Constraint Handling Rules (CHR): Rule-Based Constraint Solving and Deduction
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Transcript of Constraint Handling Rules (CHR): Rule-Based Constraint Solving and Deduction
OntologiesReasoningComponentsAgentsSimulations
Constraint Handling Rules (CHR):Constraint Handling Rules (CHR):Rule-Based Constraint Solving and Rule-Based Constraint Solving and
DeductionDeduction
Jacques Robin
OutlineOutline
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
Constraint Handling Rules (CHR):Constraint Handling Rules (CHR):Key IdeasKey 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
CHR by Example:CHR by Example:Rule Base Defining Rule Base Defining in Terms of = in Terms of =
reflexivity@ X Y <=> X = Y | true. asymmetry@ X Y, Y X <=> X=Y. % Constraint simplification (or rewriting) rules% Syntax: <ruleName>@ <simplifiedHead> <=> <guard> | <body>% Logically: Xvars(head guard) % <guard> (<head> Yvars(body - (head guard)) <body>)% 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@ X Y , Y Z ==> X Z.% Constraint propagation (or production) rule (in this case, unguarded)% Syntax: <ruleName>@ <propagatedHead> ==> guard | <body>% Logically: Xvars(head guard) % <guard> (<head> Yvars(body - (head guard)) <body>)% 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
idempotence@ X Y \ X Y <=> true.% Constraint simpagation rule (in this case, unguarded)% Syntax: <ruleName>@ <propagatedHead> \ <simplifiedHead> <=> guard | <body>% Logically: Xvars((head guard) <guard> (<propagatedhead> <simplifiedHead>% Yvars(body - (head guard)) <body> <propagatedhead>)% 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 conjunctionanswer1: 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 Aanswer2: A = B, B = C
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
A B, C A, B C A = 2
Matching Equations GuardBuilt-In Constraint StoreRule-Defined Constraint Store
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Condition for firing a rule:1. Rule head matches active constraint in RDCS
Generates set of equations between variables and constants from the head and the constraint (inserted to MEG)
2. Every other head from the rule matches against some other (partner) constraint in the RDCS Generates another set of equations (inserted to MEG)
3. 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
r? A B, C A, B C A = 2 X' = A = Y' = B
Normalizing SimplificationActive Constraint
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. (A,B A = 2 X',Y' X' = A = Y' = B), eg, B = 3 2 = A
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
r? A B, C A, B C A = 2 X' = A = Y' = B
Active Constraint
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. a@ X Y, Y X <=> X=Yt@ X Y, Y Z ==> X Z.i@ X Y \ X Y <=> true.
Rule firing order depends on 3 heuristics, with the following priority:1. Rule-defined constraint ordering to become active2. Rule ordering to try matching and entailment check with active constraint3. 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' = AActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y ( A,B,C A = 2 X',Y' X' = A Y' = B = C), eg, B = 3 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
a? A B, C A, B C A = 2 X' = A, Y' = B = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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' = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y' X' = B = C Y' = A), eg, B = 3 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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' = AActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
a? A B, C A, B C A = 2 X' = A, Y' = B, Y' = B, X' = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y' X' = A = C Y' = B), eg, C = 3 2 = A
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
a? A B, C A, B C A = 2 X' = A = C, Y' = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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' = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y' X' = B Y' = A = C), eg, C = 3 2 = A
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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 = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
t? A B, C A, B C A = 2 X' = A, Y' = B, Y' = C, Z' = AActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z. (A,B,C A = 2 X',Y', Z' X' = Z' = A Y' = B = C), eg, B = 3 4 = C
i@ X Y \ X Y <=> true.
