Weak and Strong Links
This article continues from Introducing Chains and Links. Recommended reading if you have not done so first.
Here I clarify certain terms scattered about the strategies and put them in one place for easy reference and well as grow the theory on how chains are formed. These terms include Weak and Strong Links, or more accurately, links with weak or strong "inference". They were introduced in the X Cycles strategy but have wide application.
!A => B (if not A, then B) OFF implies ON
Weak links are the opposite:
A => !B (if A, then not B) ON implies OFF
Back to the diagram. Consider the columns 3 and 8 which are part of the X-Wing and are marked with thin double lines. Because there are more than two 9s in the columns we can't draw a strong inference. We can only draw a weak inference, that is, if one of those 9s is the solution all the other 9s are eliminated.
Strong can be Weak
So far the rough and ready distinction between Strong and Weak links is to do with how many candidates are in a unit – namely, Strong links are formed when only two are present, while three or more imply a Weak link. However, this is not the case.
From a strong link we can infer that
if not A, then B
From a weak link, we can infer only that
if A then not B, C, D according to how many candidates there are in a unit
However, the following is also true that for a strong link:
if A, then not B
So, some Strong links can be reversed to give us a "link with weak inference" - if the occasion calls for it. It is perfectly logical to assert on a unit with two candidates of X both:
In Figure 4 we have an array of 6 candidates on a board. A number of strategies can show that the 6 on H9 can be eliminated. I have coloured some cells using Singles Chains Rule 5 which link up some pairs on the board - either all of the yellow cells will be 6 or all of the cyan cells will be 6. Since H9 can see C9 (yellow) and H5 (cyan) it cannot be a 6 since it can see cells with both colours.
This, amazingly, is not the end of links in chains. Apart from bi-value and bi-location links there are other more exotic ways to form a link in a chain.
You should read the article on Grouped X-Cycles to see how a group of candidates in several cells can be used to form a link.
Also, Almost Locked Sets can be made into links. This method needs to be documented but it is present in the solver.