Updating Agents

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Constraint Handling Rules

• Pierangelo DellAcqua
• Dept. of Science and Technology
• Linköping University
• pier_at_itn.liu.se

Constraint programming 2001 November 13th
2001 http//www.ida.liu.se/labs/logpro/ulfni/cp200
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Overview

• Motivation
• Language CHR
• Declarative and operational semantics
• Properties
• Examples of CHR constraint solvers

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Based on

• Theory and Practice of Constraint Handling Rules
• Thom Frühwirth, J. Logic Programming 199419,
  201-679
• Examples CHR constraint solvers available at
• www.informatik.uni-muenchen.de/fruehwir/chr/

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Black-box vs Glass-box solvers

• In most systems constraint solving is hard-wired
  in a built-in constraint solver in a low-level
  language black-box approach
• efficiency
• non-extensible, unpredictable, hard to debug
• Some systems facilitate defining new constraints
  and solvers glass-box approach
• improved control of propagation and search
• examples CHR, HAL, ...

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Constraint Handling Rules (CHR)

• Declarative programming language for the
  specification and implementation of constraint
  solvers and programs

Application

• CHR-constraint solvers are open and flexible, can
  be maintained, debugged and analysed

CHR-constraints
CHR-solver
built-in constraints
Black-box
Host language (Prolog, Lisp, )
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CHR by example

• The partial order relation X ? Y as a
  user-defined constraint

XltY ltgt XY true. reflexivity XltY ? YltX
ltgt XY. antisymmetry XltY ? YltZ gt
XltZ. transitivity
Computation AltB ? CltA ? BltC CltA ? AltB
propagates CltB by transitivity CltB ? BltC
simplifies to BC by antisymmetry AltB ? CltA
simplifies to AB by antisymmetry since BC AB ?
BC
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CHR syntax
Head H conjunction of CHR-constraints Guard G conj
unction of built-in constraints Body B conjunction
of built-in and CHR-constraints
A CHR-program is a finite set of CHR-rules. There
are three kinds of CHR-rules
Simplification H ltgt G B Propagation H gt G
B Simpagation H1 \ H2 ltgt G B
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Declarative semantics
Simplification rule H ltgt G B ?h (?g (G) ? ( H
? ?b ( B ) ) ) Propagation rule H gt G B ?h
(?g (G) ? ( H ? ?b ( B ) ) ) Simpagation rule H1
\ H2 ltgt G B ?h1?h2 (?g (G) ? (H1?H2 ? ?b
(H1?B)))
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Declarative semantics (2)
Declarative semantics of a CHR-program P
Sem(P) LP, CT where LP is
the logical reading of the CHR-rules in P and CT
is a theory for built-in constraints
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Operational semantics
A state is a tuple ?F,E,D? where F is a
conjunction of CHR- and built-in constraints
(goal store) E is a conjunction of
CHR-constraints (CHR-store) D is a conjunction of
built-in constraints (built-in constraints store)
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CHR transitions
Solve ?C?F, E, D? ? ?F, E, D2? if C is a
built-in constraint and CT (C?D) ? D2
Introduce ?H?F, E, D? ? ?F, H?E, D? if H is a
CHR-constraint
Simplify ?F, H2?E, D? ? ?B?F, E, HH2?D? if (H
ltgt G B) in P and CT D ? ?h (HH2 ? G)
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CHR transitions (2)
Propagate ?F, H2?E, D? ? ?B?F, H2?E,
HH2?D? if (H gt G B) in P and CT D ? ?h
(HH2 ? G)
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Initial and final states
An initial state consists of a goal G and empty
constraint stores
?G,true,true?
A final state is either of the form
(i) ?F,E,false? failed final state or of the
form (ii) ?true,E,D? successful final state
where no transition is applicable and D
? false
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CHR computations
A computation of a goal G is a sequence S0, S1,
of states with Si ? Si1 beginning with the
initial state S0 ?G,true,true? and ending with
a final state or diverging
An answer of a goal G is the final state of a
computation for G
The logical meaning of a state ?F,E,D?, which
occurs in a computation for G, is ?x (F?E?D),
where x are the variables in ?F,E,D? but not in G
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Example CHR calculus

