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Post-finalization note: Because of the extremely high cost of
conforming to the first and third conditions of default-hash
,
implementers may disregard those conditions and examine only a
bounded portion of the argument.
This SRFI provides comparators, which bundle a type test predicate, an equality predicate, an ordering predicate, and a hash function (the last two are optional) into a single Scheme object. By packaging these procedures together, they can be treated as a single item for use in the implementation of data structures.
The four procedures above have complex dependencies on one another, and it is inconvenient to have to pass them individually to other procedures that might or might not make use of all of them. For example, a set implementation by its nature requires only an equality predicate, but if it is implemented using a hash table, an appropriate hash function is also required if the implementation does not provide one; alternatively, if it is implemented using a tree, procedures specifying a total order are required. By passing a comparator rather than a bare equality predicate, the set implementation can make use of whatever procedures are available and useful to it.
This SRFI is a simplified and enhanced rewrite of SRFI 114, and shares some of its design rationale and all of its acknowledgements. The largest change is the replacement of the comparison procedure with the ordering procedure. This allowed most of the special-purpose comparators to be removed. In addition, many of the more specialized procedures, as well as all but one of the syntax forms, have been removed as unnecessary.
Special thanks to Taylan Ulrich Bayırlı/Kammer, whose insistence
that SRFI 114 was unacceptable inspired this redesign.
Jörg Wittenberger added Chicken-specific type declarations,
which I have moved to comparators.scm
, as it is
a Chicken-specific library.
He also provided Chicken-specific metadata and setup commands.
Comments from Shiro Kawai, Alex Shinn, and Kevin Wortman
guided me to the current design for bounds and salt.
The procedures in this SRFI are in the (srfi 128)
library (or (srfi :128)
on R6RS), but the sample implementation currently places them in the (comparators)
library. This means it can't be used alongside SRFI 114, but there's no reason for anyone to do that.
A comparator is an object of a disjoint type. It is a bundle of procedures that are useful for comparing two objects in a total order. It is an error if any of the procedures have side effects. There are four procedures in the bundle:
The type test predicate returns #t
if its argument has the correct type to be passed as an argument to the other three procedures, and #f
otherwise.
The equality predicate returns #t
if the two objects are the same in the sense of the comparator, and #f
otherwise. It is the programmer's responsibility to ensure that it is reflexive, symmetric, transitive, and can handle any arguments that satisfy the type test predicate.
The ordering predicate returns #t
if the first object precedes the second in a total order, and #f
otherwise. Note that if it is true, the equality predicate must be false. It is the programmer's responsibility to ensure that it is irreflexive, antisymmetric, transitive, and can handle any arguments that satisfy the type test predicate.
The hash function takes an object and returns an exact non-negative integer. It is the programmer's responsibility to ensure that it can handle any argument that satisfies the type test predicate, and that it returns the same value on two objects if the equality predicate says they are the same (but not necessarily the converse).
It is also the programmer's responsibility to ensure that all four procedures provide the same result whenever they are applied to the same object(s) (in the sense of eqv?
), unless the object(s) have been mutated since the last invocation. In particular, they must not depend in any way on memory addresses in implementations where the garbage collector can move objects in memory.
The comparator objects defined in this SRFI are not applicable to circular structure or to NaNs, or to objects containing any of these. Attempts to pass any such objects to any procedure defined here, or to any procedure that is part of a comparator defined here, is an error except as otherwise noted.
Predicates: comparator? comparator-ordered? comparator-hashable?
Constructors: make-comparator make-pair-comparator make-list-comparator make-vector-comparator make-eq-comparator make-eqv-comparator make-equal-comparator
Standard hash functions: boolean-hash char-hash char-ci-hash string-hash string-ci-hash symbol-hash number-hash
Bounds and salt: hash-bound hash-salt
Default comparators: make-default-comparator default-hash comparator-register-default!
Accessors and invokers: comparator-type-test-predicate comparator-equality-predicate comparator-ordering-predicate comparator-hash-function comparator-test-type comparator-check-type comparator-hash
Comparison predicates: =? <? >? <=? >=?
