
  constraint, before trying the next rule that
  matches the active constraint

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' A, Y' B, Y' C, X' A
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?( ?A,B,C A 2 ?
  ?X',Y' X' A ? Y' B C), eg, B 3 ? 4 C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' A, Y' B C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' C, Y' A, Y' A, X' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2
  ? ?X',Y' X' B C ? Y' A), eg, B 3 ? 4 C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' B C, Y' A
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' A, Y' B, Y' B, X' C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2 ?
  ?X',Y' X' A C ? Y' B), eg, C 3 ? 2 A
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' A C, Y' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' B, Y' C, Y' A, X' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2 ?
  ?X',Y' X' B ? Y' A C), eg, C 3 ? 2 A
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
a? A ? B, C ? A, B ? C A 2 X' B, Y' A C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t? A ? B, C ? A, B ? C A 2 X' A, Y' B, Y' C, Z' A
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?(?A,B,C A 2 ?
  ?X',Y', Z' X' Z' A ? Y' B C), eg, B 3
  ? 4 C
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t? A ? B, C ? A, B ? C A 2 X' Z' A, Y' B C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t? A ? B, C ? A, B ? C A 2 X' C, Y' A, Y' A, Z' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?A,B,C A 2 ?
  ?X',Y',Z' X' C ? Y' A ? Z' B, e.g.,
  X'C,Y'2,Z'B
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t? A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
A ? B, C ? A, B ? C, C ? B A 2
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• For a given active constraint
• a matching multi-headed propagation rule is
  reapplied with all matching partner constraints,
  before any other rule is tried
• in contrast, a matching multi-headed
  simplification or simpagation rule is applied
  only once with the first matching partner
  constraint, and then engine moves on to the next
  rule

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t? A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Y' B, Z' C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?A,B,C A 2
  ? ?X',Y',Z' X' A ? Y' B ? Z' B, e.g.,
  X'A,Y'B, Z'C
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t? A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
A ? B, C ? A, B ? C, C ? B, A ? C A 2
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Attempt to reapply same propagation rule matching
  same pair of active and partner constraints with
  same head pair but swapped assignments
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' B, Y' C, Y' A, Z' B
Active Constraint
Partner Constraint
27
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?(?A,B,C A 2 ?
  ?X',Y', Z' X' Z' B ? Y' A C), eg, A 2
  ? 4 C
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Z' B, Y' A C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' B, Y' C, Z' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?(?A,B,C A 2
  ? ?X',Y', Z' X' A ? Y' Z' B C), eg, B
  3 ? 4 C
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' Z' B C
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B, Y' A, Z' B
Active Constraint
Partner Constraint
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CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?(?A,B,C A
  2 ? ?X',Y', Z' X' C ? Y' Z' A B), eg, A
  2 ? 3 B
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' Z' A B
Active Constraint
Partner Constraint
32
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' B, Y' A, Z' C
Active Constraint
Partner Constraint
33
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?(?A,B,C A
  2 ? ?X',Y', Z' X' Y' A B ? Z' C ), eg, A
  2 ? 3 B
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' A B, Z' C
Active Constraint
Partner Constraint
34
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' C, Y' A, Z' B
Active Constraint
Partner Constraint
35
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z. ?(?A,B,C A 2
  ? ?X',Y', Z' X' Y' A C ? Z' B ), eg, A
  2 ? 4 C
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
t? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' A C, Z' B
Active Constraint
Partner Constraint
36
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' B, X' C, Y' A
Active Constraint
Partner Constraint
37
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2 ?
  ?X',Y', Z' X' Y' Z' A B C ), eg, A 2
  ? 4 C

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' A B C
Active Constraint
Partner Constraint
38
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' A, X' A, Y' B
Active Constraint
Partner Constraint
39
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2 ?
  ?X',Y', Z' X' Y' Z' A B C ), eg, A 2
  ? 4 C
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' A B C
Active Constraint
Partner Constraint
40
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y B, X B, Y C
Active Constraint
Partner Constraint
41
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2
  ? ?X',Y X' Y' A B C ), eg, A 2 ? 4 C
  

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A B Y C
Active Constraint
Partner Constraint
42
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' B, Y' C, X A, Y B
Active Constraint
Partner Constraint
43
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2 ?
  ?X',Y' X' Y' A B C ), eg, A 2 ? 4 C
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' A B C
Active Constraint
Partner Constraint
44
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y B, X C, Y B
Active Constraint
Partner Constraint
45
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2
  ? ?X',Y X' A C, Y B), eg, A 2 ? 4 C