Rule RDCS BICS MEG
t? A B, C A, B C A = 2 X' = Z' = A, Y' = B = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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' = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> 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@ X Y \ X Y <=> 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' = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> 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@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z. (A,B,C A = 2 X',Y', Z' X' = Z' = B Y' = A = C), eg, A = 2 4 = C
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z. (A,B,C A = 2 X',Y', Z' X' = A Y' = Z' = B = C), eg, B = 3 4 = C
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z. (A,B,C A = 2 X',Y', Z' X' = C Y' = Z' = A = B), eg, A = 2 3 = B
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z. (A,B,C A = 2 X',Y', Z' X' = Y' = A = B Z' = C ), eg, A = 2 3 = B
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z. (A,B,C A = 2 X',Y', Z' X' = Y' = A = C Z' = B ), eg, A = 2 4 = C
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. a@ X Y, Y X <=> X=Y t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. a@ X Y, Y X <=> X=Y t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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'
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. (A,B,C A = 2 X',Y' X' = Y' = B = C ), eg, B = 3 4 = C
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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,
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y' X' = Y' = A = B = C), eg, B = 3 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y' X' = A = C Y' = B), eg, A = 2 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y' X' = Y' = A = B = C), eg, A = 2 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2 X',Y’ X' = Y' = A = B = C), eg, A = 2 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y A,B,C A = 2 X',Y' X' = C’ Y’ = B), eg, A = 2, X’ = C, Y’ = B
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
PartnerConstrain
t
ActiveConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y A,B,C A = 2 X',Y' X' = C’ Y’ = B), eg, A = 2, X’ = C, Y’ = B
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. (A,B,C A = 2, B = C X',Y’ X' = Y' = A = B = C), eg, A = 2 4 = C a@ X Y, Y X <=> X=Y t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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 = CActive
Constraint
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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’ = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2, B = C X',Y’ X' = Y' = A = B = C), eg, A = 2 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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 = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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’ = BActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y (A,B,C A = 2, B = C X',Y’ X' = Y' = A = B = C), eg, A = 2 4 = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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 = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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’ = AActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y A,B,C A = 2, B = C X',Y’ X' = A Y’ = C), eg, X’ = 2, Y’ = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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 = CActive
Constraint
PartnerConstrain
t
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y A,B,C A = 2, B = C X',Y’ X' = A Y’ = C), eg, X’ = 2, Y’ = C
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true.
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. A,B,C A = B = C = 2 X',Y’ X' = Y’ = A = B), eg, X’ = 2, Y’ = 2
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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 = BActive
Constraint
CHR by Example:CHR by Example: Rule Base Defining Rule Base Defining in Terms of = in Terms of =
r@ X Y <=> X = Y | true. A,B,C A = B = C = 2 X',Y’ X' = Y’ = A = B), eg, X’ = 2, Y’ = 2
a@ X Y, Y X <=> X=Y
t@ X Y, Y Z ==> X Z.
i@ X Y \ X Y <=> 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
Body:Rule-Defined and
Built-In Constraints
Guard:Built-In Constraints (from host language)
Head: Rule-Defined Constraints
• Simplification rule: sh1(X,a), sh2(b,Y) <=> g1(X,Y), g2(a,b,c) | b1(X,c), b2(Y,c).• Propagation rule: ph1(X,Y), ph2(d) ==> g3(X), g4(d,Y) | b3(X,d), b4(X,Y).• Simpagation rule: ph3(X), ph4(Y,Z) \ sh3(X,U), sh4(Y,V) <=> 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 <=> g, h | b, c. is equivalent to p, r, s, t <=> g, h | p, r, b, c.
2..*And Formula
CHR: Syntax OverviewCHR: Syntax Overview
CHR Base
* CHR Rule
guard
simplified head
propagated head
body
LogicalFormula
0..1
0..1
0..1
Atomic Formula
SimpagationRule
SimplificationRule
PropagationRule
{non-overlapping, complete}
Built-InConstraint
Rule DefinedConstraint
CHR: Complete Abstract SyntaxCHR: Complete Abstract Syntax
SimpagationRule
SimplificationRule
PropagationRule
CHR Base
* CHR Rule
guard
simplified head
propagated head
body
LogicalFormula
0..1
0..1
0..12..*
{non-overlapping, complete}
Non-GroundTerm
GroundTerm
{non-overlapping, complete}
And Formula
Atomic Formulaarg
*
Term
ConstraintSymbol
FunctionalTerm
Non-FunctionalTerm
{non-overlapping, complete}
arg
*
FunctionSymbol
ConstraintDomain*
*
*
Built-InConstraint
Rule DefinedConstraint
true false
VariableConstantSymbol
Rule DefinedConstraint
Symbol
Built-InConstraint
Symbol
CHR: Derivation Data StructuresCHR: Derivation Data Structures
SimpagationRule
SimplificationRule
PropagationRule
CHR Base
* CHR Rule
guard
simplified head
propagated head
body
CHRLogicalFormula
Atomic Formula
Built-InConstraint
Rule DefinedConstraint
0..1
0..1
0..12..*
Term
And Formula
arg
*
*
**
{ordered}
Rule DefinedConstraint Store
Built-InConstraint Store
UsedRule
DerivationState
*
CHRDerivation
CHR: Declarative Semantics inCHR: Declarative Semantics inClassical First-Order Logic (CFOL)Classical First-Order Logic (CFOL)
Simplification rule: sh1, ... , shi <=> 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 ==> 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)
CHR: Constraint and RuleCHR: Constraint and RulePriority HeuristicsPriority 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
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> X Y | Z = X
r2@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | X = Z.
r5@ min(X,Y,Z) ==> Z X, Z Y.
