Alternating finite automaton

In automata theory, an alternating finite automaton (AFA) is a nondeterministic finite automaton whose transitions are divided into existential and universal transitions. For example, let A be an alternating automaton.

• For an existential transition ${\displaystyle (q,a,q_{1}\vee q_{2})}$, A nondeterministically chooses to switch the state to either ${\displaystyle q_{1}}$ or ${\displaystyle q_{2}}$, reading a. Thus, behaving like a regular nondeterministic finite automaton.
• For a universal transition ${\displaystyle (q,a,q_{1}\wedge q_{2})}$, A moves to ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$, reading a, simulating the behavior of a parallel machine.

Note that due to the universal quantification a run is represented by a run tree. A accepts a word w, if there exists a run tree on w such that every path ends in an accepting state.

A basic theorem states that any AFA is equivalent to a deterministic finite automaton (DFA), hence AFAs accept exactly the regular languages.

An alternative model which is frequently used is the one where Boolean combinations are in disjunctive normal form so that, e.g., ${\displaystyle \{\{q_{1}\},\{q_{2},q_{3}\}\}}$ would represent ${\displaystyle q_{1}\vee (q_{2}\wedge q_{3})}$. The state tt (true) is represented by ${\displaystyle \{\emptyset \}}$ in this case and ff (false) by ${\displaystyle \emptyset }$. This representation is usually more efficient.

Alternating finite automata can be extended to accept trees in the same way as tree automata, yielding alternating tree automata.