Final answer:
To prove that every c.e. set is m-reducible to the halting problem, we need to show that there exists a computable function that transforms elements of the c.e. set to elements of the halting problem. This can be demonstrated by constructing a Turing machine that simulates the behavior of the c.e. set and mapping its inputs to inputs for the halting problem.
Step-by-step explanation:
To prove that every c.e. set is m-reducible to the halting problem, we need to show that there exists a computable function that transforms elements of the c.e. set to elements of the halting problem. Let's consider a specific example:
- Assume we have a c.e. set A, represented by a Turing machine. This machine halts if and only if its input is in the set A.
- We can construct a Turing machine B that simulates the behavior of A. If A halts, B halts as well. Otherwise, B runs indefinitely.
- Now, we can define a computable function f(x) that maps an input x to an input for the halting problem. If x is in A, f(x) returns the input for B that leads to halting. Otherwise, f(x) returns the input for B that runs indefinitely.
Therefore, we have shown that every c.e. set is m-reducible to the halting problem.