Rule RDCS BICS MEG
min(1,2,M)
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> X Y | Z = X M true |= X'=1,Y'=2,Z'=M X' = 1, Y' = 2, Z' = M, 1 2
r2@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> Z X, Z Y.
Rule RDCS BICS MEG
r? min(1,2,M) true X' = 1, Y' = 2, Z' = M, X' Y'
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> X Y | Z = X M true |= X'=1,Y'=2,Z'=M X' = 1, Y' = 2, Z' = M, X' = 1 2 = Y'
r2@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> 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
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> X Y | Z = X M true |= X'=1,Y'=2,Z'=M X' = 1, Y' = 2, Z' = M, X' = 1 2 = Y'
r2@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> 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))
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> X Y | Z = X
r2@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> Z X, Z Y.
Rule RDCS BICS MEG
min(A,B,M) A B
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> 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@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> Z X, Z Y.
Rule RDCS BICS MEG
r1? min(A,B,M) A B X' = A, Y' = B, Z' = M, X' Y'
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> 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@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> 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
CHR Base Example: CHR Base Example: Defining min in Terms of Defining min in Terms of , , and = and =
r1@ min(X,Y,Z) <=> 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@ min(X,Y,Z) <=> Y X | Z = Y.
r3@ min(X,Y,Z) <=> Z < X | Y = Z.
r4@ min(X,Y,Z) <=> Z < Y | Y = Z.
r5@ min(X,Y,Z) ==> 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))
CHR Bases as ComponentCHR 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
<<Component>>HostPlatform
<<Interface>>Min
min(X:Real, Y:Real, Z:Real)
<<Component>>CHRDEngine
X Y X = Y | trueX Y Y X X = YX Y Y Z X ZX Y \ X Y true X X falseX Y Y Z X Y Y Z | X ZY Z X Y X Y Y Z | X Z X Y Y Z X Y Y Z | X Z
<<Component>>LoeStlCHRDBase
<<Interface>>LoeStl
(X:Real, Y:Real): Boolean(X:Real, Y:Real): Boolean
«uses»
min(X,Y,Z) X Y | Z = Xmin(X,Y,Z) Z Y | Z = Xmin(X,Y,Z) Y Z | Z = Ymin(X,Y,Z) Z X | Z = Ymin(X,Y,Z) Z X Z Y
<<Component>>MinCHRDBase
<<Interface>>CHRDEngine
derive()
«uses»
Example CHR Base Component Example CHR Base Component AssemblyAssembly
«uses» <<Interface>>EqNeq
= (X:Real, Y:Real): Boolean (X:Real, Y:Real): Boolean
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C.
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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 number Host Programming Language Type Checking Function
Rule RDCS BICS MEG
?Y == 2,?X + ?Y == 3, ?U - ?V == 2, ?U + ?V == 0
true
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C ?Y, true |= ?P=?Y,C=2 ?P = ?Y, C = 2
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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?
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
?X,?Y,?U,?V ?Y = 2 |= <?P,?Q,C,R> = <?X,2,3,1> ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
?X,?Y ?Y = 2 |= <?P,?Q,C,R> = <?X,2,3,1> ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> R := (C + D) / 2 | ?P = R
?X,?Y,?U,?V ?X = 1, ?Y = 2 |= <?P,?Q,C,D,R> = <?U,?V,0,2,1> ?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 = ?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, D = 2, R = 1
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> R := (C + D) / 2 | ?P = R.
