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

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

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

CHR by Example: Rule Base Defining  in Terms of =
X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y. % Constraint simplification (or rewriting) rules % Syntax: <simplifiedHead> <=> <guard> | <body> % Logically: Xvars(head  guard) % <guard>  (<head>  Yvars(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 X  Y , Y  Z ==> X  Z. % Constraint propagation (or production) rule (in this case, unguarded) % Syntax: <propagatedHead> ==> guard | <body> % <guard>  (<head>  Yvars(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

X  Y \ X  Y <=> true. % Constraint simpagation rule (in this case, unguarded) % Syntax: <propagatedHead> \ <simplifiedHead> <=> guard | <body> % Logically: Xvars((head  guard) <guard>  (<propagatedhead>  <simplifiedHead> %  Yvars(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 conjunction answer1: A = 2, B = 2, C = 2, % Final constraint store = initial constraint store % simplified through repeated rule application until no rule neither simplifies nor % propagates any new constraint query2: A  B, B  C, C  A answer2: A = B, B = C

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Rule RDCS BICS MEG A  B, C  A, B  C A = 2 Rule-Defined Constraint Store Built-In Constraint Store Matching Equations  Guard

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Condition for firing a rule: Rule head matches active constraint in RDCS Generates set of equations between variables and constants from the head and the constraint (inserted to MEG) Every other head from the rule matches against some other (partner) constraint in the RDCS Generates another set of equations (inserted to MEG) Rule r fires iff: X1,...,Xi  vars(MEG  BICS - r) BICS  Y1,...,Yj  vars(r) MEG Rule RDCS BICS MEG r? A  B, C  A, B  C A = 2 X' = A, Y' = B, X' = Y' Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Rule RDCS BICS MEG r? A  B, C  A, B  C A = 2 X' = A = Y' = B Active Constraint Normalizing Simplification

X  Y <=> X = Y | true (A,B A = 2  X',Y' X' = A = Y' = B), eg, B = 3  2 = A X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Rule RDCS BICS MEG r? A  B, C  A, B  C A = 2 X' = A = Y' = B Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Rule firing order depends on 3 heuristics, with the following priority: Rule-defined constraint ordering to become active Rule ordering to try matching and entailment check with active constraint Rule-defined constraint ordering to become partner constraints Engine first tries matching and entailment check: All rules with current active constraint, before trying any rule with the next constraint in the RDCS; For all elements of the RDCS as partner for the first multi-headed rule that matches the active constraint, before trying the next rule that matches the active constraint; Rule RDCS BICS MEG a? A  B, C  A, B  C A = 2 X' = A, Y' = B, Y' = C, X' = A Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y ( A,B,C A = 2  X',Y' X' = A  Y' = B = C), eg, B = 3  4 = C X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Rule RDCS BICS MEG a? A  B, C  A, B  C A = 2 X' = A, Y' = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y' X' = B = C  Y' = A), eg, B = 3  4 = C X  Y, Y  Z ==> X  Z. 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' = A Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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' = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y' X' = A = C  Y' = B), eg, C = 3  2 = A X  Y, Y  Z ==> X  Z. X  Y \ X  Y <=> true. Rule RDCS BICS MEG a? A  B, C  A, B  C A = 2 X' = A = C, Y' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y' X' = B  Y' = A = C), eg, C = 3  2 = A X  Y, Y  Z ==> X  Z. 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 = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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' = A Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 X  Y \ X  Y <=> true. Rule RDCS BICS MEG t? A  B, C  A, B  C A = 2 X' = Z' = A, Y' = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Y' = B, Z' = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 X' = A, Y' = B, Z' = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C A  B, C  A, B  C, C  B, A  C

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = B, Y' = C, Y' = A, Z' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = Z' = B, Y' = A = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = A, Y' = B, Y' = C, Z' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = A, Y' = Z' = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B, Y' = A, Z' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = C, Y' = Z' = A = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = A, Y' = B, Y' = A, Z' = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = Y' = A = B, Z' = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = A, Y' = C, Y' = A, Z' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y 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 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C t? A  B, C  A, B  C, C  B, A  C X' = Y' = A = C, Z' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y' = B, X' = C, Y' = A Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = Y' = A = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = C, Y' = A, X' = A, Y' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = Y' = A = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y’ = B, X’ = B, Y’ = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A = B = Y’ = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = B, Y' = C, X’ = A, Y’ = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = Y' = A = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y’ = B, X’ =C, Y’ = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A = C, Y’ = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B, X’ = A, Y’ = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A = C, Y' = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y’ = B, X’ =A, Y’ = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y’ = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y' = C, X’ = A, Y’ = B Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C i? A  B, C  A, B  C, C  B, A  C X' = A, Y' = B = C Active Constraint Partner Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C r? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B, X' = Y' Active Constraint

