First Assignment

Due Friday October 8

1. Are the following problems decidable or undecidable? Give proof!
   1. Given a Turing machine M and a string x, will M ever write the symbol @ (part of M's alphabet) on its tape?
   2. M takes fewer than 1,000,000 steps on some input.
   3. M takes fewer than 1,000,000 steps on all inputs.
   4. M moves its head to the left more than 1,000,000 times on input 0.

2. An oracle Turing machine is a Turing machine with a special oracle tape, and three special states: "QUERY", "YES", and "NO". Given an arbitrary language (set of strings) B, M can obtain the information whether x is in B by writing x in its oracle tape, and entering the QUERY state. In the next configuration, the oracle tape of M is erased, and M will be in state YES if x is in B, and in state NO if it isn't.

   Let HALT be the language {N: (ordinary) Turing machine N with blank input eventually halts.}

   1. Sketch an oracle Turing machine that with oracle HALT decides whether a Turing machine P halts on input y.
   2. Prove that the problem "does oracle Turing machine M with oracle HALT halt on input x" is not decidable by oracle Turing machines with oracle HALT.
   3. (Extra Credit) Let HALT2 be the language {N: oracle Turing machine with oracle HALT eventually halts.} Prove the analogs of 1, 2 above for oracle Turing machines with oracle HALT2. Generalize.

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   Janos Simon