Constraint Handling Rules

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Constraint Handling Rules

May, 2005Jairson Vitorino and Marcos Aurélio

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Contents

• CHR– Introduction – Syntax– Operational Semantics– Example

• CHR-– Introduction – Operational Semantics– Example

• References

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CHR

Constraint Handling Rules

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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)

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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

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Syntax

• Constraints– User-defined

• Ex: noattack(Q1,Q2), connected(X,Y,3)

– Built-In• Ex: A>B, A=B, ground(A)

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Syntax: typical rule

H1,...,Hn <=>G1,...Gn | B1,...,Bn

MultipleHeads

Guards Body

Commited choice: When the engine picks a rulethere will not be any backtracking, so in CHR order matters!

(Only user-defined)(Only built-in) Both constraint

categories

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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

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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 ProgramOr Base Rule

Final State: true or false

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• 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.”

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• Transitions (Solve)

X=Y, G1, G2Goal

U1, ..., UnUDCS

B1, ..., BnBICS

Solve:

G1, G2Goal

U1, ..., UnUDCS

B1, ..., Bn, X=YBICS

Built-in

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• Transitions (Introduce)

C(a,b), G1, ..., GnGoal

U1, ..., UnUDCS

B1, ..., BnBICS

Solve:

G1, G2Goal

C(a,b), U1, ..., UnUDCS

B1, ..., BnBICS

User-defined

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• Transitions (Simplify)U1,U2 B1| U3,U4

U5..., UnGoal

U1, U2 ,V1, ..., VnUDCS

B1, ..., BnBICS

Simplify:

U5,..., UnGoal

U3, U4, V1, ..., VnUDCS

B1, ..., BnBICS

Rule in CHR program

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• Transitions (Propagate)U1,U2 B1| U3,U4

UnGoal

U1, U2, ..., UnUDCS

B1, ..., BnBICS

Propagate:

U1, U2,..., UnGoal

U3, U4, U1, U2, ..., UnUDCS

B1, ..., BnBICS

Rule in CHR program

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• Transitions (Simpagate)U1,U2 / U5B1| U3,U4

U4,..., Un

GoalU1, U2,U5, ..., Un

UDCSB1, ..., Bn

BICS

Simpagate:

U6, ..., Un

GoalU3,U4, U1,U2 ..., Un

UDCSB1, ..., Bn

BICS

Rule in CHR program

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Example

CHR ProgramX=<Y <=> X=Y | true. X=<Y,Y=<X <=> X=Y. X=<Y,Y=<Z ==> X=<Z.

Initial state:

A<=B, C<=A, B<=CGoal

trueUDCS

trueBICS

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ExampleIntroduce:

C<=A, B<=CGoal

A<=BUDCS

trueBICS

Introduce:

B<=CGoal

C<=A, A<=BUDCS

trueBICS

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ExampleIntroduce:

trueGoal

C<=A, A<=B, B<=C

UDCStrue

BICS

Propagate: X<=Y,Y<=Z ==> X<=Z.

trueGoal

C<=B, C<=A, A<=B, B<=C

UDCStrue

BICS

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ExampleIntroduce:

trueGoal

C<=A, A<=BB<=C, C<=B

UDCStrue

BICS

Simplify: X<=Y,Y<=X <=> X=Y.

trueGoal

C<=A, A<=BUDCS

B=CBICS

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ExampleSolve:

trueGoal

C<=A, A<=BUDCS

B=CBICS

Simplify: X<=Y,Y<=X <=> X=Y.

Goaltrue

UDCSB=C, A=B

BICS

(Logo: C<=A equivale a B<=A)

true

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ExampleSolve:

trueGoal

trueUDCS

B=C, A=BBICS

Final State

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CHR-

Constraint Handling Rules with Disjunctions

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Introduction

• Allows disjunctions in the bodies of rules and in goals

• It can be used as a general-purpose logic programming language

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• State– Minimal set of subgoals

• Ex: a (b (c d)) e

– Logical meaning: y A– Initial State– Final State

• Successful• Failed

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• Transitions (Solve)

Solve:

a (b c) (b d) eGoal

a (b (c d)) eGoal

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• Solve– “To normalize the predefined means

to produce a new state G’ that is (according to the logical theory T0) logically equivalent to the state G”

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• Transitions (Simplify)– b c f g

Simplify:

a (f g) (b d) eGoal

a (b c) (b d) eGoal

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• Transitions (Split)

Simplify:

(b (a d e f)) (c (a d e f))Goal

(b c) (a d e f)Goal

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Example

bird albatross penguin.penguin flies false.

Initial state:

bird flies

Goal

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ExampleSimplify: bird albatross penguin.

(albatross penguin) flies

Goal

Split:

(albatross flies) (penguin flies)Goal

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Example

Propagate: penguin flies false.

(albatross flies) falseGoal

Final State (no derivation step can be applied)

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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).

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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).

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Reference

• Theory and practice of Constraint Handling Rules (http://www.pst.informatik.uni-muenchen.de/personen/fruehwir/jlp-chr1/jlp-chr1.html)

• A languagem for experimenting with Declarative Paradigms (http://citeseer.ist.psu.edu/611754.html)

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Reference

• XSB Prolog– http://xsb.sf.net

• The ECLiPSe Constraint Logic Programming System– http://www.icparc.ic.ac.uk/eclipse/

• WebCHR– http://bach.informatik.uni-ulm.de/~webchr/

• JACK – Java Constraint Kit– http://www.pms.ifi.lmu.de/software/jack/

Constraint Handling Rules

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