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Submitted by : (unknown) at: 2007-11-17T22:30:51-08:00 (9 years ago)
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Axiom Version :
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map confuses sets.

axiom
A:Set Integer:=set [-2,-1,0]

\label{eq1}\left\{ - 2, \: - 1, \: 0 \right\}(1)
Type: Set(Integer)
axiom
B:Set Integer:=set [0,1,4]

\label{eq2}\left\{ 0, \: 1, \: 4 \right\}(2)
Type: Set(Integer)
axiom
C:=map(x +-> x^2,A)

\label{eq3}\left\{ 0, \: 1, \: 4 \right\}(3)
Type: Set(Integer)
axiom
test(C=B)

\label{eq4} \mbox{\rm true} (4)
Type: Boolean

somehow a sort is missing after applying map.

Proposed Fix

If S has OrderedSet then map_! should include 'sort':

   map_!(f,s) ==
     map_!(f,s)$Rep
     sort_!(s)$Rep
     removeDuplicates_!

See diff

See also: SandBoxSetAny for a more ambitious fix that defines an ordering for all Sets.

spad
)abbrev domain SET Set
++ Author: Michael Monagan; revised by Richard Jenks
++ Date Created: August 87 through August 88
++ Date Last Updated: May 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ A set over a domain D models the usual mathematical notion of a finite set
++ of elements from D.
++ Sets are unordered collections of distinct elements
++ (that is, order and duplication does not matter).
++ The notation \spad{set [a,b,c]} can be used to create
++ a set and the usual operations such as union and intersection are available
++ to form new sets.
++ In our implementation, \Language{} maintains the entries in
++ sorted order.  Specifically, the parts function returns the entries
++ as a list in ascending order and
++ the extract operation returns the maximum entry.
++ Given two sets s and t where \spad{#s = m} and \spad{#t = n},
++ the complexity of
++   \spad{s = t} is \spad{O(min(n,m))}
++   \spad{s < t} is \spad{O(max(n,m))}
++   \spad{union(s,t)}, \spad{intersect(s,t)}, \spad{minus(s,t)}, \spad{symmetricDifference(s,t)} is \spad{O(max(n,m))}
++   \spad{member(x,t)} is \spad{O(n log n)}
++   \spad{insert(x,t)} and \spad{remove(x,t)} is \spad{O(n)}
Set(S:SetCategory): FiniteSetAggregate S == add
   Rep := FlexibleArray(S)
   # s       == _#$Rep s
   brace()   == empty()
   set()     == empty()
   empty()   == empty()$Rep
   copy s    == copy(s)$Rep
   parts s   == parts(s)$Rep
   inspect s == (empty? s => error "Empty set"; s(maxIndex s))
extract_! s == x := inspect s delete_!(s, maxIndex s) x
find(f, s) == find(f, s)$Rep
map(f, s) == map_!(f,copy s)
reduce(f, s) == reduce(f, s)$Rep
reduce(f, s, x) == reduce(f, s, x)$Rep
reduce(f, s, x, y) == reduce(f, s, x, y)$Rep
if S has ConvertibleTo InputForm then convert(x:%):InputForm == convert [convert("set"::Symbol)@InputForm, convert(parts x)@InputForm]
if S has OrderedSet then s = t == s =$Rep t max s == inspect s min s == (empty? s => error "Empty set"; s(minIndex s))
map_!(f,s) == map_!(f,s)$Rep sort_!(s)$Rep removeDuplicates_! s
construct l == zero?(n := #l) => empty() a := new(n, first l) for i in minIndex(a).. for x in l repeat a.i := x removeDuplicates_! sort_! a
insert_!(x, s) == n := inc maxIndex s k := minIndex s while k < n and x > s.k repeat k := inc k k < n and s.k = x => s insert_!(x, s, k)
member?(x, s) == -- binary search empty? s => false t := maxIndex s b := minIndex s while b < t repeat m := (b+t) quo 2 if x > s.m then b := m+1 else t := m x = s.t
remove_!(x:S, s:%) == n := inc maxIndex s k := minIndex s while k < n and x > s.k repeat k := inc k k < n and x = s.k => delete_!(s, k) s
-- the set operations are implemented as variations of merging intersect(s, t) == m := maxIndex s n := maxIndex t i := minIndex s j := minIndex t r := empty() while i <= m and j <= n repeat s.i = t.j => (concat_!(r, s.i); i := i+1; j := j+1) if s.i < t.j then i := i+1 else j := j+1 r
difference(s:%, t:%) == m := maxIndex s n := maxIndex t i := minIndex s j := minIndex t r := empty() while i <= m and j <= n repeat s.i = t.j => (i := i+1; j := j+1) s.i < t.j => (concat_!(r, s.i); i := i+1) j := j+1 while i <= m repeat (concat_!(r, s.i); i := i+1) r
symmetricDifference(s, t) == m := maxIndex s n := maxIndex t i := minIndex s j := minIndex t r := empty() while i <= m and j <= n repeat s.i < t.j => (concat_!(r, s.i); i := i+1) s.i > t.j => (concat_!(r, t.j); j := j+1) i := i+1; j := j+1 while i <= m repeat (concat_!(r, s.i); i := i+1) while j <= n repeat (concat_!(r, t.j); j := j+1) r
subset?(s, t) == m := maxIndex s n := maxIndex t m > n => false i := minIndex s j := minIndex t while i <= m and j <= n repeat s.i = t.j => (i := i+1; j := j+1) s.i > t.j => j := j+1 return false i > m
