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CHR: Constraint Handling Rules

This chapter is written by Tom Schrijvers, K.U. Leuven for the hProlog system. Adjusted by Jan Wielemaker to fit the SWI-Prolog documentation infrastructure and remove hProlog specific references.

The CHR system of SWI-Prolog is the K.U.Leuven CHR system. The runtime environment is written by Christian Holzbaur and Tom Schrijvers while the compiler is written by Tom Schrijvers. Both are integrated with SWI-Prolog and licenced under compatible conditions with permission from the authors. Porting and maintenance on YAP is the entire responsability of Vítor Santos Costa.

The main reference for SWI-Prolog's CHR system is:

T. Schrijvers, and B. Demoen, The K.U.Leuven CHR System: Implementation and Application, First Workshop on Constraint Handling Rules: Selected Contributions (Fruwirth, T. and Meister, M., eds.), pp. 1--5, 2004.

Introduction

Constraint Handling Rules (CHR) is a committed-choice bottom-up language embedded in Prolog. It is designed for writing constraint solvers and is particularily useful for providing application-specific constraints. It has been used in many kinds of applications, like scheduling, model checking, abduction, type checking among many others.

CHR has previously been implemented in other Prolog systems (SICStus, Eclipse, Yap), Haskell and Java. This CHR system is based on the compilation scheme and runtime environment of CHR in SICStus.

In this documentation we restrict ourselves to giving a short overview of CHR in general and mainly focus on elements specific to this implementation. For a more thorough review of CHR we refer the reader to [Freuhwirth:98]. More background on CHR can be found at the CHR web site.

Syntax and Semantics

We present informally the syntax and semantics of CHR.

CHR Syntax

The syntax of CHR rules in hProlog is the following:

rules --> rule, rules.
rules --> [].

rule --> name, actual_rule, pragma, [atom(`.`)].

name --> atom, [atom(`@`)].
name --> [].

actual_rule --> simplification_rule.
actual_rule --> propagation_rule.
actual_rule --> simpagation_rule.

simplification_rule --> constraints, [atom(`<=>`)], guard, body.
propagation_rule --> constraints, [atom(`==>`)], guard, body.
simpagation_rule --> constraints, [atom(`\`)], constraints, [atom(`<=>`)],
                     guard, body.

constraints --> constraint, constraint_id.
constraints --> constraint, [atom(`,`)], constraints.

constraint --> compound_term.

constraint_id --> [].
constraint_id --> [atom(`#`)], variable.

guard --> [].
guard --> goal, [atom(`|`)].

body --> goal.

pragma --> [].
pragma --> [atom(`pragma`)], actual_pragmas.

actual_pragmas --> actual_pragma.
actual_pragmas --> actual_pragma, [atom(`,`)], actual_pragmas.

actual_pragma --> [atom(`passive(`)], variable, [atom(`)`)].

Additional syntax-related terminology:

head: the constraints in an actual_rule before the arrow (either <=> or ==>)

Semantics Semantics

In this subsection the operational semantics of CHR in Prolog are presented informally. They do not differ essentially from other CHR systems.

When a constraint is called, it is considered an active constraint and the system will try to apply the rules to it. Rules are tried and executed sequentially in the order they are written.

A rule is conceptually tried for an active constraint in the following way. The active constraint is matched with a constraint in the head of the rule. If more constraints appear in the head they are looked for among the suspended constraints, which are called passive constraints in this context. If the necessary passive constraints can be found and all match with the head of the rule and the guard of the rule succeeds, then the rule is committed and the body of the rule executed. If not all the necessary passive constraint can be found, the matching fails or the guard fails, then the body is not executed and the process of trying and executing simply continues with the following rules. If for a rule, there are multiple constraints in the head, the active constraint will try the rule sequentially multiple times, each time trying to match with another constraint.

This process ends either when the active constraint disappears, i.e. it is removed by some rule, or after the last rule has been processed. In the latter case the active constraint becomes suspended.

A suspended constraint is eligible as a passive constraint for an active constraint. The other way it may interact again with the rules, is when a variable appearing in the constraint becomes bound to either a nonvariable or another variable involved in one or more constraints. In that case the constraint is triggered, i.e. it becomes an active constraint and all the rules are tried.

Rules

There are three different kinds of rules, each with their specific semantics:

simplification The simplification rule removes the constraints in its head and calls its body.
propagation The propagation rule calls its body exactly once for the constraints in its head.
simpagation The simpagation rule removes the constraints in its head after the \ and then calls its body. It is an optimization of simplification rules of the form: [constraints_1, constraints_2 <=> constraints_1, body ] Namely, in the simpagation form:

constraints1 \ constraints2 <=> body

constraints1 constraints are not called in the body.

