Final answer:
A Turing machine accepting the language where the number of 1s in a string determines the position to check for another 1 is complex and involves counting and position-matching mechanics.
Step-by-step explanation:
The student asked for the formal description of a Turing machine that accepts the language the #(1, x)-th symbol of x is 1 with Σ = {0, 1}. The description of such a Turing machine can be quite technical, involving a set of states, an alphabet, transition functions, a start state, accept and reject states. To accept the language, the machine could initially scan the input string to count the number of 1's encountered.
After counting the occurrences of 1, the Turing machine would then proceed to the corresponding position in the input string that matches the count and check if that symbol is indeed a 1. If the symbol is 1, the machine would go into an accept state, and if not, it would go into a reject state.