?X,?Y,?U,?V ?X = 1, ?Y = 2 |= <?P,?Q,C,D,R> = <?U,?V,0,2,1> ?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 = ?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, D = 2, R = 1
?U + ?V == 0 ?Y = 2, ?X = 1, ?U = 1
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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, D = 2, R = 1
r1? ?U + ?V == 0 ?Y = 2, ?X = 1, ?U = 1
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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, D = 2, R = 1
r2? ?U + ?V == 0 ?Y = 2, ?X = 1, ?U = 1 ?P = ?U, ?Q = ?V, C = 0, ?Q.number
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
?X,?Y,?U,?V ?X = 1, ?Y = 2, ?U = 1 |= <?P,?Q,C,R> = <1,?V,0,-1> ?P = ?U, ?Q = ?V, C = 0, ?P.number, R = -1
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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, D = 2, R = 1
r3? ?U + ?V == 0 ?Y = 2, ?X = 1, ?U = 1 ?P = ?U, ?Q = ?V, C = 0, ?P.number, R = -1
CHR Base Example: Restricted FormCHR Base Example: Restricted Formof Real Linear Equations Solverof Real Linear Equations Solver
r1@ ?P == C <=> P = C
r2@ ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R.
r3@ ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R.
?X,?Y,?U,?V ?X = 1, ?Y = 2, ?U = 1 |= <?P,?Q,C,R> = <1,?V,0,-1> ?P = ?U, ?Q = ?V, C = 0, ?P.number, R = -1
r4@ ?P + ?Q == C \ ?P - ?Q == D <=> 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, D = 2, R = 1
r3! ?U + ?V == 2 ?Y = 2, ?X = 1, ?U = 1 ?P = ?U, ?Q = ?V, C = 0, ?P.number, R = -1
true ?Y = 2, ?X = 1, ?U = 1, ?V = -1
CHRCHR : Abstract Syntax : Abstract Syntax
OrAnd Formula
connective: enum{or,and}
SimpagationRule
SimplificationRule
PropagationRule
CHR Base
* CHR Rule
guard
simplified head
propagated head
body
And Formula
Atomic Formula
Constraint
Built-InConstraint
Rule DefinedConstraint
2..*
true false
0..1
0..1
0..1
Built-InConstraint Store
Rule DefinedConstraint Store
*
FiredRule
DerivationState
*
CHRDerivation
*
{ordered} * *
TriedAlternative
Body
*
*
CHRCHR: Declarative Semantics in: Declarative Semantics inClassical First-Order Logic (CFOL)Classical First-Order Logic (CFOL)
Simplification rule: sh1, ... , shi <=> g1, ..., gj | b11, ..., bk
p ; ... ; b11, ..., bl
q.
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
1 ... bk
p) ... (b11 ... bk
q))
Propagation rule: ph1, ... , phi ==> g1, ..., gj | b11, ..., bk
p ; ... ; b11, ..., bl
q.
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
1 ... bk
p) ... (b11 ... bk
q))
CHRCHR: Operational Semantics: Operational Semantics
When rule R with disjunctive body B1 ; ... ; Bk is fired Update both constraint stores using B1
Start next matching-updating cycle
When BICS = false or when no rule matches the RDCS Backtrack to last alternative body Bi
Restore both constraint stores to their states prior to their update with Bi
Update both constraint stores using Bi+1
Start next matching-updating cycle
Exhaustively try all alternative bodies of all fired rules through backtracking
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
d4@ d(r4,C) ==> (C = r ; C = b).
d3@ d(r3,C) ==> (C = r ; C = b).
d2@ d(r2,C) ==> (C = b ; C = g).
d5@ d(r5,C) ==> (C = r ; C = g).
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]). true
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m? m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]). true
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2? l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
n? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
n? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
d1a? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r1, C = C1
Already fired w/ same constraint. Not repeated to avoid trivial non-termination
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. C1,C7 Ri',Rj',Ci',Cj' C1=r | Ri=r1, Rj=r7, Ci=C1, Cj=C7, Ci=Cj
l1@ l([ ],[ ]) <=> true. eg., C1 = b r = C7
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
n? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r1, Rj = r7, Ci = C1, Cj = C7, Ci = Cj
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7a? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = r
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
n? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = r
Ri = r1, Rj = r7, Ci = r, Cj = r
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
n! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = r
Ri = r1, Rj = r7, Ci = r, Cj = r
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
false
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
n! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = r
Ri = r1, Rj = r7, Ci = r, Cj = r
bt n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
false
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7b? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7b! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = b
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. C1,C7 Ri',Rj',Ci',Cj' C1=r | Ri=r1, Rj=r7, Ci=C1, Cj=C7, Ci=Cj
l1@ l([ ],[ ]) <=> true. eg., Cj = b r = Ci
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7b! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
n? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = b
Ri = r1, Rj = r7, Ci = C1, Cj = C7, Ci = Cj
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7b! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
l2? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = b
R = r4, Rs = [r3,r2,r5,r6],C = C4, Cs = [C3,C2,C5,C6]
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7b! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), d(r4,C4), l([r3,r2,r5,r6],[C3,C2,C5,C6])
C1 = r,C7 = b
R = r4, Rs = [r3,r2,r5,r6],C = C4, Cs = [C3,C2,C5,C6]
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
...
