The Rules
Consider two ALS, A and B.
If A and B have only one common candidate then no reduction is possible.
If A and B have two or more common candidates, a candidate x is a weak link (aka restricted common candidate) if all instances of x in ALS A can see all the instances of x in ALS B. If A and B do not have at least one weak link then no reduction is possible.
If A and B have only one weak link, then the other common candidate(s) can be eliminated from all cells (that are not cells of A or B) that can see all instances of the common candidate in both A and B.
If A and B have two weak links x and y, then x can be eliminated from any cells (that are not cells of A or B) that can see all the x candidates in both A and B. y can be eliminated from any cells (that are not cells of A or B) that can see all the y candidates in both A and B. A non-common candidate z contained in A can be eliminated from any cells (that are not cells of A or B) that contain an x and can see all the z candidates in A. A non-common candidate z contained in A can be eliminated from any cells (that are not cells of A or B) that contain a y and can see all the z candidates in A. This same rule applies to ALS B.
The Examples
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Example of two ALS with a single weak link
|---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 | A 238 | B 689 | 269 || A 368 | 4 | 5 || 7 | A 23 | 1
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 | 357 | B 67 | 567 || 1 | 36 | 2 || 9 | 8 | 4
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 | 238 | 1 | 4 || 9 | 7 | 38 || 5 | 23 | 6
===========================||=======================||=======================
r4 | 1 | 3 | 579 || 567 | 69 | 67 || 2 | 4 | 8
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 | 25 | 4 | 8 || 25 | 13 | 13 || 6 | 9 | 7
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 | 6 | B 79 | 29 || 278 | 89 | 4 || 1 | 5 | 3
===========================||=======================||=======================
r7 | 78 | 678 | 67 || 4 | 2 | 9 || 3 | 1 | 5
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 | 9 | 5 | 3 || 78 | 18 | 178 || 4 | 6 | 2
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 | 4 | 2 | 1 || 36 | 5 | 36 || 8 | 7 | 9
.............................................................................
A={2368} at r1c1, r1c4, r1c8
B={6789} at r1c2, r2c2, r6c2
1. The common candidates of A and B are 6 and 8.
2. A and B are weakly linked by 8 because all the 8 in A can see all the 8 in B. They can not be weakly linked by 6 because the 6 of B in r2c2 can not see the 6 of A in r1c4. This also explains why r6c2 had to be used in B instead of r7c2. If r7c2 was used, the 6 and 8 in r7c2 would not be able to see all (or any in this case) of the 6 and 8 in A.
3. The other common candidate of A and B is 6. The list of cells (that are not in A or B) that can see the 6 in all cells of both A and B is r1c3. The 6 can be removed from r1c3.
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Example of two ALS with two weak links. This example is from Bob Hanson's web page (http://www.stolaf.edu/people/hansonr/sudoku/explain.htm).
|---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 | 37 | 9 | 8 || 6 | 1 | 2 || A 37 | 4 | 5
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 | 5 | 6 | 2 || 3 | 48 | 478 || A 19 | A 178 | A 789
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 | 347 | 1 | 34 || 78 | 5 | 9 || B 237 | 3678 | 2678
===========================||=======================||=======================
r4 | 489 | 258 | 459 || 478 | 2348 | 34578 || 6 | 378 | 1
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 | 48 | 3 | 6 || 1478 | 248 | 1478 || 5 | 9 | 2478
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 | 1 | 258 | 7 || 9 | 6 | 3458 || B 23 | 38 | 248
===========================||=======================||=======================
r7 | 3689 | 58 | 359 || 2 | 349 | 34 || 19 | 167 | 679
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 | 2 | 7 | 19 || 18 | 89 | 6 || 4 | 5 | 3
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 | 369 | 4 | 139 || 5 | 7 | 13 || 8 | 2 | 69
.............................................................................
A = {13789} at r1c7, r2c7, r2c8, r2c9
B = {237} at r3c7, r6c7
1. The common candidates of A and B are 3 and 7.
2. A and B are weakly linked by 3 because all the cells of A and B that contain a 3 can see each other (col 7). A and B are weakly linked by 7 because all the cells of A and B than contain a 7 can see each other (block 3).
3. The list of all cells (not in A or B) that can see all the 3 in A and B is r45789c7. There is not a 3 in any of those cells so no reduction is possible.
4. The list of all cells (not in A or B) that can see all the 7 in A and B is r1c89, r3c89. The 7 can be removed from r3c8 and r3c9.
5. The non-common candidates of A are {189}. The list of cells (not in A or B) that can see all instances of 1 in A are r2c123456, r1c89, r3c89. Of those cells, the ones that contain a 3 or 7 are r2c6, r3c8, r3c9. None of these cells contain a 1 so no reduction is possible.
6. The list of cells (not in A or B) that can see all instances of 8 in A are r2c123456, r1c89, r3c89. Of those cells, the ones that contain a 3 or 7 are r2c6, r3c8, r3c9. 8 can be removed from r2c6, r3c8, r3c9.
7. The list of cells (not in A or B) that can see all instances of 9 in A are r2c123456, r1c89, r3c89. Of those cells, the ones that contain a 3 or 7 are r2c6, r3c8, r3c9. None of these cells contain a 9 so no reduction is possible.
8. The non-common candidate of B is 2. The list of cells (not in A or B) that can see all instances of 2 in B is r45789c2. Of those cells, none contain a 3 or 7 so no reduction is possible.
But I believe "valid" examples exist.
