Update automata.md

embedded-system-design/theory-of-computation/automata.md

@@ -65,5 +65,5 @@ Formal Notation:
 > 
 > 4) _Deterministic Automaton_: a deterministic machine defines a unique transition function for each pair of state $q_i$ and input $\sigma_{i + 1}$; where $$i \in N$$ is the index of the current state in the collection $$Q$$, and $$(i + 1) \in N$$ is an arbitrary number representing the position of the input $$\sigma$$ in the collection $$\Sigma$$, the output of the function is deterministically the next state $q_{i + 1}$; where $$0 <= i < n$$ and $$n \in N$$ is an arbitrary number representing the index of the final accepting state; thus the following holds and can be used to define the set for the output states: $Q_{out} = [\bigcup_{i = 0}^{n - 1} \delta(q_i, \sigma_{i + 1}) \rightarrow [q_{i + 1}]] = [q \in Q | \delta(q_i, \sigma_{i + 1}) = q_{i + 1};\ where\ 0 <= i < n \land n \in N]$
 > 
-> 5) _Non-deterministic Automaton_: a non-deterministic machine defines a non-unique transition function for each pair of state $q_i$ and input $\sigma_{i + 1}$, in other words the transition from the state $q_i$ with the input $\sigma_{i + 1}$ is not pre-determined, thus we can define the transition function as $$\Delta (q_{i}, \sigma_{i + 1}) \rightarrow P(Q)$$, and the set for the output states of this machine as $Q_{out} = [\bigcup_{i = 0}^{n - 1} \Delta(q_i, \sigma_{i + 1}) \rightarrow P(Q)] = [q \in P(Q) | \delta(q_i, \sigma_{i + 1}) = q_{i + 1};\ where\ 0 <= i < n \land n \in N]$.
+> 5) _Non-deterministic Automaton_: a non-deterministic machine defines a non-unique transition function for each pair of state $q_i$ and input $\sigma_{i + 1}$, in other words the transition from the state $q_i$ with the input $\sigma_{i + 1}$ is not pre-determined, thus we can define the transition function as $$\delta (q_{i}, \sigma_{i + 1}) \rightarrow P(Q)$$; where $$P(Q)$$ is the power set of Q of cardinality $$|P(Q)| = 2^{|Q|}$$, and the set for the output states of this machine as $Q_{out} = [\bigcup_{i = 0}^{n - 1} \delta(q_i, \sigma_{i + 1}) \rightarrow P(Q)] = [q \in P(Q) | \delta(q_i, \sigma_{i + 1}) = q_{i + 1};\ where\ 0 <= i < n \land n \in N]$.