Giorgi Japaridze Theory of Computability Reducibility Chapter 5
Giorgi Japaridze Theory of Computability Reducibility Chapter 5
Giorgi Japaridze Theory of Computability Reducibility Chapter 5
5. 1. a Giorgi Japaridze Theory of Computability The undecidability of the halting problem Let HALTTM = {<M, w> | M is a TM and M halts on input w} HALTTM is called the halting problem. Theorem 5. 1: HALTTM is undecidable. Proof idea: Assume, for a contradiction, that HALTTM is decidable. I. e. there is a TM R that decides HALTTM. Construct the following TM S: S = “On input <M, w>, an encoding of a TM M and a string w: 1. Run R on input <M, w>. 2. If R rejects, reject. 3. If R accepts, simulate M on w until it halts. 4. If M has accepted, accept; if M has rejected, reject. ” • If M works forever on w, what will S do on <M, w>? Expicitly reject • If M accepts w, what will S do on input <M, w>? Accept • If M explicitly rejects w, what will S do on <M, w>? Expicitly reject Thus, S decides the language ATM. But this is impossible (Theorem 4. 11)
5. 3. a Giorgi Japaridze Theory of Computability Definition of mapping reducibility Let A and B be languages over an alphabet . A mapping reduction of A to B is a computable function f: * * such that, for every w *, w A iff f(w) B. We say that A is mapping reducible to B, written A m. B, if there is a mapping reduction of A to B. * * A B f f
5. 3. a Giorgi Japaridze Theory of Computability Definition of mapping reducibility Let A and B be languages over an alphabet . A mapping reduction of A to B is a computable function f: * * such that, for every w *, w A iff f(w) B. We say that A is mapping reducible to B, written A m. B, if there is a mapping reduction of A to B. * * A B f f
5. 3. a Giorgi Japaridze Theory of Computability Definition of mapping reducibility Let A and B be languages over an alphabet . A mapping reduction of A to B is a computable function f: * * such that, for every w *, w A iff f(w) B. We say that A is mapping reducible to B, written A m. B, if there is a mapping reduction of A to B. * * A B f f
5. 3. c Giorgi Japaridze Theory of Computability Using mapping reducibility for proving decidability/undecidability Theorem 5. 22: If A m. B and B is decidable, then A is decidable. Proof: Let DB be a decider for B and f be a mapping reduction from A to B. We describe a decider DA for A as follows. DA= “On input w: 1. Compute f(w). 2. Run DB on input f(w) and do whatever DB does. ” Corollary 5. 23: If A m. B and A is undecidable, then B is undecidable. Theorem 5. 22 remains valid with “Turing recognizable” instead of “decidable”. So does Corollary 5. 23.
5. 3. d Giorgi Japaridze Theory of Computability A mapping reduction of ATM to HALTTM For a TM M, let M* be the TM obtained from M by adding a dead-end (infinite loop) state and re-directing to it all transitions that go to the reject state. Thus, • If M accepts input x, then M* • If M explicitly rejects x, then M* • If M never halts on x, then M* To summarize, M accepts x iff M* accepts x never halts on x Let then f be the function defined by f(<M, w>)=<M*, w>. Is f computable? Yes! Obviously <M, w> ATM iff f(<M, w>) HALTTM i. e. f is a mapping reduction of ATM to HALTTM So, since ATM is undecidable, HALTTM is undecidable as well.
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