Update automata.md

Author: pavly-gerges

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

@@ -69,15 +69,15 @@ Formal notation of the deterministic machines:
 
 > [!TIP]
 > Some vague terms are ought to be defined so far:
-> 1) _The Language of the Automaton_: the set of all strings that the machine _M_ accepts, and is better be defined formally using the set builder notation (e.g.: $$L(M) = [s\ |\ s$$ _is a string composed of the alphabets_ $$\Sigma \land Rule_1 \land ...]$$).
+> 1) _The Language of the Automaton_: the set of all strings that the machine _M_ accepts, and is better be defined formally using the set builder notation (e.g.: $$L(M) = \\{s\ |\ s$$ _is a string composed of the alphabets_ $$\Sigma \land Rule_1 \land ...\\}$$).
 >
 > 2) _Finite State Recognizers_: abstract machines used to determine whether a given input string belongs to a particular language, as defined by a formal grammar.
 >  
 > 3) _State-State Relations_: are defined by the transition functions, which are functions in case of the deterministic machines and relations and/or partial functions in case of the non-deterministic machines.
 > 
-> 4) _Deterministic Automaton (Revisited)_: 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: $Q_{out} = [\bigcup_{i = 0}^{n - 1} \delta(q_i, \sigma_{i + 1}) \rightarrow [q_{i + 1}]] = [q \in Q | \delta(q_i, \sigma_{i + 1}) = q_{i + 1};\ where\ 0 <= i < n \land n \in N]$
+> 4) _Deterministic Automaton (Revisited)_: 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 (Revisited)_: 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 $Q_{out} = [\bigcup_{i = 0}^{n - 1} \delta(q_i, \sigma_{i + 1}) \rightarrow P(Q)] = [q \in P(Q) | \delta(q_i, \sigma_{i + 1}) = q_{i + 1};\ where\ 0 <= i < n \land n \in N]$.
+> 5) _Non-deterministic Automaton (Revisited)_: 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 $Q_{out} = \\{\bigcup_{i = 0}^{n - 1} \delta(q_i, \sigma_{i + 1}) \rightarrow P(Q)\\} = \\{q \in P(Q) | \delta(q_i, \sigma_{i + 1}) = q_{i + 1};\ where\ 0 <= i < n \land n \in N\\}$.
 >
 > 6) A quick guess to a typical GNU/C99 prototype abstraction that models the automaton constructs; it essentially uses the _proof by construction_ to construct the automaton by putting the most peculiar rule in-mind; which states that _an automaton uses a limited memory and is a quintuple sequence_:
 > ```c