I am working on a problem whose complexity is unknown.
Due to the nature of the problem, I cannot use long edges as I want, so 3SAT and its variants are almost impossible to use.
Finally, I have decided to opt for the most primitive method: Turing machines.
Oddly enough, I could not find any examples of NP hardness reduction performed directly by modeling the problem as a language and showing that a deterministic Turing machine cannot decide if a given instance belongs to that language (it could have spoiled the terminology problem here).
So, assuming there are no problems to perform a reduction in NP hardness, how do you prove that a problem is NP-hard? Is there any publication that does this?