Orcas Constraint Handling Rules May, 2005 Jairson Vitorino and Marcos Aurélio ORCAS Orcas

Contents CHR –Introduction –Syntax –Operational Semantics –Example CHR- –Introduction –Operational Semantics –Example References

Orcas CHR Constraint Handling Rules

Orcas Introduction Constraint Logic Programming –Logic Programming + Constraint Solving CHR: Language for writing Constraint Solvers: –rule-based –Flexible to define new domains –Constraint resolution via rules (declarative)

Orcas Introduction CHR –Rewrite rules (pattern matching + simplify rules) –Plus propagation rules –Similiar to production rules (forward chaining), but with a well defined semantics –CHR + disjunction is turing-complete, first- order knowledge language and monotonic Current CSP software packages have fixed domains and constraints, answer: CHR

Orcas Syntax Constraints –User-defined Ex: noattack(Q1,Q2), connected(X,Y,3) –Built-In Ex: A>B, A=B, ground(A)

Orcas Syntax: typical rule H1,...,Hn G1,...Gn | B1,...,Bn Multiple Heads GuardsBody Commited choice: When the engine picks a rule there will not be any backtracking, so in CHR order matters! (Only user-defined) (Only built-in) Both constraint categories

Orcas Syntax Rules –Simplification H1,...,Hn G1,...Gn | B1,...,Bn –Propagation H1,...,Hn =>G1,...Gn | B1,...,Bn –Simpagation H1,...,Hn / Hn1,...Hnn G1,...Gn | B1,...,Bn

Orcas Operational Semantics State Goal or query User defined Constraints Store (UDCS) Built-in Constraint Store (BICS) CHR ENGINE Logical meaning: y Goal  UDCS  BICS CHR Program Or Base Rule Final State: true or false

Orcas Operational Semantics Transitions –Solve and Introduce –Propagate –Simplify Commited-choice and Confluence “The confluence property of a program guarantees that any computation starting from an arbitrary given initial state, i.e. any possible order of rule applications, results in the same final state.”

Orcas Operational Semantics Transitions (Solve) X=Y, G1, G2 Goal U1,..., Un UDCS B1,..., Bn BICS Solve: G1, G2 Goal U1,..., Un UDCS B1,..., Bn, X=Y BICS Built-in

Orcas Operational Semantics Transitions (Introduce) C(a,b), G1,..., Gn Goal U1,..., Un UDCS B1,..., Bn BICS Solve: G1, G2 Goal C(a,b), U1,..., Un UDCS B1,..., Bn BICS User-defined

Orcas Operational Semantics Transitions (Simplify) U1,U2  B1| U3,U4 U5..., Un Goal U1, U2,V1,..., Vn UDCS B1,..., Bn BICS Simplify: U5,..., Un Goal U3, U4, V1,..., Vn UDCS B1,..., Bn BICS Rule in CHR program

Orcas Operational Semantics Transitions (Propagate) U1,U2  B1| U3,U4 Un Goal U1, U2,..., Un UDCS B1,..., Bn BICS Propagate: U1, U2,..., Un Goal U3, U4, U1, U2,..., Un UDCS B1,..., Bn BICS Rule in CHR program

Orcas Operational Semantics Transitions (Simpagate) U1,U2 / U5  B1| U3,U4 U4,..., Un Goal U1, U2,U5,..., Un UDCS B1,..., Bn BICS Simpagate: U6,..., Un Goal U3,U4, U1,U2..., Un UDCS B1,..., Bn BICS Rule in CHR program

Orcas Example CHR Program X= X=Y | true. X= X=Y. X= X=<Z. Initial state: A<=B, C<=A, B<=C Goal true UDCS true BICS

Orcas Example Introduce: C<=A, B<=C Goal A<=B UDCS true BICS Introduce: B<=C Goal C<=A, A<=B UDCS true BICS

Orcas Example Introduce: true Goal C<=A, A<=B, B<=C UDCS true BICS Propagate: X X<=Z. true Goal C<=B, C<=A, A<=B, B<=C UDCS true BICS

Orcas Example Introduce: true Goal C<=A, A<=B B<=C, C<=B UDCS true BICS Simplify: X X=Y. true Goal C<=A, A<=B UDCS B=C BICS

Orcas Example Solve: true Goal C<=A, A<=B UDCS B=C BICS Simplify: X X=Y. Goal true UDCS B=C, A=B BICS (Logo: C<=A equivale a B<=A) true

Orcas Example Solve: true Goal true UDCS B=C, A=B BICS Final State

Orcas CHR- Constraint Handling Rules with Disjunctions

Orcas Introduction Allows disjunctions in the bodies of rules and in goals It can be used as a general-purpose logic programming language

Orcas Operational Semantics State –Minimal set of subgoals Ex: a  (b  (c  d))  e –Logical meaning: y A –Initial State –Final State Successful Failed

Orcas Operational Semantics Transitions (Solve) Solve: a  (b  c)  (b  d)  e Goal a  (b  (c  d))  e Goal

Orcas Operational Semantics Solve –“To normalize the predefined means to produce a new state G’ that is (according to the logical theory T 0 ) logically equivalent to the state G”

Orcas Operational Semantics Transitions (Simplify) –b  c  f  g Simplify: a  (f  g)  (b  d)  e Goal a  (b  c)  (b  d)  e Goal

Orcas Operational Semantics Transitions (Split) Simplify: (b  (a  d  e  f))  (c  (a  d  e  f)) Goal (b  c)  (a  d  e  f) Goal

Orcas Example bird  albatross  penguin. penguin  flies  false. Initial state: bird  flies Goal

Orcas Example Simplify: bird  albatross  penguin. ( albatross  penguin)  flies Goal Split: ( albatross  flies)  (penguin  flies) Goal

Orcas Example Propagate: penguin  flies  false. ( albatross  flies)  false Goal Final State (no derivation step can be applied)

Orcas Example: K-coloring map main() true | color(X1), color(X2), color(X3), color(X4), connected(X1,X2), connected(X1,X3), connected(X3,X4), connected(X2,X4). connected(X1,X2) ground(X1), ground(X2), X1=X2 |false. color(X) ==> true | (X=1 ; X=2 ; X=3).

Orcas Example: 4-Queens main() true | c(X1), c(X2), c(X3), c(X4), // rainhas na(X1,X2,1), na(X1,X3,2), na(X1,X4,3), na(X2,X3,1), na(X2,X4,2), na(X3,X4,1), sol(X1,X2,X3,X4). na(X,Y,D) ground(X), ground(Y) | P=Y+D, M=Y~D, ne(X,Y), ne(X,P), ne(X,M). c(X) true | (X=1 ; X=2 ; X=3 ; X=4). sol(X1,X2,X3,X4) ground(X1), ground(X2), ground(X3), ground(X4) | write(X1), write(X2), write(X3), writeLn(X4).

Orcas Reference Theory and practice of Constraint Handling Rules ( muenchen.de/personen/fruehwir/jlp-chr1/jlp- chr1.html) muenchen.de/personen/fruehwir/jlp-chr1/jlp- chr1.html A languagem for experimenting with Declarative Paradigms (

Orcas Reference XSB Prolog – The ECLiPSe Constraint Logic Programming System – WebCHR – ulm.de/~webchr/ ulm.de/~webchr/ JACK – Java Constraint Kit –