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 tells that any AFA is equivalent to an non-deterministic finite automaton (NFA) by performing a similar kind of powerset construction as it is used for the transformation of an NFA to a deterministic finite automaton (DFA). This construction converts an AFA with k states to an NFA with up to ${\displaystyle 2^{k}}$ states.

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.

Formal Definition

An alternating finite automaton (AFA) is a 6-tuple, ${\displaystyle (S(\exists ),S(\forall ),\Sigma ,\delta ,P_{0},F)}$, where

References

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