Rule Names

Naming a rule is optional and has no semantical meaning. It only functions as documentation for the programmer.

Pragmas

The semantics of the pragmas are:

passive(Identifier) The constraint in the head of a rule Identifier can only act as a passive constraint in that rule.

Additional pragmas may be released in the future.

CHR_Options Options

It is possible to specify options that apply to all the CHR rules in the module. Options are specified with the option/2 declaration:

                option(Option,Value).

Available options are:

check_guard_bindings This option controls whether guards should be checked for illegal variable bindings or not. Possible values for this option are on, to enable the checks, and off, to disable the checks.
optimize This is an experimental option controlling the degree of optimization. Possible values are full, to enable all available optimizations, and off (default), to disable all optimizations. The default is derived from the SWI-Prolog flag optimise, where true is mapped to full. Therefore the commandline option -O provides full CHR optimization. If optimization is enabled, debugging should be disabled.
debug This options enables or disables the possibility to debug the CHR code. Possible values are on (default) and off. See debugging for more details on debugging. The default is derived from the prolog flag generate_debug_info, which is true by default. See -nodebug. If debugging is enabled, optimization should be disabled.
mode This option specifies the mode for a particular constraint. The value is a term with functor and arity equal to that of a constraint. The arguments can be one of -, + or ?. The latter is the default. The meaning is the following:

The corresponding argument of every occurrence of the constraint is always unbound.

The corresponding argument of every occurrence of the constraint is always ground.

? The corresponding argument of every occurrence of the constraint can have any instantiation, which may change over time. This is the default value.

The declaration is used by the compiler for various optimizations. Note that it is up to the user the ensure that the mode declaration is correct with respect to the use of the constraint. This option may occur once for each constraint.

type_declaration This option specifies the argument types for a particular constraint. The value is a term with functor and arity equal to that of a constraint. The arguments can be a user-defined type or one of the built-in types:
int The corresponding argument of every occurrence of the constraint is an integer number.
float ...{} a floating point number.
number ...{} a number.
natural ...{} a positive integer.
any The corresponding argument of every occurrence of the constraint can have any type. This is the default value.

Currently, type declarations are only used to improve certain optimizations (guard simplification, occurrence subsumption, ...{}).

type_definition This option defines a new user-defined type which can be used in type declarations. The value is a term of the form type( name, list), where name is a term and list is a list of alternatives. Variables can be used to define generic types. Recursive definitions are allowed. Examples are

type(bool,[true,false]).
type(complex_number,[float + float * i]).
type(binary_tree(T),[ leaf(T) | node(binary_tree(T),binary_tree(T)) ]).
type(list(T),[ [] | [T | list(T)]).

The mode, type_declaration and type_definition options are provided for backward compatibility. The new syntax is described below.

CHR in Prolog Programs

The CHR constraints defined in a particulary chr file are associated with a module. The default module is user. One should never load different chr files with the same CHR module name.

Constraint Declarations

Every constraint used in CHR rules has to be declared. There are two ways to do this. The old style is as follows:

option(type_definition,type(list(T),[ [] , [T|list(T)] ]).
option(mode,foo(+,?)).
option(type_declaration,foo(list(int),float)).
:- constraints foo/2, bar/0.

The new style is as follows:

:- chr_type list(T) ---> [] ; [T|list(T)].
:- constraints foo(+list(int),?float), bar.

Compilation

The SWI-Prolog CHR compiler exploits term_expansion/2 rules to translate the constraint handling rules to plain Prolog. These rules are loaded from the library chr. They are activated if the compiled file has the chr extension or after finding a declaration of the format below.

:- constraints ...

It is adviced to define CHR rules in a module file, where the module declaration is immediately followed by including the chr library as examplified below:

:- module(zebra, [ zebra/0 ]).
:- use_module(library(chr)).

:- constraints ...

Using this style CHR rules can be defined in ordinary Prolog pl files and the operator definitions required by CHR do not leak into modules where they might cause conflicts.

CHR Debugging

The CHR debugging facilities are currently rather limited. Only tracing is currently available. To use the CHR debugging facilities for a CHR file it must be compiled for debugging. Generating debug info is controlled by the CHR option debug, whose default is derived from the SWI-Prolog flag generate_debug_info. Therefore debug info is provided unless the -nodebug is used.

