a reader of PBCS wrote: In a single sentence, of all the ideas introduced in your books and implemented in CSP-Rules/SudoRules, which would you name as the most innovative
wow, wow: a single sentence? That's much too easy. Indeed, I can answer with a single character. But I'll have to first list some contenders.
Contender #1 : the whole framework of epistemic logic, reducible to intuitionnistic logic, with a formal definition of candidates, resolution rules (with precise conditions sufficient to prove their conclusion generically, once and for all, without any additional ad hoc reasoning in each instantiation), resolution theories, resolution paths within these theories, all this allowing to prove precise results. Not sure it's a main contender, as all the rest can be understood informally without it, but it remains the fundamental theoretical background for all the rest.
Contender #2 : the view of a pattern, especially a chain pattern, as a "static" pattern visible on the current resolution state (i.e. PM); this view was since the beginning and still is in a radical opposition to the classical view of a chain as a "chain of inferences". Of course, a "static" chain pattern can be the support of inferences, but it is not, in and of itself, a chain of inferences. See #3 to #5 for very concrete consequences of this.
Contender #3 : the idea of the z- and t-candidates in chains. However, in isolation, this is not enough to make it the main innovative idea: in a sense, T&E implicitly uses z- and t- candidates; similarly, Sudoku Explainer implicitly uses z- and t- candidates. The innovative idea is that such candidates are not part of my chains. This is justified because z- and t- candidates have no impact on the possibilities of extending a partial-chain. A practical consequence is, any whip/braid... can be drawn exactly as a bivalue-chain with no added, useless complications.
Contender #4 : some specific types of chains: whips, g-whips, braids, g-braids; the precise definition of T&E; the clear relationship between T&E, braids, whips or between gT&E, g-braids, g-whips. Such chains have very specific properties wrt to the vague "contradiction chains [of inferences]" and saying that they can be reduced to such chains is utter nonsense.
Contender #5 : the definition of the complexity of a pattern as the number of CSP-Variables its definition involves. This applies consistently to Subsets, any types of chains, any exotic patterns...
This definition relies on the previous 4 points, it can be formulated in pure logic terms, which guarantees the essential property that the associated ratings are invariant under isomorphism.
Notice that this point is what distinguishes most radically the various ratings defined in [PBCS] from the "number of nodes" used in Sudoku Explainer and other systems based on "(forcing) contradiction chains [of inferences]". This is also what makes any claim that whips or braids can be reduced to such chains or forcing nets utter nonsense.
Contender #6 : the various theorems about several resolution theories having the confluence property. These theorems allow to find the simplest solution after following a single resolution path. This is the most important result in practice and they justify the simplest-first strategy.`
Contender #7 : the introduction of additional CSP-Variables. In Sudoku, this appears as the super-symmetric view, in which all the types of CSP-Variables (rc-, rn-, cn, and bn-) are considered on the same footing.
This is just a quick list of contenders, without thinking too much about it. With more time, I should be able to find more. But let's do with this.
Notice that all these ideas were already in [HLS, 2007].
Now my answer to the question in a single character, as promised : 5.