图灵机协助

Question

I've been practicing with designing Turing machines, and one that I recently solved was for a language L = { w : n _a (w) = n _b (w)}, where the number of a's in w are equal to the number of b's in w.

However, how would I go about designing a Turing machine for a language where the number of a's in w does not equal the number of b's in w? (eg. L = {w : n _a (w) ≠ n _b (w)}?

The book and websites that I've been using for reference define a Turing machine as follows:

M = (Q, Σ, ℾ, δ, q ₀ , ☐, F), where Q is the states, Σ is the input alphabet, ℾ is the tape alphabet, δ is the transitions, q ₀ is the start state, ☐ is a blank space that is an element of ℾ (☐ ∊ ℾ), and F is the final state designated as q _f .

Answer 1

If you've recently solved the case where the number of a and b are equal, an immediate solution for the case where they're not equal is this:

• if you used to crash (have no valid transition) then add a transition to halt_accept instead;
• if you used to explicitly reject by going to halt_reject, go to halt_accept instead;
• if you used to accept by entering halt_accept, go to halt_reject instead.

This new TM (1) accepts anything that would not have been accepted by your TM for the case of equal numbers of a and b, and (2) rejects anything that would have been accepted by your TM for the case of equal numbers of a and b.

In fact, this procedure works for any TM whose job is simply to decided membership of the input string in the language of the machine; it would not work to enumerate strings in the complementary language, but that's not necessarily something you need (and you could get enumeration by using a machine constructed like this as a sub-machine in another that enumerates all strings and checks them).