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CSC 363H              Tutorial Exercises for Week  2            Summer 2006
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 -  Exercise 3.7 (p.160).
    Explain why the following is not description of a legitimate Turing
    machine.
    M_bad = "The input is a polynomial p over variables x_1, ..., x_k.
            1. Try all possible settings of x_1, ..., x_k to integer values.
	    2. Evaluate p on all of these settings.
	    3. If any of these settings evaluates to 0, accept; otherwise
	       reject."

 -  Show that TMs with "stay put" head movement are equivalent to regular
    TMs.  Give a detailed answer.

 -  For some alphabet S, say that a function f : S* -> S* is "computable"
    if there exists a TM M such that for all strings w in S*, M accepts w
    with final configuration q_accept f(w).  In other words, when M is
    started with w on its tape, it eventually enters its accepting state
    with only f(w) on the tape and its head on the leftmost symbol of f(w).
    
    Show that a function f is computable in this sense iff the language
        L_f = { w#f(w) : w in S* }
    is decidable (where # is a symbol that does not belong to S).
    
    (Note that this question is slightly more involved than the typical
    tutorial exercise you can expect, but thinking it through will prove to
    be very useful to you later on.)