Alternating finite automaton


In automata theory, an alternating finite automaton 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, 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 would represent. The state tt is represented by in this case and ff by.
This clause representation is usually more efficient.

Formal definition

An alternating finite automaton is a 6-tuple,
, where
The model was introduced by Chandra, Kozen and Stockmeyer.

State complexity

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. proved that converting an -state AFA to an equivalent DFA
requires states in the worst case. Another construction by Fellah, Jürgensen and Yu. converts an AFA with states to a nondeterministic finite automaton with up to states by performing a similar kind of powerset construction as used for the transformation of an NFA to a DFA.

Computational complexity

The membership problem asks, given an AFA and a word, whether accepts. This problem is P-complete. This is true even on a singleton alphabet, i.e., when the automaton accepts a unary language.
The non-emptiness problem, the universality problem, and the equivalence problem are PSPACE-complete for AFAs.