Syntax: comparator-if<=>
(comparator?
obj)
Returns #t
if obj is a comparator, and #f
otherwise.
(comparator-ordered?
comparator)
Returns #t
if comparator has a supplied ordering predicate, and #f
otherwise.
(comparator-hashable?
comparator)
Returns #t
if comparator has a supplied hash function, and #f
otherwise.
The following comparator constructors all supply appropriate type test predicates, equality predicates, ordering predicates, and hash functions based on the supplied arguments. They are allowed to cache their results: they need not return a newly allocated object, since comparators are pure and functional. In addition, the procedures in a comparator are likewise pure and functional.
(make-comparator
type-test equality ordering hash)
Returns a comparator which bundles the type-test, equality, ordering, and hash procedures provided. However, if ordering or hash is #f
, a procedure is provided that signals an error on application. The predicates comparator-ordered?
and/or comparator-hashable?
, respectively, will return #f
in these cases.
Here are calls on make-comparator
that will return useful comparators for standard Scheme types:
(make-comparator boolean? boolean=? (lambda (x y) (and (not x) y)) boolean-hash)
will return a comparator for booleans, expressing the ordering #f
< #t
and the standard hash function for booleans.
(make-comparator real? = < (lambda (x) (exact (abs x))))
will return a comparator expressing the natural ordering of real numbers and a plausible (but not optimal) hash function.
(make-comparator string? string=? string<? string-hash)
will return a comparator expressing the implementation's ordering of strings and the standard hash function.
(make-comparator string? string-ci=? string-ci<? string-ci-hash
will return a comparator expressing the implementation's case-insensitive ordering of strings and the standard case-insensitive hash function.
(make-pair-comparator
car-comparator cdr-comparator)
This procedure returns comparators whose functions behave as follows.
The type test returns #t
if its argument is a pair,
if the car satisfies the type test predicate of car-comparator, and the
cdr satisfies the type test predicate of cdr-comparator.
#t
if the cars are equal
according to car-comparator and the cdrs are equal according
to cdr-comparator, and #f
otherwise.The ordering function first compares the cars of its pairs using the equality predicate of car-comparator. If they are equal, then the ordering predicate of car-comparator is applied to the cars and its value is returned. Otherwise, the predicate compares the cdrs using the equality predicate of cdr-comparator. If they are not equal, then the ordering predicate of cdr-comparator is applied to the cdrs and its value is returned.
(make-list-comparator
element-comparator type-test empty? head tail)
This procedure returns comparators whose functions behave as follows:
The type test returns #t
if its argument satisfies
type-test and the elements satisfy the type test predicate
of element-comparator.
The total order defined by the equality and ordering functions is as follows (known as lexicographic order):
The hash function computes the hash values of the elements using the hash function of element-comparator and then hashes them together in an implementation-defined way.
(make-vector-comparator
element-comparator type-test length ref)
This procedure returns comparators whose functions behave as follows:
The type test returns #t
if its argument satisfies
type-test and the elements satisfy the type test predicate
of element-comparator.
The equality predicate returns #t
if both of the following
tests are satisfied in order: the lengths of the vectors are the
same in the sense of =, and the elements of the vectors
are the same in the sense of the equality predicate of
element-comparator.
The ordering predicate returns #t
if
the results of applying length to the first vector is less than
the result of applying length to the second vector.
If the lengths are equal, then the elements are examined pairwise using
the ordering predicate of element-comparator. If any pair of elements
returns #t
, then that is the result of the list comparator's
ordering predicate;
otherwise the result is #f
Here is an example, which returns a comparator for byte vectors:
(make-vector-comparator (make-comparator exact-integer? = < number-hash) bytevector? bytevector-length bytevector-u8-ref)
(make-eq-comparator)
(make-eqv-comparator)
(make-equal-comparator)
These procedures return comparators whose functions behave as follows:
The type test returns #t
in all cases.
eq?
, eqv?
,
and equal?
respectively.The ordering function is implementation-defined, except that it must conform to the rules for ordering functions. It may signal an error instead.
The hash function is default-hash
.