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A C, Y B
Active Constraint
Partner Constraint
46
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B, X A, Y B
Active Constraint
Partner Constraint
47
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2 ?
  ?X',Y' X' A C,Y B ), eg, A 2 ? 4 C
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A C, Y' B
Active Constraint
Partner Constraint
48
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y B, X A, Y C
Active Constraint
Partner Constraint
49
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2
  ? ?X',Y X' A, Y B C), eg, B 3 ? 4 C

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y B C
Active Constraint
Partner Constraint
50
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' C, X A, Y B
Active Constraint
Partner Constraint
51
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true. ?(?A,B,C A 2 ?
  ?X',Y' X' A, Y B C), eg, B 3 ? 4 C
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
i? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A, Y' B C
Active Constraint
Partner Constraint
52
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Heuristic to choose next active constraint after
  processing of active constraint A added to the
  store constraints N1, ... Nn
• N1, ... , Nn in order
• Constraints O1, ... , Om present in the store
  before processing A

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B, X' Y'
Active Constraint
53
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true. ?(?A,B,C A
  2 ? ?X',Y' X' Y' B C ), eg, B 3 ? 4 C
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' B C
Active Constraint
54
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B, Y' A, X' B,
Partner Constraint
Active Constraint
55
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A
  2 ? ?X',Y' X' Y' A B C), eg, B 3 ? 4
  C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y' A C B
Partner Constraint
Active Constraint
56
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 Y' C, X' B, X' A, Y' B
Partner Constraint
Active Constraint
57
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A
  2 ? ?X',Y' X' A C ? Y' B), eg, A 2 ? 4
  C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' A C, Y' B
Partner Constraint
Active Constraint
58
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B, Y C, X A
Partner Constraint
Active Constraint
59
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2 ?
  ?X',Y' X' Y' A B C), eg, A 2 ? 4 C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' Y A, B C
Partner Constraint
Active Constraint
60
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 Y C, X B, X C, Y A
Partner Constraint
Active Constraint
61
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2 ?
  ?X',Y X' Y' A B C), eg, A 2 ? 4 C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X Y A B C
Partner Constraint
Active Constraint
62
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B, Y B, X C
Partner Constraint
Active Constraint
63
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?A,B,C A 2
  ? ?X',Y' X' C ? Y B), eg, A 2, X C,
  Y B
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a? A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
Partner Constraint
Active Constraint
64
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?A,B,C A 2
  ? ?X',Y' X' C ? Y B), eg, A 2, X C,
  Y B
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
a! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
A ? B, C ? A, A ? C A 2, B C
65
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
r? A ? B, C ? A, A ? C A 2, B C X A, Y C, X Y
Active Constraint
66
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true. ?(?A,B,C A
  2, B C ? ?X',Y X' Y' A B C), eg, A 2
  ? 4 C
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
r? A ? B, C ? A, A ? C A 2, B C X Y A C
Active Constraint
67
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a? A ? B, C ? A, A ? C A 2, B C X A, Y C, Y A, X B
Active Constraint
Partner Constraint
68
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2, B
  C ? ?X',Y X' Y' A B C), eg, A 2 ? 4
  C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a? A ? B, C ? A, A ? C A 2, B C X Y A B C
Active Constraint
Partner Constraint
69
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.
• Alternate matching combination
• Active constraint matched against rightmost head
• Partner constraint matched against leftmost head

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a? A ? B, C ? A, A ? C A 2, B C Y A, X C, X A, Y B
Active Constraint
Partner Constraint
70
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?(?A,B,C A 2, B
  C ? ?X',Y X' Y' A B C), eg, A 2 ? 4
  C
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a? A ? B, C ? A, A ? C A 2, B C X Y A B C
Active Constraint
Partner Constraint
71
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a? A ? B, C ? A, A ? C A 2, B C X A, Y C, Y C, X A
Active Constraint
Partner Constraint
72
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?A,B,C A 2, B
  C ? ?X',Y X' A ? Y C), eg, X 2, Y C
  
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a? A ? B, C ? A, A ? C A 2, B C X Y A C
Active Constraint
Partner Constraint
73
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY ?A,B,C A 2, B
  C ? ?X',Y X' A ? Y C), eg, X 2, Y C
  
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a! A ? B, C ? A, A ? C A 2, B C X Y A C
A ? B A 2, B C, A C
74
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true.
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a! A ? B, C ? A, A ? C A 2, B C X Y A C
r? A ? B A B C 2 X A, Y B, X Y
Active Constraint
75
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true. ?A,B,C
  A B C 2 ? ?X',Y X' Y A B), eg, X
  2, Y 2
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a! A ? B, C ? A, A ? C A 2, B C X Y A C
r? A ? B A B C 2 X Y A B
Active Constraint
76
CHR by Example Rule Base Defining ? in Terms of

• r_at_ X ? Y ltgt X Y true. ?A,B,C
  A B C 2 ? ?X',Y X' Y A B), eg, X
  2, Y 2
• a_at_ X ? Y, Y ? X ltgt XY
• t_at_ X ? Y, Y ? Z gt X ? Z.
• i_at_ X ? Y \ X ? Y ltgt true.