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m! m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), true
l2! l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]), n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7)
true R = r1, Rs = [r7,r4,r3,r2,r5,r6],C = C1, Cs = [C7,C4,C3,C2,C5,C6]
d1a! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
true C = C1
l2! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), l([r7,r4,r3,r2,r5,r6],[C7,C4,C3,C2,C5,C6])
C1 = r R = r7, Rs = [r4,r3,r2,r5,r6],C = C7, Cs = [C4,C3,C2,C5,C6]
d7b! n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r Ri = r7, C = C7
l2? n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r7,C7), d(r4,C4), l([r4,r3,r2,r5,r6],[C4,C3,C2,C5,C6])
C1 = r,C7 = b
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
d7@ d(r7,C) ==> (C = r ; C = b).
d4@ d(r4,C) ==> (C = r ; C = b).
d3@ d(r3,C) ==> (C = r ; C = b).
d2@ d(r2,C) ==> (C = b ; C = g).
d5@ d(r5,C) ==> (C = r ; C = g).
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Rule RDCS BICS MEG
m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]). true
n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7), d(r1,C1), d(r2,C2), d(r2,C2), d(r1,C3), d(r4,C4), d(r5,C5), d(r6,C6), d(r7,C7)
C1 = g,C2 = b,C3 = r,C4 = r,C5 = g,C6 = r,C7 = b
CHRCHR Base Example: Base Example:Map Coloring ProblemMap Coloring Problem
% More efficient version with forward checkingd1@ c(r1,r), c(r1,b), c(r1,g) ==> false.
d1@ d(r1,C), c(r1,r), c(r1,b) ==> C = g.
d1@ d(r1,C), c(r1,r), c(r1,g) ==> C = b.
d1@ d(r1,C), c(r1,b), c(r1,g) ==> C = r.
d1@ d(r1,C), c(r1,b) ==> (C = r ; C = g).
d1@ d(r1,C), c(r1,g) ==> (C = r ; C = b).
d1@ d(r1,C), c(r1,r) ==> (C = b ; C = g).
d1@ d(r1,C) ==> (C = r ; C = b ; C = g).
...
d6@ d(r6,C) ==> (C = r ; C = g; C = t).
m@ m <=> n(r1,r2), n(r1,r3), n(r1,r4), n(r1,r7), n(r2,r6), n(r3,r7), n(r4,r5), n(r4,r7), n(r5,r6), n(r5,r7).
fcr@ n(Ri,Rj), d(Rj,Cj) ==> c(Ri,Cj).
n@ n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false.
l1@ l([ ],[ ]) <=> true.
l2@ l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).
r2b g r6 r g t
r3
r b
r4
r b
r5
b g
r1
r b g
r7
r b
Implementing a Rewriting SystemImplementing a Rewriting Systemin CHRin CHR
Map each conditional rewrite system rule of the formCondition | LHS RHSonto a CHR simplification rule of the formLHS Condition | RHSi.e., map the rewrite rule condition onto the CHR guard the rewrite rule LHS onto the CHR head the rewrite rule RHS onto the CHR body
Replace each functional terms ti appearing in a Condition, LHS or RHS of the rewrite rule by: a new variable Vi, and
a new equational atom Vi = ti in the guard, head or body (respectively) of the CHR
For example: fib(suc(suc(N)) plus(fib(suc(N)),fib(N)), becomes fib(U,V) <=> U = suc(W), W = suc(N) |fib(N,Y), fib(W,X),
plus(X,Y,V).
Example Term RewritingExample Term Rewritingas as CHRCHR Solving:Solving:fibonaccifibonacci
a) plus(X,0) X
b) plus(X,suc(Y)) suc(plus(X,Y))
c) fib(0) suc(0)
d) fib(suc(0)) suc(0)e) fib(suc(suc(N))
plus(fib(suc(N)),fib(N))
a@ plus(X,U,V) <=> U = 0 | V = X.b@ plus(X,U,V) <=> U = suc(Y) | V = suc(W), plus(X,Y,W).c@ fib(U,V) <=> U = 0 | V = suc(0).d@ fib(U,V) <=> U = suc(0) | V = suc(0).e@ fib(U,V) <=> U = suc(W), W = suc(N) | fib(N,Y), fib(W,X), plus(X,Y,V).