These comparators accept circular structure (in the case of
equal-comparator
, provided the implementation's equal?
predicate
does so) and NaNs.
These are hash functions for some standard Scheme types, suitable for
passing to make-comparator
. Users may write their own
hash functions with the same signature. However, if programmers wish their hash
functions to be backward compatible with the reference implementation
of SRFI 69,
they are advised to write their hash functions to accept a second argument
and ignore it.
(boolean-hash
obj)
(char-hash
obj)
(char-ci-hash
obj)
(string-hash
obj)
(string-ci-hash
obj)
(symbol-hash
obj)
(number-hash
obj)
These are suitable hash functions for the specified types. The hash functions char-ci-hash
and string-ci-hash
treat their argument case-insensitively.
Note that while symbol-hash
may return the hashed value of
applying symbol->string
and then string-hash
to the symbol, this is not a requirement.
The following macros allow the callers of hash functions to affect their behavior without interfering with the calling signature of a hash function, which accepts a single argument (the object to be hashed) and returns its hash value. They are provided as macros so that they may be implemented in different ways: as a global variable, a SRFI 39 or R7RS parameter, or an ordinary procedure, whatever is most efficient in a particular implementation.
(hash-bound)
[syntax]
Hash functions should be written so as to return a number between 0 and the largest reasonable number of elements (such as hash buckets) a data structure in the implementation might have. What that value is depends on the implementation. This value provides the current bound as a positive exact integer, typically for use by user-written hash functions. However, they are not required to bound their results in this way.
(hash-salt)
[syntax]
A salt is random data in the form of a non-negative exact integer used as an additional input to a hash function in order to defend against dictionary attacks, or (when used in hash tables) against denial-of-service attacks that overcrowd certain hash buckets, increasing the amortized O(1) lookup time to O(n). Salt can also be used to specify which of a family of hash functions should be used for purposes such as cuckoo hashing. This macro provides the current value of the salt, typically for use by user-written hash functions. However, they are not required to make use of the current salt.
The initial value is implementation-dependent, but must be less
than the value of (hash-bound)
, and should be distinct
for distinct runs of a program unless otherwise specified by the
implementation. Implementations may provide a means to specify the salt
value to be used by a particular invocation of a hash function.
(make-default-comparator)
Returns a comparator known as a default comparator that accepts Scheme values and orders them in some implementation-defined way, subject to the following conditions:
Given disjoint types a and b, one of three conditions must hold:
The empty list must be ordered before all pairs.
When comparing booleans, it must use the total order #f
< #t
.
When comparing characters, it must use char=?
and char<?
.
Note: In R5RS, this is an implementation-dependent order that is typically the same as Unicode codepoint order; in R6RS and R7RS, it is Unicode codepoint order.
When comparing pairs, it must behave the same as a comparator returned by make-pair-comparator
with default comparators as arguments.
When comparing symbols, it must use an implementation-dependent total order. One possibility is to use the order obtained by applying symbol->string
to the symbols and comparing them using the total order implied by string<?
.
When comparing bytevectors, it must behave the same as a comparator created by the expression (make-vector-comparator (make-comparator < = number-hash) bytevector? bytevector-length bytevector-u8-ref)
.
When comparing numbers where either number is complex, since non-real numbers cannot be compared with <
, the following least-surprising ordering is defined: If the real parts are < or >, so are the numbers; otherwise, the numbers are ordered by their imaginary parts. This can still produce somewhat surprising results if one real part is exact and the other is inexact.
When comparing real numbers, it must use =
and <
.
When comparing strings, it must use string=?
and string<?
.
Note: In R5RS, this is lexicographic order on the implementation-dependent order defined by char<?
; in R6RS it is lexicographic order on Unicode codepoint order; in R7RS it is an implementation-defined order.
When comparing vectors, it must behave the same as a comparator returned by (make-vector-comparator (make-default-comparator) vector? vector-length vector-ref)
.
When comparing members of types registered with comparator-register-default!
, it must behave in the same way as the comparator registered using that function.
Default comparators use default-hash
as their hash function.
