Most recent developments in the logic of Sudoku have been related to AIC, or Alternating Inference Chains. They reduce the basic logic of the grid to simple relations between candidates or groups of candidates, which can be chained together to prove more complicated relationships and eliminate other candidates. Most "Forcing Chains" and guess-and-check logic can instead be expressed as a neat AIC. AIC are equally as valid and powerful, perhaps even more powerful, in Calcudoku puzzles. I'll give a basic overview of the logic as well as a few examples.
When the puzzle is in an unsolved state every cell has a list of possible digits it can be, known as candidates. You solve the puzzle by proving candidates are FALSE until only one remains, which must be TRUE and therefore must be the solution to that cell. AIC use two types of basic inferences, or "links", between any two candidates, or sets of candidates.
"Strong inference": At least one must be true. Can also be understood as "if A is false, B is true", and vice versa.
A cell with two candidates (bivalue) can be said to be strongly linked. If one of the candidates is false, there's only one remaining, so of course the remainder must be true. Similarly if a digit candidate only appears within a row/column twice (bilocal), then these candidates are strongly linked. In diagrams these are typically drawn in red.
"Weak inference": At most one can be true. Can also be understood as "if A is true, B is false", and vice versa.
This is the most common inference; every candidate is weakly linked to the other candidates in its cell, as well as all matching candidates in its row and column. In diagrams these are typically drawn in blue.
If you build a chain that alternates between strong and weak inferences, starting & ending with a strong link, you effectively prove that each end of the chain is strongly linked: if one end is false, the other end must be true and vice versa. This means you can eliminate any candidate that sees both ends of the chain.
An example:
The 1 in c1 (referred to as 1c1 from here) is either TRUE or FALSE. If it's true, the 1 in c3 will be eliminated.
If it's false, 4c1 is true, 4a1 is false, 3a1 is true, 3a3 is false, 1a3 is true. In this case, 1 in c3 will be eliminated.
So, no matter if 1c1 is true or not, the 1 in c3 will be eliminated, therefore we can eliminate it.
It is important to note that this if-then logic does not need to be followed in order to prove an AIC - that's the beauty of it. It is sufficient to prove that each strong/weak inference is valid. As long as all the candidate relationships are correct, and the chain is unbroken and alternates between strong & weak with strong inferences at either end, then you know that at least one end of the chain must be true.
This specific example only uses strong links in bivalue cells, and is commonly known as an XY-Wing in Sudoku.
Another example, this time with a bilocal strong link:
This time without exhaustively following the logic, we can see that this is a valid AIC composed of alternating strong/weak links. 3c3 and 3b4 are connected by the AIC so we know at least one must be true. 3b3 sees both ends of the chain, so it can be eliminated. This is better known as a W-Wing.
The real solving power comes when you consider links within cages too. A 5+ cage in a 4x4 puzzle has two possible solutions: 1+4 and 2+3. We can say that these combinations are strongly linked: if one is false the other is true and vice versa. Of course, I have another example:
I've coloured the 5+ combinations in green and yellow. This AIC eliminates the 4 in b2.
It is possible to define more types of strong links. In Sudoku we use many different constructs as nodes, such as ALS, AHS, Fish, UR Guardians, Impossible Pattern Guardians, MSLS, ALC, and more. These go outside of the scope of this post so you can research them at your own leisure.
You can read more about AIC from these links:
Original AIC Thread for Sudoku
Eureka Notation (used in Sudoku, will be used in my solver's output when I'm finished implementing AIC).
As an aside I've released a stable version of my solver here: Direct Link, ReadMe. AIC is not yet implemented in this version, it should be done in a few days/weeks though.