Constraint Handling Rules

1
Constraint Handling Rules

• May, 2005
• Jairson Vitorino and Marcos Aurélio
• ORCAS

Orcas
2
Contents

• CHR
• Introduction
• Syntax
• Operational Semantics
• Example
• CHR-?
• Introduction
• Operational Semantics
• Example
• References

3
CHR

• Constraint Handling Rules

4
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)

5
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
6
Syntax

• Constraints
• User-defined
• Ex noattack(Q1,Q2), connected(X,Y,3)
• Built-In
• Ex AgtB, AB, ground(A)

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Syntax typical rule
H1,...,Hn ltgtG1,...Gn B1,...,Bn
Multiple Heads
Guards
Body
Both constraint categories
(Only built-in)
(Only user-defined)
Commited choice When the engine picks a
rule there will not be any backtracking, so in
CHR order matters!
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Syntax

• Rules
• Simplification
• H1,...,Hn ltgt G1,...Gn B1,...,Bn
• Propagation
• H1,...,Hn gtG1,...Gn B1,...,Bn
• Simpagation
• H1,...,Hn / Hn1,...HnnltgtG1,...Gn B1,...,Bn

9
Operational Semantics
State
User defined Constraints Store (UDCS)
Logical meaning ?y Goal ? UDCS ? BICS
Goal or query
Built-in Constraint Store (BICS)
CHR Program Or Base Rule
CHR ENGINE
Final State true or false
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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.

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Operational Semantics

• Transitions (Solve)

Built-in
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Operational Semantics

• Transitions (Introduce)

User-defined
13
Operational Semantics

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

Rule in CHR program
14
Operational Semantics

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

Rule in CHR program
15
Operational Semantics

• Transitions (Simpagate)
• U1,U2 / U5?B1 U3,U4

Rule in CHR program
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Example

• CHR Program
• XltY ltgt XY true.
• XltY,YltX ltgt XY.
• XltY,YltZ gt XltZ.

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Example
18
Example
19
Example
20
Example
(Logo CltA equivale a BltA)
true
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Example
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

24
Operational Semantics

• 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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Operational Semantics

• Transitions (Solve)

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Operational Semantics

• 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

27
Operational Semantics

• Transitions (Simplify)
• b ? c ?f ? g

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Operational Semantics

• Transitions (Split)

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Example

• bird ? albatross ? penguin.
• penguin ? flies ? false.

30
Example
31
Example
Final State (no derivation step can be applied)
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Example K-coloring map
main() ltgt true color(X1),               
 color(X2),                 color(X3),         
       color(X4),               
 connected(X1,X2),               
 connected(X1,X3),               
 connected(X3,X4),               
 connected(X2,X4).connected(X1,X2) ltgt
ground(X1), ground(X2), X1X2 false.color(X)  
      gt true (X1 X2 X3).
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Example 4-Queens
main() ltgt 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) ltgt
ground(X), ground(Y) PYD, MYD, ne(X,Y),
ne(X,P),ne(X,M).c(X) ltgt true (X1 X2
X3 X4).sol(X1,X2,X3,X4) ltgt 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/persone
  n/fruehwir/jlp-chr1/jlp-chr1.html)
• A languagem for experimenting with Declarative
  Paradigms (http//citeseer.ist.psu.edu/611754.html
  )

35
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/