X  Y <=> X = Y | true (A,B,C A = 2  X',Y' X' = Y' = B = C ), eg, B = 3  4 = C X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r? A  B, C  A, B  C, C  B, A  C X' = Y' = B = C Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B, Y' = A, X' = B, Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y' X' = Y' = A = B = C), eg, B = 3  4 = C X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = Y' = A = C = B Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C Y' = C, X' = B, X' = A, Y' = B Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y' X' = A = C  Y' = B), eg, A = 2  4 = C X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = A = C, Y' = B Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B, Y’ = C, X’ = A Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y' X' = Y' = A = B = C), eg, A = 2  4 = C X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = Y’ = A, = B = C Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C Y’ = C, X’ = B, X’ = C, Y’ = A Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y (A,B,C A = 2  X',Y’ X' = Y' = A = B = C), eg, A = 2  4 = C X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X’ = Y’ = A = B = C Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B, Y’ = B, X’ = C Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y A,B,C A = 2  X',Y' X' = C’  Y’ = B), eg, A = 2, X’ = C, Y’ = B X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a? A  B, C  A, B  C, C  B, A  C X' = C, Y' = B Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y A,B,C A = 2  X',Y' X' = C’  Y’ = B), eg, A = 2, X’ = C, Y’ = B X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C a! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B A  B, C  A, A  C A = 2, B = C

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B r? A  B, C  A, A  C A = 2, B = C X’ = A, Y’ = C, X’ = Y’ Active Constraint

X  Y <=> X = Y | true (A,B,C A = 2, B = C  X',Y’ X' = Y' = A = B = C), eg, A = 2  4 = C X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B r? A  B, C  A, A  C A = 2, B = C X’ = Y’ = A = C Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a? A  B, C  A, A  C A = 2, B = C X’ = A, Y’ = C, Y’ = A, X’ = B Partner Constraint Active Constraint

X  Y <=> X = Y | true. 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 X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a? A  B, C  A, A  C A = 2, B = C X’ = Y’ = A = B = C Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 A  B, C  A, B  C, C  B X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a? A  B, C  A, A  C A = 2, B = C Y’ = A, X’ = C, X’ = A, Y’ = B Partner Constraint Active Constraint

X  Y <=> X = Y | true. 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 X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a? A  B, C  A, A  C A = 2, B = C X’ = Y’ = A = B = C Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a? A  B, C  A, A  C A = 2, B = C X’ = A, Y’ = C, Y’ = C, X’ = A Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y A,B,C A = 2, B = C  X',Y’ X' = A  Y’ = C), eg, X’ = 2, Y’ = C X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a? A  B, C  A, A  C A = 2, B = C X’ = Y’ = A = C Partner Constraint Active Constraint

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y A,B,C A = 2, B = C  X',Y’ X' = A  Y’ = C), eg, X’ = 2, Y’ = C X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C 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

X  Y <=> X = Y | true. X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C 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

X  Y <=> X = Y | true A,B,C A = B = C = 2  X',Y’ X' = Y’ = A = B), eg, X’ = 2, Y’ = 2 X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a! A  B, C  A, A  C A = 2, B = C X’ = Y’ = A = C r? A  B A = B = C = 2 X’ = Y’ = A = B Active Constraint

X  Y <=> X = Y | true A,B,C A = B = C = 2  X',Y’ X' = Y’ = A = B), eg, X’ = 2, Y’ = 2 X  Y, Y  X <=> X=Y X  Y, Y  Z ==> X  Z. 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 X' = A, Y' = B, Z' = C r! A  B, C  A, B  C, C  B, A  C X' = C, Y' = B a! A  B, C  A, A  C A = 2, B = C X’ = Y’ = A = C A  B A = B = C = 2 X’ = Y’ = A = B Constraints Simplified Final Normalized Solved Form

{non-overlapping, complete}
CHR: Syntax Overview guard simplified head propagated head body Logical Formula 0..1 * CHR Rule 2..* And Formula CHR Base Simpagation Rule Simplification Propagation {non-overlapping, complete} Atomic Formula Built-In Constraint Rule Defined 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. Head: Rule-Defined Constraints Guard: Built-In Constraints (from host language) Body: Rule-Defined and Built-In Constraints

