Let T = {<M> | M is a TM that accepts w r whenever it accepts w}.
Show that T is undecidable.
I have two answers to this question - San Diego :
5.9
Let T = { <M> | M is a TM that accepts w r whenever it accepts w }.Assume T is decidable and let decider R decide T. Reduce from A TM by constructing a TM S as follows:
- S: on input <M,w>
- create a TM Q as follows:
On input x:
- if x does not have the form 01 or 10 reject.
- if x has the form 01, then accept.
- else (x has the form 10), Run M on w and accept if M accepts w.
- Run R on
- Accept if R accepts, reject if R rejects.
Because S decides A TM , which is known to be undecidable, we then know that T is not decidable
Undisclosed source:
5.12 We show that A TM ≤ m S by mapping ‹ M , w › to ‹ M' › where M' is the following TM:
- M' = “On input x :
- If x = 01 then accept .
- If x ≠ 10 then reject .
- If x = 10 simulate M on w .
If M accepts w then accept ; if M halts and rejects then reject .”If ‹ M , w › ∈ A TM then M accepts w and L ( M' ) = {01,10}, so ‹ M' › ∈ S .
Conversely, if ‹ M , w › ∉ A TM then L ( M' ) = {01}, so ‹ M' › ∉ S . Therefore,
‹ M , w › ∈ A TM ⇔ ‹ M' › ∈ S .
But I do not understand the following:
1- what is the relation between x and w?
2- why we consider the 2 cases ‹ M , w › ∈ A TM and ‹ M , w › ∉ A TM ?
3- why if A is mapping reducible to S this makes S undecidable?
could anyone clarify these points for me?
I think it is not suitable for asking in SO because it is not a educational website, but I answered it.
1- what is the relation between x and w?
Answer 1: x is a symbol that used for using a symbol for operate. This symbol should not be in alphabet of language, just it. It hasn't any relation to w.
2- why we consider the 2 cases ‹M, w› ∈ ATM and ‹M, w› ∉ ATM?
Answer 2: For proofing a language like L is decidable or not, we need to determine a string like w is member of language or not. So we have to consider two type of string w∉L and w∈L.
3- why if A is mapping reducible to S this makes S undecidable?
Answer 3: It means the process of checking a string is in language in A and S is similar and if we can't find a algorithm for checking this for A, we can't find any algorithm for S.
The technical post webpages of this site follow the CC BY-SA 4.0 protocol. If you need to reprint, please indicate the site URL or the original address.Any question please contact:yoyou2525@163.com.