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 $(q,\; a,\; q\_1\; vee\; q\_2)$, "A" nondeterministically chooses to switch the state to either $q\_1$ or $q\_2$, reading "a". Thus, behaving like a regular nondeterministic finite automaton.
* For a universal transition $(q,\; a,\; q\_1\; wedge\; q\_2)$, "A" moves to $q\_1$ and $q\_2$, 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 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 $2^k$ 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 DNF so that $\{\{q\_1\}\{q\_2,q\_3\}\}$ would represent $q\_1\; vee\; (q\_2\; wedge\; q\_3)$. The state tt (true) is represented by $\{\{\}\}$ in this case and ff (false) by $emptyset$.This clause representation is usually more efficient.

Formal Definition

An alternating finite automaton (AFA) is a 6-tuple,$(S(exists),\; S(forall),\; Sigma,\; delta,\; P\_0,\; F)$, where

*$S(exists)$ is a finite set of existential states. Also commonly represented as $S(vee)$.
*$S(forall)$ is a finite set of universal states. Also commonly represented as $S(wedge)$.
*$Sigma$ is a finite set of input symbols.
*$delta$ is a set of transition functions to next state $(S(exists)\; cup\; S(forall))\; imes\; (Sigma\; cup\; \{\; varepsilon\; \}\; )\; o\; 2^\{S(exists)\; cup\; S(forall)\}$.
*$P\_0$ is the initial (start) state, such that $P\_0\; in\; S(exists)\; cup\; S(forall)$.
*$F$ is a set of accepting (final) states such that $F\; subseteq\; S(exists)\; cup\; S(forall)$.