(default-hash
obj)
This is the hash function used by default comparators, which accepts a Scheme value and hashes it in some implementation-defined way, subject to the following conditions:
default-hash
when applied to the car and the cdr.boolean-hash
,
char-hash
, string-hash
,
symbol-hash
, or number-hash
respectively.default-hash
when applied
to each of the elements.(comparator-register-default!
comparator)
Registers comparator for use by default comparators, such that if the objects being compared
both satisfy the type test predicate of comparator, it will be employed by default comparators
to compare them. Returns an unspecified value.
It is an error if any value satisfies both the type test predicate
of comparator and any of the following type test predicates:
boolean?
, char?
, null?
, pair?
,
symbol?
, bytevector?
, number?
,
string?
, vector?
, or
the type test predicate of a comparator that has already been registered.
This procedure is intended only to extend default comparators into
territory that would otherwise be undefined, not to override their
existing behavior.
In general, the ordering of calls to comparator-register-default!
should be irrelevant. However, implementations that support inheritance
of record types may wish to ensure that default comparators always check
subtypes before supertypes.
(comparator-type-test-predicate
comparator)
(comparator-equality-predicate
comparator)
(comparator-ordering-predicate
comparator)
(comparator-hash-function
comparator)
Return the four procedures of comparator.
(comparator-test-type
comparator obj)
Invokes the type test predicate of comparator on obj and returns what it returns.
More convenient than comparator-type-test-predicate
, but less efficient when the predicate is called repeatedly.
(comparator-check-type
comparator obj)
Invokes the type test predicate of comparator on obj and returns true if it returns true, but signals an error otherwise.
More convenient than comparator-type-test-predicate
, but less efficient when the predicate is called repeatedly.
(comparator-hash
comparator obj)
Invokes the hash function of comparator on obj and returns what it returns.
More convenient than comparator-hash-function
, but less efficient when the function is called repeatedly.
Note: No invokers are required for the equality and ordering predicates,
because =?
and <?
serve this function.
(=?
comparator object_{1} object_{2} object_{3} ...)
(<?
comparator object_{1} object_{2} object_{3} ...)
(>?
comparator object_{1} object_{2} object_{3} ...)
(<=?
comparator object_{1} object_{2} object_{3} ...)
(>=?
comparator object_{1} object_{2} object_{3} ...)
These procedures are analogous to the number, character, and string comparison predicates of Scheme. They allow the convenient use of comparators to handle variable data types.
These procedures apply the equality and ordering predicates of comparator to the objects as follows. If the specified relation returns #t
for all object_{i} and object_{j} where n is the number of objects and 1 <= i < j <= n, then the procedures return #t
, but otherwise #f
.
Because the relations are transitive, it suffices to compare each object with its successor.
The order in which the values are compared is unspecified.
(comparator-if<=>
[ <comparator> ] <object_{1}> <object_{2}> <less-than> <equal-to> <greater-than>)
It is an error unless <comparator> evaluates to a comparator and <object_{1}> and <object_{2}> evaluate to objects that the comparator can handle. If the ordering predicate returns true when applied to the values of <object_{1}> and <object_{2}> in that order, then <less-than> is evaluated and its value returned. If the equality predicate returns true when applied in the same way, then <equal-to> is evaluated and its value returned. If neither returns true, <greater-than> is evaluated and its value returned.
If <comparator> is omitted, a default comparator is used.
The sample implementation is found in the repository of this SRFI. It contains the following files.
comparators-impl.scm
- the record type definition and most of the proceduresdefault.scm
- a simple implementation of the default constructor, which should be improved by implementers to handle records and implementation-specific typesr7rs-shim.scm
- procedures for R7RS compatibility, including a trivial implementation of bytevectors on top of SRFI 4 u8vectorscomplex-shim.scm
- a trivial implementation of real-part
and imag-part
for Schemes that don't have complex numberscomparators.meta
- Chicken-specific metadatacomparators.setup
- Chicken-specific executable setupcomparators.sld
- an R7RS librarycomparators.scm
- a Chicken librarycomparators-test.scm
- a test file using the Chicken test
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