Re: Empty interval representations & Motion 13...
Dan & P1788
I have gone through Dan's work on comparisons with the empty set (below), checking against the set theory definitions given in my mail earlier today.
I think they all agree, except the two I have marked (*). Namely I think
empty interior empty
empty strictLess empty
should both give True.
Pretty good overall.
John
On 22 Apr 2010, at 03:46, Dan Zuras Intervals wrote:
> But if we use empty = [+oo,-oo], how does that come out
> for Ulrich's comparisons?
>
> a equals b <==> a1 = b1 && a2 = b2
> a subset b <==> b1 <= a1 && a2 <= b2
> a lessEqual b <==> a1 <= b1 && a2 <= b2
> a precedeTouch b <==> a2 <= b1
> a interior b <==> b1 < a1 && a2 < b2
> a strictLess b <==> a1 < b1 && a2 < b2
> a preceed b <==> a2 < b1
>
> Let's let any = [b1,b2] be some otherwise non-empty
> interval. I believe we have that
>
> empty equals any = False
> empty subset any = True
> empty lessEqual any = False (1)
> empty precedeTouch any = True (2)
> empty interior any = True
> empty strictLess any = False
> empty preceed any = True (3)
>
> as well as
>
> any equals empty = False
> any subset empty = False
> any lessEqual empty = False (1)
> any precedeTouch empty = True (2)
> any interior empty = False
> any strictLess empty = False
> any preceed empty = True (3)
>
> and, for completeness sake
>
> empty equals empty = True
> empty subset empty = True
> empty lessEqual empty = True (4)
> empty precedeTouch empty = True (2)
> empty interior empty = False (*)
> empty strictLess empty = False (*)
> empty preceed empty = True (4)
>
>
> Which are all correct with the following qualifications
>
> (1) (a lessEqual b) tests for mere overlap in one direction
> or the other. Therefore, it is acceptable to the programmer
> that this be true if SOME a is less than or equal to SOME b.
> This has the flavor of an existential quantification & if
> this test is written in this way, empty tests correct.
>
> (2) Similarly, (a precedeTouch b) test to make sure that NO
> element of a exceeds ANY element of b. If it is written
> with universal quantification (essentially not (any a >
> any b)) then empty tests correct.
>
> (3) This one is True except for (empty preceed [-oo,x]) or
> ([x,+oo] preceed empty) in which case it tests False. This
> may have to be special cased.
>
> (4) My interpretation is that these two are False with the
> quantification I suggest. So they may have to be special
> cased as well.