union(s:%, t:%) == m := maxIndex s n := maxIndex t i := minIndex s j := minIndex t r := empty() while i <= m and j <= n repeat s.i = t.j => (concat_!(r, s.i); i := i+1; j := j+1) s.i < t.j => (concat_!(r, s.i); i := i+1) (concat_!(r, t.j); j := j+1) while i <= m repeat (concat_!(r, s.i); i := i+1) while j <= n repeat (concat_!(r, t.j); j := j+1) r
else map_!(f,s) == map_!(f,s)$Rep removeDuplicates_! s
insert_!(x, s) == for k in minIndex s .. maxIndex s repeat s.k = x => return s insert_!(x, s, inc maxIndex s)
remove_!(x:S, s:%) == n := inc maxIndex s k := minIndex s while k < n repeat x = s.k => return delete_!(s, k) k := inc k s
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/7072998645375249993-25px002.spad using 
      old system compiler.
   SET abbreviates domain Set 
------------------------------------------------------------------------
   initializing NRLIB SET for Set 
   compiling into NRLIB SET 
   compiling exported # : $ -> NonNegativeInteger
;;; *** |SET;#;$Nni;1| REDEFINED Time: 0.28 SEC.
compiling exported brace : () -> $
;;; *** |SET;brace;$;2| REDEFINED Time: 0 SEC.
compiling exported set : () -> $
;;; *** |SET;set;$;3| REDEFINED Time: 0 SEC.
compiling exported empty : () -> $
;;; *** |SET;empty;$;4| REDEFINED Time: 0 SEC.
compiling exported copy : $ -> $
;;; *** |SET;copy;2$;5| REDEFINED Time: 0 SEC.
compiling exported parts : $ -> List S
;;; *** |SET;parts;$L;6| REDEFINED Time: 0 SEC.
compiling exported inspect : $ -> S
;;; *** |SET;inspect;$S;7| REDEFINED Time: 0.06 SEC.
compiling exported extract! : $ -> S
;;; *** |SET;extract!;$S;8| REDEFINED Time: 0 SEC.
compiling exported find : (S -> Boolean,$) -> Union(S,failed)
;;; *** |SET;find;M$U;9| REDEFINED Time: 0 SEC.
compiling exported map : (S -> S,$) -> $
;;; *** |SET;map;M2$;10| REDEFINED Time: 0 SEC.
compiling exported reduce : ((S,S) -> S,$) -> S
;;; *** |SET;reduce;M$S;11| REDEFINED Time: 0.01 SEC.
compiling exported reduce : ((S,S) -> S,$,S) -> S
;;; *** |SET;reduce;M$2S;12| REDEFINED Time: 0 SEC.
compiling exported reduce : ((S,S) -> S,$,S,S) -> S
;;; *** |SET;reduce;M$3S;13| REDEFINED Time: 0 SEC.
****** Domain: S already in scope augmenting S: (ConvertibleTo (InputForm)) compiling exported convert : $ -> InputForm
;;; *** |SET;convert;$If;14| REDEFINED Time: 0.28 SEC.
****** Domain: S already in scope augmenting S: (OrderedSet) compiling exported = : ($,$) -> Boolean
;;; *** |SET;=;2$B;15| REDEFINED Time: 0.01 SEC.
compiling exported max : $ -> S
;;; *** |SET;max;$S;16| REDEFINED Time: 0 SEC.
compiling exported min : $ -> S
;;; *** |SET;min;$S;17| REDEFINED Time: 0 SEC.
compiling exported map! : (S -> S,$) -> $
;;; *** |SET;map!;M2$;18| REDEFINED Time: 0.01 SEC.
compiling exported construct : List S -> $
;;; *** |SET;construct;L$;19| REDEFINED Time: 0.01 SEC.
compiling exported insert! : (S,$) -> $
;;; *** |SET;insert!;S2$;20| REDEFINED Time: 0.01 SEC.
compiling exported member? : (S,$) -> Boolean
;;; *** |SET;member?;S$B;21| REDEFINED Time: 0.01 SEC.
compiling exported remove! : (S,$) -> $
;;; *** |SET;remove!;S2$;22| REDEFINED Time: 0.01 SEC.
compiling exported intersect : ($,$) -> $
;;; *** |SET;intersect;3$;23| REDEFINED Time: 0 SEC.
compiling exported difference : ($,$) -> $
;;; *** |SET;difference;3$;24| REDEFINED Time: 0 SEC.
compiling exported symmetricDifference : ($,$) -> $
;;; *** |SET;symmetricDifference;3$;25| REDEFINED Time: 0 SEC.
compiling exported subset? : ($,$) -> Boolean
;;; *** |SET;subset?;2$B;26| REDEFINED Time: 0.15 SEC.
compiling exported union : ($,$) -> $
;;; *** |SET;union;3$;27| REDEFINED Time: 0.02 SEC.
compiling exported map! : (S -> S,$) -> $ Time: 0.01 SEC.
compiling exported insert! : (S,$) -> $ Time: 0.01 SEC.
compiling exported remove! : (S,$) -> $ Time: 0.02 SEC.
****** Domain: S already in scope augmenting S: (Evalable S) ****** Domain: S already in scope augmenting S: (ConvertibleTo (InputForm)) ****** Domain: S already in scope augmenting S: (Finite) ****** Domain: S already in scope augmenting S: (OrderedSet) (time taken in buildFunctor: 10)
;;; *** |Set| REDEFINED
;;; *** |Set| REDEFINED Time: 0.01 SEC.
Cumulative Statistics for Constructor Set Time: 0.91 seconds
finalizing NRLIB SET Processing Set for Browser database: --------constructor--------- ; compiling file "/var/zope2/var/LatexWiki/SET.NRLIB/SET.lsp" (written 01 AUG 2011 08:06:27 AM):
; /var/zope2/var/LatexWiki/SET.NRLIB/SET.fasl written ; compilation finished in 0:00:00.532 ------------------------------------------------------------------------ Set is now explicitly exposed in frame initial Set will be automatically loaded when needed from /var/zope2/var/LatexWiki/SET.NRLIB/SET
>> System error: The bounding indices 163 and 162 are bad for a sequence of length 162. See also: The ANSI Standard, Glossary entry for "bounding index designator" The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