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Rule RDCS BICS MEG
f(s(s(0)),R) true
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Guard Entailment Condition:R true N1,U1,V1,W1 U1=s(s(0)) V1=R U1=s(W1) W1=s(N1),
e.g., N1=0, U1=s(s(0)), V1=R, W1=s(0)
Rule RDCS BICS MEG
e? f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification): U1=s(s(0)) U1=s(W1) W1=s(0)
W1=s(0) W1=s(N1) N1=0
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification): U1=s(s(0)) U1=s(W1) W1=s(0)
W1=s(0) W1=s(N1) N1=0
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Guard Entailment Condition:R,N1,U1,V1,Y1,W1 R=V1 N1=0 U1=s(s(0)) W1=s(0) U2,V2 U2=N1 V2=Y1 U2=0,e.g., U2=0, V2=Y1
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c? f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification):V2=Y1 V2=s(0) Y1=s(0)
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0), U2=N1, V2=Y1, U2=0, V2=s(0)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Guard Entailment Condition: R,N1,U1,V1,Y1,W1,U2,V2 R=V1 N1=U2=0 U1=s(s(0)) W1=Y1=V2=s(0) U3,V3 U3=W1 V3=X1 U3=s(0) e.g., U3=s(0), V3=X1
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d? f(W1,X1), p(X1,Y1,V1)
R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=W1, V3=X1, U3=s(0)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification):V3=X1 V3=s(0) X1=s(0)
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=W1, V3=X1, U3=s(0)
p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Guard Entailment Condition: R,N1,U1,V1,X1,Y1,W1,U2,V2,U3,V3R=V1 N1=U2=0 U1=s(s(0)) W1=X1=Y1=V2=U3=V3=s(0) U4,V4,X4,Y4,W4 X4=X1 U4=Y1 V4=V1 U4=s(Y4)e.g., U4=s(0), V4=R, X4=s(0), Y4=0
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=Z1, V3=X1, U3=s(0)
b? p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0)
X4=X1, U4=Y1, V4=V1, U4=s(Y4)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification):U4=Y1 Y1=s(0) U4=s(0)U4=s(0) U4=s(Y4) Y4=0X4=X1 X1=s(0) X4=s(0)
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=Z1, V3=X1, U3=s(0)
b! p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0) X4=X1, U4=Y1, V4=V1, U4=s(Y4)
p(X4,Y4,W4) R=V1=V4=s(W4), N1=U2=Y4=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=U4=X4=s(0)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Guard Entailment Condition: R,N1,U1,V1,X1,Y1,W1,U2,V2,U3,V3,U4,V4,W4,X4,Y4,R=V1=V4=s(W4) N1=U2=Y4=0 ) U1=s(s(0)) ) W1=X1=Y1=V2=U3=V3=U4=X4=s(0) U5,V5,X5 X5=X4 U5=Y4 V5=W4 U5 = 0e.g., U5=0, V5=W4, X5=s(0)
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=Z1, V3=X1, U3=s(0)
b! p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0) X4=X1, U4=Y1, V4=V1, U4=Y4
a? p(X4,Y4,W4) R=V1=V4=s(W4), N1=U2=Y4=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=U4=X4=s(0)
X5=X4, U5=Y4, V5=W4, U5 = 0
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification):X5=X4 X4=s(0) X5=s(0)X5=s(0) V5=X5 V5=s(0)V5=s(0) V5=W4 W4=s(0)W4=s(0) R=s(W4) R=s(s(0))
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=Z1, V3=X1, U3=s(0)
b! p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0) X4=X1, U4=Y1, V4=V1, U4=Y4
a! p(X4,Y4,W4) R=V1=V4=s(W4), N1=U2=Y4=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=U4=X4=s(0)
X5=X4, U5=Y4, V5=W4, U5 = 0
R=V1=V4=s(W4), N1=U2=Y4=U5=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=U4=X4=X5=s(0), V5=W4, V5=X5
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification):X5=X4 X4=s(0) X5=s(0)X5=s(0) V5=X5 V5=s(0)V5=s(0) V5=W4 W4=s(0)W4=s(0) R=s(W4) R=s(s(0))
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=Z1, V3=X1, U3=s(0)
b! p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0) X4=X1, U4=Y1, V4=V1, U4=Y4
a! p(X4,Y4,W4) R=V1=V4=s(W4), N1=U2=Y4=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=U4=X4=s(0)
X5=X4, U5=Y4, V5=W4, U5 = 0
R=V1=V4=s(s(0)), N1=U2=Y4=U5=0, U1=s(s(0))W1=X1=Y1=V2=U3=V3=U4=X4=W4=V5=X5=s(0)
Example Term RewritingExample Term Rewritingas as CHRCHR Solving Solving Solving: Solving: fibonacci(2) = fibonacci(2) =
??a@ p(X,U,V) <=> U = 0 | V = X.b@ p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W).c@ f(U,V) <=> U = 0 | V = s(0).d@ f(U,V) <=> U = s(0) | V = s(0).e@ f(U,V) <=> U = s(W), W = s(N) | f(N,Y), f(W,X), p(X,Y,V).
