J F Crook's solution
It has taken me many attempts and a long time to understand J F Crook's paper and your responses, (Mainly on the former as his concepts were difficult to grasp).
In your response you are dismissive of the Trial and Error approach. But looking at the accepted strategies several of these effectively do this. For example: Simple Colouring is where you select a number and cell (relatively) at random and follow through the links. If a cell can see other cells with the two colours the selcted number is eliminated from that cell. Surely this is effectively trial and error, except that there is some logic behind the elimination?
This is a very interesting area and its quite philosophical. I would distinguish between a) the 'pattern' that is found that a strategy uses to eliminate candidates (logically) and b) finding that pattern in the first place. All the documented strategies assume you can find the opportunity and they are all logical apart from the last two (which I place under the heading 'trial and error'). However, there is little help for the solver in how to find them apart from simple rules like - look for bi value (2 candidates in cell) and bi-location (two candidates in a unit) situations (since these lead to chains which are the building blocks of many strategies).
When searching for opportunities to apply strategies your path is determined by all the dead ends and in a sense these are the 'errors'. A computer has to look at every single dead end because it doesn't have a mind. But in order for a human or a computer not to take years solving a puzzle, we use logical optimizations. Even your 'random' search is not really random (trial) because you intelligently select what seem to be the best opportunities first.
Where I draw the line - because the whole 'trial and error' vs determinism - is blurry - is whether a strategy changes the board to get an answer. Nishio and Bowmans bingo do this, as does Crook fundamentally. By this I mean "If I remove/add a number here - what is the consequence?". A 'pattern' based strategy is, imho, is on the other side of the 'trial and error' divide. It asks the question "because X or Y is present/absent, what is consequence of that?"
I hope that stab at rephrasing helps and I've not repeated myself.