CHR: Complete Abstract Syntax
body guard 0..1 CHR Rule Logical Formula 2..* CHR Base * simplified head 0..1 And Formula propagated head 0..1 {non-overlapping, complete} Constraint Domain * arg * Term Constraint Symbol Atomic Formula Simplification Rule Simpagation Rule Propagation Rule Built-In Constraint Rule Defined true false Rule Defined Constraint Symbol Built-In arg * Non-Ground Term Ground {non-overlapping, complete} Functional Term Non-Functional {non-overlapping, complete} Variable Constant Symbol Function Symbol

CHR: Derivation Data Structures
body guard 0..1 CHR Rule CHR Logical Formula 2..* CHR Base * simplified head 0..1 And Formula * {ordered} Rule Defined Constraint Store Built-In Used Rule Derivation State CHR propagated head 0..1 arg Atomic Formula * Simplification Rule Simpagation Rule Propagation Rule Term Built-In Constraint Rule Defined Constraint

CHR: Declarative Semantics in 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 Rule Priority 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:

CHR Base Example: Defining min in Terms of ,  and =
min(X,Y,Z) <=> X  Y | Z = X min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | X = Z. min(X,Y,Z) ==> Z  X, Z  Y. Rule RDCS BICS MEG min(1,2,M)

min(X,Y,Z) <=> X  Y | Z = X M true |= X'=1,Y'=2,Z'=M X' = 1, Y' = 2, Z' = M, 1  2 min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. 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'

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' min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. 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' M = Z' = X' = 1

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' min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. 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' M = 1 Projection(CS,vars(Query))

min(X,Y,Z) <=> X  Y | Z = X min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. min(X,Y,Z) ==> Z  X, Z  Y. Rule RDCS BICS MEG min(A,B,M) A  B

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' min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. 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'

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' min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. 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

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' min(X,Y,Z) <=> Y  X | Z = Y. min(X,Y,Z) <=> Z < X | Y = Z. min(X,Y,Z) <=> Z < Y | Y = Z. 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 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

Example CHR Base Component Assembly
min(X,Y,Z)  X  Y | Z = X min(X,Y,Z)  Z  Y | Z = X min(X,Y,Z)  Y  Z | Z = Y min(X,Y,Z)  Z  X | Z = Y min(X,Y,Z)  Z  X  Z  Y <<Component>> MinCHRDBase <<Component>> CHRDEngine <<Interface>> Min min(X:Real, Y:Real, Z:Real) <<Interface>> CHRDEngine derive() «uses» «uses» <<Interface>> EqNeq = (X:Real, Y:Real): Boolean  (X:Real, Y:Real): Boolean <<Interface>> LoeStl (X:Real, Y:Real): Boolean (X:Real, Y:Real): Boolean «uses» X  Y  X = Y | true X  Y  Y  X  X = Y X  Y  Y  Z  X  Z X  Y \ X  Y  true X  X  false X  Y  Y  Z  X  Y  Y  Z | X  Z Y  Z  X  Y  X  Y  Y  Z | X  Z X  Y  Y  Z  X  Y  Y  Z | X  Z <<Component>> LoeStlCHRDBase <<Component>> HostPlatform

CHR Base Example: Restricted Form of Real Linear Equations Solver
?P == C <=> P = C. ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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

?P == C <=> P = C ?Y, true |= ?P=?Y,C=2 ?P = ?Y, C = 2 ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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 ?Y = 2

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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? ?Y = 2 Why r1 does not apply?

?P == C <=> P = C ?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 ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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? ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1

?P == C <=> P = C ?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 ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 ?Y = 2, ?X = 1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r1? ?Y = 2, ?X = 1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r2? ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 2, ?Q.number

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r3? ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 2, ?P.number

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4? ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 0, ?P = ?U, ?Q = ?V, D = 2, R = 1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4? ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 0, D = 2, R = 1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4! ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 0, D = 2, R = 1 ?Y = 2, ?X = 1, ?U = 1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4! ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 0, D = 2, R = 1 r1? ?Y = 2, ?X = 1, ?U = 1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R ?P + ?Q == C <=> ?P.number, R := C - ?P | ?Q = R. ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4! ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 0, D = 2, R = 1 r2? ?Y = 2, ?X = 1, ?U = 1 ?P = ?U, ?Q = ?V, C = 0, ?Q.number