Built-in First-Order Atom Syntactic Equality Solver (Unification):X5=X4 X4=s(0) X5=s(0)X5=s(0) V5=X5 V5=s(0)V5=s(0) V5=W4 W4=s(0)W4=s(0) R=s(W4) R=s(s(0))
Rule RDCS BICS MEG
e! f(s(s(0)),R) true U1=s(s(0)), V1=R, U1=s(W1), W1=s(N1)
c! f(N1,Y1), f(W1,X1), p(X1,Y1,V1)
R=V1, N1=0, U1=s(s(0)), W1=s(0) U2=N1, V2=Y1, U2=0
d! f(W1,X1), p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=Y1=V2=s(0) U3=Z1, V3=X1, U3=s(0)
b! p(X1,Y1,V1) R=V1, N1=U2=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=s(0) X4=X1, U4=Y1, V4=V1, U4=Y4
a! p(X4,Y4,W4) R=V1=V4=s(W4), N1=U2=Y4=0, U1=s(s(0)), W1=X1=Y1=V2=U3=V3=U4=X4=s(0)
X5=X4, U5=Y4, V5=W4, U5 = 0
R=s(s(0))
Projection(BICS, vars(Query))
CHRCHRVV vs. Rewriting Systems vs. Rewriting Systems
CHRV: Matching applied to atomic formula
conjunctions Rule head is matched with constraint
store sub-set, which requires the head to be more general than the sub-set
Propagation rules provide further simplification opportunities
Rewriting Systems: Unification of LHS is applied
recursively down to sub-terms Rule LHS is unified with sub-term
which allows the sub-term to be more general than the LHS
All reasoning done through rewriting (no propagation rules)
Common characteristics: Forward chains rules Requires conflict resolution strategy to choose:
Which of several matching rules to fire Non-monotonic reasoning due to:
Constraint retraction in Rule-Defined Constraint Store Retraction of substituted sub-term
Tricky confluence and termination issues
Implementing a Production SystemImplementing a Production Systemin CHRin CHR
Map each production rule of the form:IF m1 AND ... AND ml THEN a1 AND ... AND an
where: {a1 ,..., an} = {add(n1) ,..., add(ni)} {delete(o1) ,..., delete(oj)} {hplOp1(p11,..., p1n) ,..., hplOpk(pk1,..., pkm)}
onto a CHR simpagation rule of the form:p1,..., pr \ o1 ,..., oj hplOp1(p11, ..., p1n) ,..., hplOpk(pk1,..., pkm) | n1 ,..., ni.where {p1,..., pr} = {m1,..., ml} \ {o1 ,..., oj}
Valid only when: {o1 , ... , oj} \ {m1, ... , ml} = , and
C{hplOp1(p11,..., p1n),...,hplOpk(pk1,..., pkm)}, O{o1,...,oj}, N{n1,...,ni} C occurs before O and N in a1 and ... and an
i.e., there no direct way in CHR to: delete facts (ground constraints) not matched in the rule head call host programming language operations after some matched facts have
been deleted or add to the fact base (constraint store) two possibilities allowed in production systems that make the resulting rule
base operational behavior hard to comprehend, verify and maintain
CHRCHRVV vs. Production Systems vs. Production Systems
CHRV: Constraint store contains arbitrary atoms
including functional, non-ground atoms Simplification rules allow straightforward
modeling for goal-driven reasoning, with rewriting simulating Prolog-like backward chaining
Disjunctive bodies Built-in backtracking search
Production Systems: Fact base only contains
ground Datalog atoms Cumbersome modeling to
implement goal-driven reasoning
No disjunctions in RHS No built-in search
Common characteristics: Forward chains rules Requires conflict resolution strategy to choose:
Which of several matching rules to fire Non-monotonic reasoning due to:
Constraint retraction in Rule-Defined Constraint Store Fact retraction in the RHS
Tricky confluence and termination issues
Implementing a Prolog Program in Implementing a Prolog Program in CHRCHR
Map Prolog fact base of the form {f1. ... fn.} onto a fact introduction CHR propagation rule: facts f1 ,..., fn.
