Essentially they generate functions that map a program state to another program state and counts those function calls and compares with e.g. the number of function calls in quick sort. They "cheat" by feeding how all elements compares the their neighbours at each step. Like, if you give the algorithm that input for free what's the point.
Notably, none of the popular sorting algorithms decide which elements to swap by looking at the whole input at each execution step. On the contrary, the decisions are typically made based on local evidence only. ... We implement this principle for the sorting task by employing a small set of k(independent of n) index variables, and include in the input the information about how A[i] compares with A[i+ 1] and A[i−1]for each index variable i.