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?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 ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4! ?Y = 2, ?X = 1 ?P = ?U, ?Q = ?V, C = 0, D = 2, R = 1 r3? ?Y = 2, ?X = 1, ?U = 1 ?P = ?U, ?Q = ?V, C = 0, ?P.number, R = -1

?P == C <=> P = C ?P + ?Q == C <=> ?Q.number, R := C - ?Q | ?P = R. ?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 ?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! ?Y = 2 ?P = ?X, ?Q = ?Y, C = 3, ?Q.number, R = 1 r4! ?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 ?Y = 2, ?X = 1, ?U = 1, ?V = -1

connective: enum{or,and}
CHR : Abstract Syntax body OrAnd Formula connective: enum{or,and} 2..* guard 0..1 CHR Rule And Formula CHR Base * simplified head Tried Alternative Body 0..1 propagated head 0..1 * Atomic Formula Fired Rule Simplification Rule Simpagation Rule Propagation Rule Constraint * {ordered} Derivation State CHR Derivation Rule Defined Constraint Store * Rule Defined Constraint * Built-In Constraint * Built-In Constraint Store * * true false

CHR: Declarative Semantics in Classical First-Order Logic (CFOL)
Simplification rule: sh1, ... , shi <=> g1, ..., gj | b11, ..., bkp ; ... ; b11, ..., blq. where: {X1, ..., Xn} = vars(sh1  ...  shi  g1  ...  gj) and {Y1, ... , Ym} = vars(b1  ...  bk) \ {X1, ..., Xn} X1, ..., Xn g1  ...  gj  (sh1  ...  shi  Y1, ... , Ym ((b11  ...  bkp)  ...  (b11  ...  bkq)) Propagation rule: ph1, ... , phi ==> g1, ..., gj | b11, ..., bkp ; ... ; b11, ..., blq. where: {X1, ..., Xn} = vars(ph1  ...  phi  g1  ...  gj) and {Y1, ... , Ym} = vars(b1  ...  bk) \ {X1, ..., Xn} X1, ..., Xn g1  ...  gj  (ph1  ...  phi  Y1, ... , Ym ((b11  ...  bkp)  ...  (b11  ...  bkq))

CHR: 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 Exhaustively try all alternative bodies of all fired rules through backtracking

CHR Base Example: Map Coloring Problem
t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). d(r4,C) ==> (C = r ; C = b). d(r3,C) ==> (C = r ; C = b). d(r2,C) ==> (C = b ; C = g). d(r5,C) ==> (C = r ; C = g). d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). Rule RDCS BICS MEG m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]). true

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). Rule RDCS BICS MEG m? m, l([r1,r7,r4,r3,r2,r5,r6],[C1,C7,C4,C3,C2,C5,C6]). true

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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)

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) R = r1, Rs = [r7,r4,r3,r2,r5,r6], C = C1, Cs = [C7,C4,C3,C2,C5,C6]

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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])

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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])

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 C1 = r

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 n? C1 = r

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 d1a? C1 = r R = r1, C = C1 Already fired w/ same constraint. Not repeated to avoid trivial non-termination

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 l2? C1 = r R = r7, Rs = [r4,r3,r2,r5,r6], C = C7, Cs = [C4,C3,C2,C5,C6]

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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])

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(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 l([ ],[ ]) <=> true eg., C1 = b  r = C7 l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r1, Rj = r7, Ci = C1, Cj = C7, Ci = Cj

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 C1 = r, C7 = r

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 n? C1 = r, C7 = r Ri = r1, Rj = r7, Ci = r, Cj = r

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 n! C1 = r, C7 = r Ri = r1, Rj = r7, Ci = r, Cj = r false

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 n! C1 = r, C7 = r Ri = r1, Rj = r7, Ci = r, Cj = r bt false

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 C1 = r, C7 = b

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(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 l([ ],[ ]) <=> true eg., Cj = b  r = Ci l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 n? C1 = r, C7 = b Ri = r1, Rj = r7, Ci = C1, Cj = C7, Ci = Cj

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) Ri = r7, C = C7 l2? C1 = r, C7 = b R = r4, Rs = [r3,r2,r5,r6], C = C4, Cs = [C3,C2,C5,C6]

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) 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), 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]

