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Summary: Theory of Computation
Lecture 14: Reducibility
Max Alekseyev
University of South Carolina
March 1, 2012
Halting Problem
ATM = { M, w | M is a TM and M accepts w}
It is easy to establish that ATM is recognizable: for a given pair
M, w we need simply to simulate M on the input w, and accept
or reject depending on whether M accepts or rejects. Note that M
may loop in which case our simulation will loop as well, making it
just a recognizer not a decider.
Halting Problem
ATM = { M, w | M is a TM and M accepts w}
It is easy to establish that ATM is recognizable: for a given pair
M, w we need simply to simulate M on the input w, and accept
or reject depending on whether M accepts or rejects. Note that M
may loop in which case our simulation will loop as well, making it
just a recognizer not a decider.
However, there is a decider that relies on another decider. On
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