Ports

vFor CHR constraints the four standard ports are defined:

call A new constraint is called and becomes active.
exit An active constraint exits: it has either been inserted in the store after trying all rules or has been removed from the constraint store.
fail An active constraint fails.
redo An active constraint starts looking for an alternative solution.

In addition to the above ports, CHR constraints have five additional ports:

wake A suspended constraint is woken and becomes active.
insert An active constraint has tried all rules and is suspended in the constraint store.
remove An active or passive constraint is removed from the constraint store, if it had been inserted.
try An active constraints tries a rule with possibly some passive constraints. The try port is entered just before committing to the rule.
apply An active constraints commits to a rule with possibly some passive constraints. The apply port is entered just after committing to the rule.

Tracing

Tracing is enabled with the chr_trace/0 predicate and disabled with the chr_notrace/0 predicate.

When enabled the tracer will step through the call, exit, fail, wake and apply ports, accepting debug commands, and simply write out the other ports.

The following debug commans are currently supported:

        CHR debug options:

                <cr>    creep           c       creep
		s	skip
		g	ancestors
                n       nodebug
		b	break
                a       abort
                f       fail
                ?       help            h       help

Their meaning is:

creep Step to the next port.
skip Skip to exit port of this call or wake port.
ancestors Print list of ancestor call and wake ports.
nodebug Disable the tracer.
break Enter a recursive Prolog toplevel. See break/0.
abort Exit to the toplevel. See abort/0.
fail Insert failure in execution.
help Print the above available debug options.

CHR Debugging Predicates

The chr module contains several predicates that allow inspecting and printing the content of the constraint store.

chr_trace Activate the CHR tracer. By default the CHR tracer is activated and deactivated automatically by the Prolog predicates trace/0 and notrace/0.

CHR_Examples Examples

Here are two example constraint solvers written in CHR.

The program below defines a solver with one constraint, leq/2, which is a less-than-or-equal constraint.

:- module(leq,[cycle/3, leq/2]).
:- use_module(library(chr)).

:- constraints leq/2.
reflexivity  @ leq(X,X) <=> true.
antisymmetry @ leq(X,Y), leq(Y,X) <=> X = Y.
idempotence  @ leq(X,Y) \ leq(X,Y) <=> true.
transitivity @ leq(X,Y), leq(Y,Z) ==> leq(X,Z).

cycle(X,Y,Z):-
        leq(X,Y),
        leq(Y,Z),
        leq(Z,X).

The program below implements a simple finite domain constraint solver.

:- module(dom,[dom/2]).
:- use_module(library(chr)).

:- constraints dom/2.

dom(X,[]) <=> fail.
dom(X,[Y]) <=> X = Y.
dom(X,L1), dom(X,L2) <=> intersection(L1,L2,L3), dom(X,L3).

intersection([],_,[]).
intersection([H|T],L2,[H|L3]) :-
        member(H,L2), !,
        intersection(T,L2,L3).
intersection([_|T],L2,L3) :-
        intersection(T,L2,L3).

Compatibility with SICStus CHR

There are small differences between CHR in SWI-Prolog and newer YAPs and SICStus and older versions of YAP. Besides differences in available options and pragmas, the following differences should be noted:

[The handler/1 declaration] In SICStus every CHR module requires a handler/1 declaration declaring a unique handler name. This declaration is valid syntax in SWI-Prolog, but will have no effect. A warning will be given during compilation.
[The rules/1 declaration] In SICStus, for every CHR module it is possible to only enable a subset of the available rules through the rules/1 declaration. The declaration is valid syntax in SWI-Prolog, but has no effect. A warning is given during compilation.
[Sourcefile naming] SICStus uses a two-step compiler, where chr files are first translated into pl files. For SWI-Prolog CHR rules may be defined in a file with any extension.

Guidelines

In this section we cover several guidelines on how to use CHR to write constraint solvers and how to do so efficiently.

[Set semantics] The CHR system allows the presence of identical constraints, i.e. multiple constraints with the same functor, arity and arguments. For most constraint solvers, this is not desirable: it affects efficiency and possibly termination. Hence appropriate simpagation rules should be added of the form:

{constraint \ constraint <=> true}.

[Multi-headed rules] Multi-headed rules are executed more efficiently when the constraints share one or more variables.
[Mode and type declarations] Provide mode and type declarations to get more efficient program execution. Make sure to disable debug (-nodebug) and enable optimization (-O).

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