A turing machine is a simple hypothetical computational engine.
It is comprised of a finite set of states, a finite alphabet of characters which
may initially appear on the tape (input alphabet), a finite alphabet of characters which may be
written to the tape (output alphabet), and a transition function mapping (State, Character) tuples to
(State, Character, Direction) tuples, where Direction is either LEFT or RIGHT.
A turing machine has a single tape comprised of cells, extending infinitely far to the right, with a single read/write head that can, at any time, be in exactly one position on the tape. A turing machine has exactly one current state, which at any time is one of the states in its set of states.
Initially, the head is at the left most cell of the tape and the state is the INIT state. Some input string made up of characters from the input alphabet. At any point in the computation, the transition function is used to find the mapping from the current state and the character currently under the head to a new state, a new character and a direction. The turing machine's state is updated to this new state, the new character is written to the tape overwriting the old one, and the head moves one position in that direction (unless the head is at the left most position and the direction is LEFT).
This is both the input and output alphabets for this turing machine. No distinction is made between input and output alphabets. That is, any character that can be read can also be written. It is up to the user to enforce an output alphabet as some strict subset of the input alphabet if they choose to.
This is a list of all the possible states the turing machine can be in. The three default states are ACCEPT, REJECT and INIT. The ACCEPT and REJECT states are both halt states. That is, once the turing machine enters either state, it will stop computing. The INIT state is the state in which the turing machine will begin.