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.

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 represented as clauses. For instance, one could assume the combinations to be in disjunctive normal form so that ${\displaystyle \{\{q_{1}\},\{q_{2},q_{3}\}\}}$ would represent ${\displaystyle q_{1}\vee (q_{2}\wedge q_{3})}$. The state tt (true) is represented by ${\displaystyle \{\{\}\}}$ in this case and ff (false) by ${\displaystyle \varnothing }$. This clause representation is usually more efficient.

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