CS30 - Spring 2016 - Class 25
Lecture notes
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- assignment 11
Nim tournament!
halting problem
- Could we write a Turing machine that simulates the running of another Turing machine?
- take as input two things:
1) some representation of the Turing machine
2) some input to run on the Turing machine input (i.e. the input Turing machine from 1)
- would then "run" the Turing machine it was simulating on the input
- this is called a "universal Turing machine"
- Consider a TM H that does the following:
H(M,w) =
- accept if M finishes (accepts or rejects) with input w
- reject if M never finishes with input w, i.e. runs forever
- this is called the halting problem
- this program comes up a lot in CS
- you run your program
- it doesn't finish
- will it or is it stuck in an infinite loop?
- for particular types of problems we can definitely answer this problem
- can we in general, though?
- Could we ever write this Turing machine?
To prove that we *cannot* we'll do a proof by contradiction
- We assume that we can, i.e. that we could construct H above
- And show that this results in a contradiction, meaning it couldn't be the case
- Let's construct the following Turing machine
D(M) =
Run H(M, [M]) where [M] is the string representation of M
- if it accepts (i.e. H(M, [M]) will finish), then loop infinitely
- if it rejects (i.e. H(M, [M] will loop forever), then stop
- What happens when we run D(D)?
- Run H(D, [D])
- if it accepts, that implies that D(D) finishes
- however, if H(D, [D]) accepts, then we know that D(D) loops infinitely
- if it rejects, that implies that D(D) never finishes
- however, if H(D, [D]) rejects, then we know that D(D) halts
- We've reached a contradiction!
- If H says D finishes, then it loops infinitely
- If H says D loops, then it halts