Re: Set Theory (Was: Do junctions support determining interesections of lists)
On Tue, Apr 04, 2006 at 11:02:55AM -0700, Jonathan Lang wrote: : Will perl6 Sets include set negation and/or a universal set? In : effect, an internal flag that says, this set contains every possible : element _except_ the ones listed? Arguably, that's what none() is. And all() is the only junction that is also a normal enumerated set. The any() junction is the set of sets that would intersect with the corresponding all() set without producing the null set. And the one() junction is the set of sets corresponding to the enumeration of all(). But this is all based on enumerated sets. Oddly missing are any Sets that are defined by rule. That would presumably take closures, though I suppose one can attempt to enumerate the closures that have to hold true and autothread through the calls to those closures... Can Russel be far behind? :-) Larry
Re: Set Theory (Was: Do junctions support determining interesections of lists)
Larry Wall wrote: On Tue, Apr 04, 2006 at 11:02:55AM -0700, Jonathan Lang wrote: : Will perl6 Sets include set negation and/or a universal set? In : effect, an internal flag that says, this set contains every possible : element _except_ the ones listed? Arguably, that's what none() is. ...except that none() is a Junction, not a Set. As such, the same arguments that say that any() and all() aren't suitable for use as Sets apply to none(), don't they? -- Jonathan Dataweaver Lang
Re: Set Theory (Was: Do junctions support determining interesections of lists)
On Tue, Apr 04, 2006 at 11:23:14AM -0700, Jonathan Lang wrote: : Larry Wall wrote: : On Tue, Apr 04, 2006 at 11:02:55AM -0700, Jonathan Lang wrote: : : Will perl6 Sets include set negation and/or a universal set? In : : effect, an internal flag that says, this set contains every possible : : element _except_ the ones listed? : : Arguably, that's what none() is. : : ...except that none() is a Junction, not a Set. As such, the same : arguments that say that any() and all() aren't suitable for use as : Sets apply to none(), don't they? You're confusing the map with the territory. We're trying to decide *how* Junctions are like Sets, not defining them into two different universes. I'm saying that all() is the Junction tha is most like a Set. A none() Junction can be viewed as the specification for an infinite set of sets that do not intersect with the corresponding all() junction. Infinite sets are a bit hard to compute with directly. A one() junction is the spec for a number of sets corresponding to the values of the corresponding all() junction, each of which contains only one element from that set. An any() Junction is all possible subsets not counting the null set. So junctional logic could be defined in terms of set theory, but not in a necessarily computable way. Larry
Re: Set Theory (Was: Do junctions support determining interesections of lists)
2006/4/4, Larry Wall [EMAIL PROTECTED]:
But this is all based on enumerated sets. Oddly missing are any
Sets that are defined by rule. That would presumably take closures,
though I suppose one can attempt to enumerate the closures that have
to hold true and autothread through the calls to those closures...
Can Russel be far behind? :-)
This is in ext/Recurrence:
use Recurrence;
# all integer numbers
$universe = Recurrence.new(
closure_next = sub { $_ + 1 },
closure_previous = sub { $_ - 1 },
:is_universe(1) );
# all even integers
$even_numbers = Recurrence.new(
closure_next = sub { 2 * int( $_ / 2 ) + 2 },
closure_previous = sub { 2 * int( ( $_ - 1 ) / 2 ) },
universe = $universe );
# all odd integers
$odd_numbers = $even_numbers.complement;
# all non-zero integers
$non_zero = Recurrence.new(
closure_next =sub ($x) { $x == -1 ?? 1 !! $x + 1 },
closure_previous =sub ($x) { $x == 1 ?? -1 !! $x - 1 },
complement_next = sub ($x) { $x 0 ?? 0 !!Inf },
complement_previous = sub ($x) { $x 0 ?? 0 !! -Inf },
universe = $universe );
- Flavio S. Glock
Re: Set Theory (Was: Do junctions support determining interesections of lists)
Larry Wall wrote: You're confusing the map with the territory. We're trying to decide *how* Junctions are like Sets, not defining them into two different universes. I'm saying that all() is the Junction tha is most like a Set. A none() Junction can be viewed as the specification for an infinite set of sets that do not intersect with the corresponding all() junction. Infinite sets are a bit hard to compute with directly. OK, then; what would be the specification for a _single_ set that contains everything that doesn't intersect with a corresponding all() Junction (the sort of thing that I'd use if I wanted to find the largest subset of A that doesn't intersect with B)? A one() junction is the spec for a number of sets corresponding to the values of the corresponding all() junction, each of which contains only one element from that set. An any() Junction is all possible subsets not counting the null set. Yeah; I got that. -- Jonathan Dataweaver Lang
Fwd: Set Theory (Was: Do junctions support determining interesections of lists)
On 4/4/06, Jonathan Lang [EMAIL PROTECTED] wrote: OK, then; what would be the specification for a _single_ set that contains everything that doesn't intersect with a corresponding all() Junction (the sort of thing that I'd use if I wanted to find the largest subset of A that doesn't intersect with B)? Can't do it. Here's a proof. Suppose you could find a _single_ set that contains everything that doesn't intersect some other set (the complement). Let S be the complement of the empty set. S is not an element of the empty set, so it must be a member of S, which is impossible. Finding the complement of a set assuming some other set is not hard though, simply by using the set difference operator. Luke
