3. suppose that we are given a turing machine m = (q, σ, γ, δ, s, t, r), and we would like to know if m will ever write a non-blank symbol when it is initialized with a completely blank tape. more formally, we ask whether there exist q ∈ q, x 6∈ { } ∗ , and j, n ∈ n such that (s, ` [infinity], 0) n−→ m (q, `x [infinity], j). (a) (10 points) prove that if m ever writes a non-blank symbol on a blank tape, then it does so within the first 2 · |q| steps of computation. (b) (10 points) assuming that the statement from (a) is true, prove that the language b = {hmi : m is a tm that never writes any non-blank symbol when run on a blank tape} is decidable.

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