Is it possible to prove that the Halting problem is undecidable using Rice's theorem? Here's what I've tried and failed:
- We want to reduce Rice's Theorem (decide if a language has the nontrivial property P) to the Halting Problem
- We want to say "if the Halting Problem can be solved, then we can decide if a language as nontrivial property P"
Here's how I thought the proof might go:
- Suppose we have a machine H that solves the halting problem. We want to use this machine to decide if a language has property P.
- We construct a machine that uses H to decide a language has property P.
Example: Take P to be "language contains the string 1".
We want to use H to decide if the language of a TM M contains the string 1, i.e that the TM halts and accepts when given input 1.
Now, if M halts on 1, that doesn't mean it accepts 1, so we don't know if L(M) contains 1. On the other hand if M does not halt on 1, that certainly means that L(M) does not contain 1.
And that's where I'm stuck. Is it actually possible to use Rice's theorem to prove that the halting problem is undecidable?