Retest

axiom
A2:Set Integer:=set [-2,-1,0]

\label{eq5}\left\{ - 2, \: - 1, \: 0 \right\}(5)
Type: Set(Integer)
axiom
B2:Set Integer:=set [0,1,4]

\label{eq6}\left\{ 0, \: 1, \: 4 \right\}(6)
Type: Set(Integer)
axiom
C2:=map(x +-> x^2,A)

\label{eq7}\left\{ 0, \: 1, \: 4 \right\}(7)
Type: Set(Integer)
axiom
test(B2=C2)

\label{eq8} \mbox{\rm true} (8)
Type: Boolean

But unfortunately the documentation lies:

  ++ In our implementation, \Language{} maintains the entries in
  ++ sorted order.  Specifically, the parts function returns the
  ++ entries as a list in ascending order and the extract operation
  ++ returns the maximum entry.

This example shows that Set is not maintained in sorterd order. So the code for Set still appears to be broken if the Set is constructed over a domain that is not an OrderedSet?.

axiom
)set message any off
showTypeInOutput true;
Type: String

axiom
Set Any has OrderedSet

\label{eq9} \mbox{\rm false} (9)
Type: Boolean
axiom
B3:Set Any:=B;B3

\label{eq10}\left\{{0 : \hbox{\axiomType{Integer}\ }}, \:{1 : \hbox{\axiomType{Integer}\ }}, \:{4 : \hbox{\axiomType{Integer}\ }}\right\}(10)
Type: Set(Any)
axiom
C3:Set Any:=C;C3

\label{eq11}\left\{{0 : \hbox{\axiomType{Integer}\ }}, \:{1 : \hbox{\axiomType{Integer}\ }}, \:{4 : \hbox{\axiomType{Integer}\ }}\right\}(11)
Type: Set(Any)
axiom
test(B3=C3)

\label{eq12} \mbox{\rm true} (12)
Type: Boolean

But why does this example work? Is set equality implemented as an order n^2 comparison if the domain is not an OrderedSet??

Re: order n^2 comparison, answer: yes --Bill Page, Thu, 05 Apr 2007 15:26:15 -0500 reply
In FiniteSetAggregate? we see:
   s = t           == #s = #t and empty? difference(s,t)

   ...

   difference(s:%, t:%) ==
     m := copy s
     for x in parts t repeat remove_!(x, m)
     m

fixed in wh-sandbox revision 489 --Bill Page, Mon, 09 Apr 2007 00:49:11 -0500 reply
Status: open => fix proposed




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