In automata theory, an alternating finite automaton (AFA) is a nondeterministic finite automaton whose transitions are divided into existential and universal transitions.
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.,
The state tt (true) is represented by
in this case and ff (false) by
Alternating finite automata can be extended to accept trees in the same way as tree automata, yielding alternating tree automata.
An alternating finite automaton (AFA) is a 5-tuple,
, we define the acceptance function
This model was introduced by Chandra, Kozen and Stockmeyer.
[1] Even though AFA can accept exactly the regular languages, they are different from other types of finite automata in the succinctness of description, measured by the number of their states.
Chandra et al.[1] proved that converting an
-state AFA to an equivalent DFA requires
states in the worst case, though a DFA for the reverse language can be constructued with only
Another construction by Fellah, Jürgensen and Yu.
states to a nondeterministic finite automaton (NFA) with up to
states by performing a similar kind of powerset construction as used for the transformation of an NFA to a DFA.
The membership problem asks, given an AFA
[3] This is true even on a singleton alphabet, i.e., when the automaton accepts a unary language.
), the universality problem (is the complement of the language of an input AFA empty?
), and the equivalence problem (do two input AFAs recognize the same language) are PSPACE-complete for AFAs[3]: Theorems 23, 24, 25 .