ALTERNATING TURING MACHINE

An alternating Turing Machine (ATM) is a kind of NTM that classifies its states into two categories:  universal and existential.  This is something that is built into an ATM.

A universal state is said to be an accepting state if all transitions at that point lead to acceptance.  In particular, if a universal state has no transitions, then it is by default an acepting state.

An existential state is said to be an accepting if there is some transition that leads to acceptance.  If there aren't any transitions, then it is not an accepting state.

Acceptance is pretty much defined recursively.  The universal states with no transitions are accepting.  The existential states with no transitions are not accepting (we could also say rejecting).  And we kind of work backward from there.

If the start state is accepting, then the string is said to be accepted by the ATM.

So, to clarify, a state is universal or existential by design of the machine.  A state is accepting (or rejecting) by virtue of how the machine operates on a specific input.

If I start in an accepting universal state, that means every valid transition from this point will take me to an accepting state.  Perhaps that second state is also universal, in which every transition will take you to an accepting state.  If I transition from a universal state to an existential state, then there is at least one transition from that existential to an accepting state, but not necessarily all transitions will take you to an accepting state.

So, to be clear, if I am in an accepting universal state, this does not mean that all paths from this point lead to acceptance, only that the next step is guaranteed to take me to an accepting state.