Turing Machines as Acceptors

This is too easy, so let's repeat it in symbols:

A Turing Machine T = (Q, , , , q₀, #, F) accepts a language L(M), where

L(M) = (w +: q₀w x_iq_fx_j
for some q_f F, x_i, x_j *}

(Notice that this definition assumes that the Turing machine starts with its tape head positioned on the leftmost symbol.)

We said a Turing machine accepts its input if it halts in a final state. There are two ways this could fail to happen:

1. The Turing machine could halt in a nonfinal state, or
2. The Turing machine could never stop (in which case we say it is in an infinite loop. )