Rule RDCS BICS MEG
t! A ? B, C ? A, B ? C A 2 X' C, Y' A, Z' B
t! A ? B, C ? A, B ? C, C ? B A 2 X' A, Y' B, Z' C
r! A ? B, C ? A, B ? C, C ? B, A ? C A 2 X' C, Y' B
a! A ? B, C ? A, A ? C A 2, B C X Y A C
r! A ? B A B C 2 X Y A B
A B C 2
Constraints Simplified
Final Normalized Solved Form
77
CHR Syntax Overview
CHR Base

• Simplification rule sh1(X,a), sh2(b,Y)
  ltgt g1(X,Y),
  g2(a,b,c) b1(X,c), b2(Y,c).
• Propagation rule ph1(X,Y), ph2(d)
  gt g3(X), g4(d,Y)
  b3(X,d), b4(X,Y).
• Simpagation rule ph3(X), ph4(Y,Z) \
  sh3(X,U), sh4(Y,V) ltgt g5(X,Z), g6(Z,Y)
  b5(X), b6(Y,Z).
• Simplification rules are conditional rewrite
  rules (condition is the guard)
• Propagation rules are event-condition-action
  rules (event is the guard)
• Simpagation rules heads are hybrid syntactic
  sugar, each can be replaced by a semantically
  equivalent simplification rule, ex, p, r \ s, t
  ltgt g, h b, c. is equivalent to p, r, s, t ltgt
  g, h p, r, b, c.

Body Rule-Defined and Built-In Constraints
Guard Built-In Constraints (from host
language)
Head Rule-Defined Constraints
78
CHR Complete Abstract Syntax
body
guard
0..1
Logical Formula
CHR Rule
2..
simplified head
CHR Base

0..1
And Formula
propagated head
0..1
non-overlapping, complete
Atomic Formula
Simpagation Rule
Simplification Rule
Propagation Rule
79
CHR Derivation Data Structures
body
guard
0..1
CHR Logical Formula
CHR Rule
2..
simplified head
CHR Base

0..1
And Formula
propagated head
0..1
arg
Atomic Formula

Simpagation Rule
Simplification Rule
Propagation Rule
Term
Built-In Constraint
Rule Defined Constraint
80
CHR Declarative Semantics inClassical
First-Order Logic (CFOL)

• Simplification rule sh1, ... , shi ltgt g1, ...,
  gj b1, ..., bk.
• where X1, ..., Xn vars(sh1 ? ... ? shi ? g1
  ? ... ? gj) and Y1, ... , Ym vars(b1 ? ... ?
  bk) \ X1, ..., Xn?X1, ..., Xn g1 ? ... ? gj
  ? (sh1 ? ... ? shi ? ?Y1, ... , Ym b1 ? ... ? bk)
• Propagation rule ph1, ... , phi gt g1, ..., gj
  b1, ..., bk.
• where X1, ..., Xn vars(ph1 ? ... ? phi ? g1
  ? ... ? gj) and Y1, ... , Ym vars(b1 ? ... ?
  bk) \ X1, ..., Xn?X1, ..., Xn g1 ? ... ? gj
  ? (ph1 ? ... ? phi ? ?Y1, ... , Ym b1 ? ... ? bk)

81
CHR Constraint and RulePriority Heuristics

• No standard, implementation dependent
• Active constraint priority heuristics
• Preferring constraints most recently inserted in
  store
• Left-to-right writing order in query
• Rule priority heuristics
• Preferring simplification rules over simpagation
  rules and simpagation over propagation rules
• Preferring simplification and simpagation rules
  with highest number of heads
• Preferring propagation rules with lowest number
  of heads
• Preferring rules whose head constraint have
  never be matched yet
• Top to bottom writing order
• Partner constraint priority heuristics
• Preferring constraints most recently inserted in
  store
• Left-to-right writing order in query

