A Myhill-Nerode theorem for automata with advice

Consider an extension of the classical model of finite automata operating on finite strings in which the machine simultaneously reads a fixed advice tape — an infinite string A. A deterministic finite-string automaton with advice A is like a deterministic finite-string automaton, except that at step n the next state depends on the current state, the nth symbol of the input w ∈ Σ∗, and the nth symbol of A ∈ Γω . The automaton halts once all of w has been read and accepts w if and only if it is in a final state (in the nonterminating model of Section 4, the automaton reads the rest of the advice tape and accepts according to a Muller condition). We now give a formal definition.