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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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 Posted: Sun Nov 16, 2008 9:16 pm Post subject: |
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I apologize for my continual engagement in this discussion. I feel this wasn't the place for this and maybe I should have messaged Danny more privately about it.
Danny,
I feel my biggest blunder here was my attempt to engage in discussion in the first place. the subject matter here is something I am still learning. therefore, my continual attempts to respond are likely seen by you and others as premature and immature.
my attempts to seek advice have strenghtened this belief and for this I apologize. |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sun Nov 16, 2008 9:34 pm Post subject: |
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[Withdrawn: I was incorrect in how I presented continuous and discontinuous loops.]
Last edited by daj95376 on Mon Nov 17, 2008 5:19 am; edited 1 time in total |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Sun Nov 16, 2008 9:54 pm Post subject: |
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| daj95376 wrote: | | Where's Asellus when you need him? |
He's traveling, visiting Los Angeles and, so far at least, avoiding fires.
As far as I know, AICs and Nice Loops are the same thing and are fairly straightforward. To be useful, all AICs are loops. A loop is continuous if all of the links alternate strong and weak around the loop. A loop is discontinuous if there are two consecutive links of the same sort at one point in the loop. The discontinuity can either be two strong links or two weak links.
Most often AICs utilize weak link discontinuities to eliminate candidates. There are a couple of ways to look at it. One, as was being done above, is to assume that the item at the weak link discontinuity is true and propogate the implications around the loop to the contradictory result that the item must be false. The only way to avoid the contradiction is if the item at the discontinuity is false.
Another way to look at it that avoids thinking is terms of these assumptions is to realize that the alternating implications around the loop mean that the item at the discontinuity is weakly linked to itself. Two items that are weakly linked cannot both be true. Thus, an item weakly linked to itself cannot be true.
In the case of a strong link discontinuity, it is the opposite. A false assumption leads to the contradiction, so it must be true. Alternately, the item at the discontinuity is strongly linked to itself. Two strongly linked items cannot both be false, so an item strongly linked to itself must be true.
In a continuous loop, there are always an even number of "nodes" and the nodes alternate true and false. If a given node is assumed true, the loop propogates in one direction; if it is assumed false, the loop propogates in the other direction. But we don't need to make any assumption. Instead, we note that all of the links must be conjugate (i.e., every link must be both weak and strong and only conjugate links satisfy this requirement). That is how a continuous loop eliminates all the "extra" candidates at each link around the loop.
As far as notation and loops in the example above, there is a little confusion. Part of that confusion comes from the sometimes practice of writing loops in Eureka notation and leaving the discontinuous node off the ends of the chain/loop. One should always remember that the discontinuity is present even if not explicitly notated.
First Danny's loop with a strong link discontinuity:
NL: [r8c6]=7=[r7c4]=1=[r7c3]-1-[r6c3]-9-[r56c2]=9=[r8c2]=7=[r8c6] => [r8c6]=7
Eureka: (7)r8c6=(7-1)r7c4=(1)r7c3 - (9=1)r6c3 - (9)r56c2=(9-7)r8c2=(7)r8c6; r8c6=7
Then, Norm's loop with a weak link discontinuity:
NL: [r8c2]-7-[r8c6]=7=[r7c4]-7-[r7c3]-1-[r6c3]-9-[r56c2]=9=[r8c2] => r8c2<>7
Eureka: (7)r8c2 - (7)r8c6 = (7)r7c4 - (7=1)r7c3 - (1=9)r6c3 - (9)r56c2=(9-7)r8c2; r8c2<>7
These two loops take slightly different paths but are otherwise the same. It is the conjugate link in the r8 <7>s that makes this so. For example, Norm's weak link discontinuity loop could be written as a strong link discontinuity loop as follows:
NL: [r8c6]=7=[r7c4]-7-[r7c3]-1-[r6c3]-9-[r56c2]=9=[r8c2]=7=[r8c6] => r8c6=7
Eureka: (7)r8c6 = (7)r7c4 - (7=1)r7c3 - (1=9)r6c3 - (9)r56c2=(9-7)r8c2=(7)r8c6; r8c6=7
In this case, the conjugate r8 link is considered as a strong link; previously, it was considered as a weak link. It works either way.
I hope this clears up whatever confusion was muddying the waters. |
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daj95376
Joined: 23 Aug 2008
Posts: 3854
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| Posted: Sun Nov 16, 2008 11:55 pm Post subject: |
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| Asellus wrote: | He's traveling, visiting Los Angeles and, so far at least, avoiding fires.
As far as I know, AICs and Nice Loops are the same thing and are fairly straightforward. To be useful, all AICs are loops. A loop is continuous if all of the links alternate strong and weak around the loop. A loop is discontinuous if there are two consecutive links of the same sort at one point in the loop. The discontinuity can either be two strong links or two weak links. |
Yes, stay away from the fires!!! They have them too often!
A great presentation. However, every XY-Chain is an AIC, but I don't believe that any XY-Chain is a Nice Loop. |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Mon Nov 17, 2008 9:31 am Post subject: |
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| daj95376 wrote: | | I don't believe that any XY-Chain is a Nice Loop. |
Well, I admit that I am no expert on Nice Loops. So, I just took a quick look at the Sudopedia entry. Any XY Chain with pincer ends that eliminate one or more victim candidates is a discontinous loop ... with the victim(s) at the weak link discontinuity ... and perfectly satisfies the Sudopedia definition of a Nice Loop as far as I can see.
From the Sudopedia entry, I would say that every Nice Loop is an AIC, but not every AIC satisfies the more restrictive Nice Loop requirements. In my opinion this is only the case because the Nice Loop definitions are unnecessarily restrictive. I don't see any reason that these two things can't be seen as alternate views of the same thing in all cases, provided one is willing to lose those unnecessary restrictions. |
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Steve R
Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England
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| Posted: Mon Nov 17, 2008 2:55 pm Post subject: |
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Aren’t we in danger of making a distinction without a difference here?
All chains are built from elementary blocks along the lines of “if one cell contains x then another does not” or “if one cell does not contains x then another does.” These blocks are then strung together until a contradiction appears. So you might write:
(1) r1c2 = 7 => r1c8 ≠ 7 => r5c8 = 7 => .. (traditional notation)
(2) r1c2 -7- r1c8 =7= r5c8 … (nice chain notation) or
(3) (7)r1c2 –(7)r1c8 = (7)r5c8 … (Eureka notation).
These are merely different ways of setting down the same thing. None adds anything to the logic so none introduces an additional elimination. It is, or should be, a matter of making the logic as readily understood by the reader as is possible.
I confess to a soft spot for nice loops because they make a pretty mathematical theory. This doesn’t make them easier to read or of greater use. The underlying puzzle is why people have abandoned the traditional style in favour of one of the others. It has the unfortunate consequence that newcomers are obliged to learn two new languages to follow what the experts say. What’s more, neither language is second nature to all the aficionados, as witness the discussion above.
Steve |
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