82
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y X Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
min(1,2,M)
83
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X
  ?M true ?X'1,Y'2,Z'M X' 1, Y' 2, Z'
  M, 1 ? 2
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
r? min(1,2,M) true X' 1, Y' 2, Z' M, X' ? Y'
84
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X
  ?M true ?X'1,Y'2,Z'M X' 1, Y' 2, Z'
  M, X' 1 ? 2 Y'
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
r! min(1,2,M) true X' 1, Y' 2, Z' M, X' ? Y'
true M Z' X' 1
85
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X
  ?M true ?X'1,Y'2,Z'M X' 1, Y' 2, Z'
  M, X' 1 ? 2 Y'
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
r! min(1,2,M) true X' 1, Y' 2, Z' M, X' ? Y'
true M 1
Projection(CS,vars(Query))
86
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
min(A,B,M) A ? B
87
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X ?A,B,M A ?
  B ?X'A,Y'B,Z'M X' A, Y' B, Z' M, X'
  A ? B Y'
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
r1? min(A,B,M) A ? B X' A, Y' B, Z' M, X' ? Y'
88
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X ?A,B,M A ?
  B ?X'A,Y'B,Z'M X' A, Y' B, Z' M, X'
  A ? B Y'
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
r1! min(A,B,M) A ? B X' A, Y' B, Z' M, X' ? Y'
true M Z' X' A, A ? B
89
CHR Base Example Defining min in Terms of ?, ?
and

• r1_at_ min(X,Y,Z) ltgt X ? Y Z X ?A,B,M A ?
  B ?X'A,Y'B,Z'M X' A, Y' B, Z' M, X'
  A ? B Y'
• r2_at_ min(X,Y,Z) ltgt Y ? X Z Y.
• r3_at_ min(X,Y,Z) ltgt Z lt X Y Z.
• r4_at_ min(X,Y,Z) ltgt Z lt Y Y Z.
• r5_at_ min(X,Y,Z) gt Z ? X, Z ? Y.

Rule RDCS BICS MEG
r1! min(A,B,M) A ? B X' A, Y' B, Z' M, X' ? Y'
true M A, A ? B
Projection(CS,vars(Query))
90
CHR Bases as Component

• Several solvers, each one implemented by a
  pair(CHR base, CHR engine)
• can be assembled in a component-based
  architecture,
• with server solvers' CHR bases defining in their
  rule heads
• the constraints used as built-ins by client
  solvers' CHR bases

91
Example CHR Base Component Assembly
92
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C.
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.
• Notation
• ?P Constraint Domain Variable and CHR
  Variable
• C Constraint Domain Constant and CHR
  Variable
• Constraint Domain Equality Predicate
• CHR Equality Predicate
• 0,1,2, ... CHR and Host Programming Language
  Constants
• Host Programming Language Variable
  Assignment Predicate, always
  returns true, performs arithmetic computation as
  side-effect
• , -, / Host Programming Language
  Arithmetic Function

Rule RDCS BICS MEG
?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true
93
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
  ??Y, true
  ??P?Y,C2 ?P ?Y, C 2
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1? ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
94
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2
95
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r1? ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2

Why r1 does not apply?
96
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• ??X,?Y,?U,?V ?Y 2 ?lt?P,?Q,C,Rgt lt?X,2,3,1gt
  ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r2? ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2 ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1

97
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• ??X,?Y ?Y 2 ?lt?P,?Q,C,Rgt lt?X,2,3,1gt ?P
  ?X, ?Q ?Y, C 3, ?Q.number, R 1
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r2! ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2 ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1
?Y 2, ?X 1
98
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R.
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r2! ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2 ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1
r1? ?U - ?V 2, ?U ?V 0 ?Y 2, ?X 1
99
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r2! ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2 ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1
r2? ?U - ?V 2, ?U ?V 0 ?Y 2, ?X 1 ?P ?U, ?Q ?V, C 2, ?Q.number
100
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r2! ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2 ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1
r3? ?U - ?V 2, ?U ?V 0 ?Y 2, ?X 1 ?P ?U, ?Q ?V, C 2, ?P.number
101
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R.

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P ?Y, C 2
r2! ?X ?Y 3, ?U - ?V 2, ?U ?V 0 ?Y 2 ?P ?X, ?Q ?Y, C 3, ?Q.number, R 1
r4? ?U - ?V 2, ?U ?V 0 ?Y 2, ?X 1 ?P ?U, ?Q ?V, C 0, ?P ?U, ?Q ?V, D 2, R 1
102
CHR Base Example Restricted Formof Real Linear
Equations Solver

• r1_at_ ?P C ltgt P C
• r2_at_ ?P ?Q C ltgt ?Q.number, R C - ?Q ?P
  R
• r3_at_ ?P ?Q C ltgt ?P.number, R C - ?P ?Q
  R.
• r4_at_ ?P ?Q C \ ?P - ?Q D ltgt R (C D)
  / 2 ?P R
• ??X,?Y,?U,?V ?X 1, ?Y 2 ?lt?P,?Q,C,D,Rgt
  lt?U,?V,0,2,1gt ?P ?U, ?Q ?V, C 0, D 2, R
  1

Rule RDCS BICS MEG
r1! ?Y 2, ?X ?Y 3, ?U - ?V 2, ?U ?V 0 true ?P