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). ... 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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) 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]) C = C1 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]) 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

t r3 r4 r5 r1 r7 d(r1,C) ==> (C = r ; C = b ; C = g). d(r7,C) ==> (C = r ; C = b). d(r4,C) ==> (C = r ; C = b). d(r3,C) ==> (C = r ; C = b). d(r2,C) ==> (C = b ; C = g). d(r5,C) ==> (C = r ; C = g). d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs). 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

t r3 r4 r5 r1 r7 % More efficient version with forward checking c(r1,r), c(r1,b), c(r1,g) ==> false. d(r1,C), c(r1,r), c(r1,b) ==> C = g. d(r1,C), c(r1,r), c(r1,g) ==> C = b. d(r1,C), c(r1,b), c(r1,g) ==> C = r. d(r1,C), c(r1,b) ==> (C = r ; C = g). d(r1,C), c(r1,g) ==> (C = r ; C = b). d(r1,C), c(r1,r) ==> (C = b ; C = g). d(r1,C) ==> (C = r ; C = b ; C = g). ... d(r6,C) ==> (C = r ; C = g; C = t). 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(Ri,Rj), d(Rj,Cj) ==> c(Ri,Cj). n(Ri,Rj), d(Ri,Ci), d(Rj,Cj) ==> Ci = Cj | false. l([ ],[ ]) <=> true. l([R|Rs],[C|Cs]) <=> d(R,C), l(Rs,Cs).

Implementing a Rewriting System in CHR
Map each conditional rewrite system rule of the form Condition | LHS  RHS onto a CHR simplification rule of the form LHS  Condition | RHS i.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 Rewriting as CHR Solving:fibonacci
plus(X,0)  X plus(X,suc(Y))  suc(plus(X,Y)) fib(0)  suc(0) fib(suc(0))  suc(0) fib(suc(suc(N))  plus(fib(suc(N)),fib(N)) plus(X,U,V) <=> U = 0 | V = X. plus(X,U,V) <=> U = suc(Y) | V = suc(W), plus(X,Y,W). fib(U,V) <=> U = 0 | V = suc(0). fib(U,V) <=> U = suc(0) | V = suc(0). fib(U,V) <=> U = suc(W), W = suc(N) | fib(N,Y), fib(W,X), plus(X,Y,V).

Example Term Rewriting as CHR Solving Solving: fibonacci(2) = ?
p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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 R=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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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=0 X4=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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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 = 0 e.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

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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)

p(X,U,V) <=> U = 0 | V = X. p(X,U,V) <=> U = s(Y) | V = s(W), p(X,Y,W). f(U,V) <=> U = 0 | V = s(0). f(U,V) <=> U = s(0) | V = s(0). 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))

CHRV vs. Rewriting Systems
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 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)

Implementing a Production System in 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

CHRV vs. Production Systems
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 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

Implementing a Prolog Program in CHR
Map Prolog fact base of the form {f fn.} onto a fact introduction CHR propagation rule: facts  f1 ,..., fn. Map each set of Prolog deductive rules of the form {p(t11,...,tn1) :- b p(t1k,...,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=tn1, b1) ;...; (X1=t1k ,..., Xn=tnk, bk). where {X1,...,Xn} is a set of fresh variables not occurring in {t11,...,tn1,b1, ... p(t1k,...,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 Program Implemented in CHR
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). parent(P,C) :- mother(P,C). sibling(C1,C2) :- not C1 = C2, parent(P,C1), parent(P,C2). CHR Translation facts  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).

CHRV 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 | b11, ..., bkp ; ... ; b11, ..., blq. where: {X1, ..., Xn} = vars(shi), and {Y1, ... , Ym} = vars(b1  ...  bk) \ {X1, ..., Xn} X1, ..., Xn true  (sh  Y1, ... , Ym ((b11  ...  bkp)  ...  (b11  ...  bkq)) Equivalent Prolog clauses: {sh :- b11, ..., bkp. , ... , sh :- b11, ..., 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 CHR CLP Engine CLP Application Rule Base
CHR Base for Domain D1 Solver Prolog Engine CHR Engine ... CHR Host Programming Language L Prolog/L Bridge CHR Base for Domain Dk Solver

... CLP with CHR CLP Application Rule Base
CHR Base for Domain D1 Solver CHR Engine ... CHR Host Programming Language L CHR Base for Domain Dk Solver

CHRV: 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

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

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

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