Map each set of Prolog deductive rules of the form{p(t1
1,...,tn1) :- b1. ... p(t1
k,...,tnk) :- bk.}
that provide the intentional part of the definition for predicate p onto a CHR simplification rule of the form
p(X1,...,Xn) (X1=t11,..., Xn=tn
1, b1) ;...; (X1=t1k ,..., Xn=tn
k, bk).where {X1,...,Xn} is a set of fresh variablesnot occurring in {t1
1,...,tn1,b1, ... p(t1
k,...,tnk), bk}
Map each set of Prolog facts of the form{p(t'11,...,t'n1). ,..., p(t'1k,...,t'nk).}that provide the extensional part of the definition for predicate p onto a CHR world closure propagation rule of the form
p(X1,...,Xn) (X1=t'11,..., Xn=t'n1) ;...; (X1=t'1k ,..., Xn=t'nk).
Valid only for pure Prolog programs
Example Prolog ProgramExample Prolog ProgramImplemented in CHRImplemented in CHR
Prolog Programfather(john,mary). father(john,peter).mother(jane,mary).
person(john,male). person(peter,male). person(jane,female). person(mary,female). person(paul, male).
parent(P,C) :- father(P,C).parent(P,C) :- mother(P,C).
sibling(C1,C2) :- not C1 = C2, parent(P,C1),
parent(P,C2).
CHR Translationfacts father(john,mary), father(john,peter), mother(jane,mary), person(john,male), person(peter,male), person(jane,female),
person(mary,female), person(paul, male).
parent(P,C) father(P,C) ; mother(P,C).
sibling(C1,C2) C1 C2 | parent(P,C1), parent(P,C2).
father(F,C) (F=john,C=mary) ; (F=john,C=peter).
mother(M,C) (M=jane,C=mary) .
person(P,G) (P=john, G=male) ; (P=peter, G=male) ;
(P=jane, G=male) ; (P=mary, G=male) ;
(P=paul, G=male).
CHRCHRVV vs. Prolog vs. Prolog
CFOL semantics of CHRV guardless, single head simplification rule, equivalent to CFOL semantics of pure Prolog clause set sharing same conclusion (Clark's completion) Simplification rule: sh <=> true | b1
1, ..., bkp ; ... ; b1
1, ..., blq.
where: {X1, ..., Xn} = vars(shi), and {Y1, ... , Ym} = vars(b1 ... bk) \ {X1, ..., Xn} X1, ..., Xn true (sh Y1, ... , Ym ((b1
1 ... bk
p) ... (b11 ... bk
q))
Equivalent Prolog clauses: {sh :- b1
1, ..., bkp. , ... , sh :- b1
1, ..., blq.}
Thus, using Clark's completion, any Prolog program can be reformulated into a semantically equivalent CHRV program
CHRV extends Prolog with: Conjunctions in the heads Guards Non-ground numerical constraints heads, guards and bodies Propagation rules
CLP with CHRCLP with CHR
CLP Engine CLP Application Rule Base
Prolog Engine
CHRHost
ProgrammingLanguage L
CHR EngineCHR Base for Domain D1 Solver
CHR Base for Domain Dk Solver
...
Prolog/L Bridge
CLP with CHRCLP with CHR
CLP Application Rule Base
CHR
HostProgrammingLanguage L
CHR EngineCHR Base for Domain D1 Solver
CHR Base for Domain Dk Solver
...
CHRCHRVV: Practical Applications: Practical Applications
Declarative, easy to extend and compose constraint solvers and all their applications Scheduling, allocation, planning, optimization, recommendation,
configuration Deductive theorem proving (propositional and first-order) and all its
applications: CASE tools, declarative programs analysis, formal methods in hardware and
software design, Hypothetical abductive reasoning and all its applications:
Diagnosis and repair, observation explanation, sensor data integration Multi-agent reasoning Spatio-temporal reasoning and robotics Hybrid reasoning integrating:
Deduction, belief revision, abduction, constraint solving and optimization with open and closed world assumption
Heterogeneous knowledge integration Semantic web services Natural language processing