• XltY ltgt XY true. reflexivity
• XltY ? YltX ltgt XY. antisymmetry
• XltY ? YltZ gt XltZ. transitivity

?AltB ? CltA ? BltC, true, true? Introduce
?3 ?true, AltB ? CltA ? BltC, true? Propagate
? ?CltB, AltB ? CltA ? BltC, true? Introduce
? ?true, AltB ? CltA ? BltC ? CltB,
true? Simplify ? ?BC, AltB ? CltA, true? Solve
? ?true, AltB ? CltA, BC ? Simplify ? ?AB,
true, BC ? Solve ? ?true, true, AB ? BC ?
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Logical equivalence of states
CHR transitions preserve the logical meaning of
states
Lemma Let P be a CHR program and G a goal. If C
is the logical meaning of a state in a
computation of G, then LP , CT ? ( G ? C)
There is no distinction between successful and
failed computations
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Correspondence between semantics
Theorem (Soundness) Let P be a CHR program and G
a goal. If G has a computation with answer C,
then LP, CT ? ( C ? G)
Theorem (Completeness) Let P be a CHR program
and G a goal with at least one finite
computation. Let C be a conjunction of
constraints. If LP, CT ?(G?C), then G has a
computation with answer C2 such that LP, CT
? ( C ? C2)
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Example completeness
The completeness theorem does not hold if G has
no finite computations
Let P be p ltgt p and G the goal p. Since LP
is p?p, it holds that LP,CT p?p, but G has
only an infinite computation
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Example failed computations
The completeness theorem is weak for failed
computations
p ltgt q, p ltgt false
Let P be
We have that LP, CT ?q, but q has no failed
computation. It has a successful derivation with
answer q.
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Confluence
Confluence The answer of a goal G is always the
same, no matter which of the applicable rules are
applied
p ltgt q, p ltgt false, q ltgt false is
confluent
p ltgt q, p ltgt false is not confluent
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Soundness and Completeness revisited

• Theorem (Strong Soundness and Completeness)
• Let P be a terminating and confluent CHR program,
  G a goal and C a conjunction of constraints.
• Then the following are equivalent
• LP, CT ?(C?G)
• G has a computation with answer C2 such that
• LP, CT ?(C?C2)
• Every computation of G has an answer C2 such
  that
• LP, CT ?(C?C2)

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CLP CHR
Any CLP language can be extended with CHR
Idea
- Allow clauses for CHR constraints introduce
choices - Regard a predicate as a constraint and
add CHR rules for it
Dont know and dont care nondeterminism combined
in a declarative way
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CLPCHR language
A CLPCHR program is a finite set of (i) CLP
clauses for predicates and CHR constraints,
and (ii) CHR rules for CHR constraints.
A CLP clause is of the form
H - B1,,Bk (k ? 0)
conjunction of atoms, CHR constraints and
built-in constraints
an atom or a CHR constraint not a built-in
constraint
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CLPCHR language (2)
The logical meaning of a CLP clause is given by
Clarks completion
Backward compatibility
Labelling declarations (see def. 6.1 of JLP
paper) are dropped, easily simulated
H - B label-with H if G
lw, H ltgt G H2, lw H2 - B
new CHR constraint
CHR constraint
new predicate
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CLPCHR transitions
Unfold (revisited) ?H2?F, E, D? ? ?B?F, E,
HH2?D? if (H-B) in P, H2 is a predicate and CT
D??h (HH2)
Label (revisited) ?F, H2?E, D? ? ?B?F, E,
HH2?D? if (H-B) in P, H2 is a CHR constraint
and CT D??h (HH2)
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Examples of CHR solvers
Several constraint solvers have been written in
CHR, including new constraint domains such as
terminological and temporal reasoning
bool.pl boolean constraints arc.pl arc-consistency
over finite domains interval.pl interval domains
over integers and reals list.pl equality
constraints over concatenation of lists
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Sicstus Prolog CHR
A CHR rule in SicstusPrologCHR is of the form
H ltgt G B H gt G B H1 \ H2
ltgt G B
where
H is a conjunction of CHR-constraints G is a
conjunction of atoms and built-in constraints B
is a conjunction of atoms, built-in and
CHR-constraints
A CHR rule can be fired if its guard G is
true Note that during the proof of the guard G
no new binding can be generated for variables
that occur also in H