Lecture 02

Finite Automata

Recall: Goal. Determine which decision we can solve

Thesis 2: "Solving a Language" $$\Longleftrightarrow$$ Finite Automata

Define: A (deterministic) finite automaton is a 5-tuple ($$Q,\Sigma,\delta,q_0,F$$)

• $$Q$$: finite set of states
• $$\Sigma$$: Alphabet
• $$\delta$$: $$Q\times\Sigma\rightarrow Q$$
• $$q_0$$: Initial state
• $$F\subseteq Q$$: set of accepting states

Define: A finite automaton accepts $$w\in\Sigma^n$$ ($$n\geq 0$$) iff there is a sequence of states $$r_0,...,r_n$$ such that

1. $$\forall i\in[n]=\{1,2,...,n\}$$ $$r_i=\delta (r_{i-1},w_i)$$
2. $$r_n\in F$$
3. $$r_0=q_0$$

Define The empty string is denoted by $$\epsilon$$

Observation: $$L_{\epsilon}=\{\epsilon\}$$, $$L_{\emptyset}=\emptyset$$, $$L_{\epsilon}\neq L_{\emptyset}$$

Example

1. $$L=\{w|$$last char of $$w$$ is 1$$\}$$
2. $$L=\{w|$$next-to-last char of $$w$$ is 1 $$\}$$
3. $$L=\{w|w$$ has the same number of $$01$$ and $$10$$ substrings$$\}$$

Define A language $$L$$ is regular iff it is recognized by some finite automaton

Non-deterministic Finite Automata

Define A non-deterministic finite automaton is a 5-tuple ($$Q,\Sigma,\delta,q_0,F$$)

• $$Q,\Sigma,q_0,F$$ as before
• $$\delta$$: $$Q\times\Sigma\rightarrow\mathcal{P}(Q)$$ where $$\mathcal{P}(Q)$$ is a set of subsets of $$Q$$

Define The NFA accepts $$w\in\Sigma^n$$ iff $$\exists r_0,...,r_n\in Q$$ such that

1. $$r_0=q_0$$
2. $$r_n\in F$$
3. $$\forall i\in[n], r_i\in\delta(r_{i-1},w_i)$$

Theorem Every language that can be recognized by an NFA is regular

Proof Given an NFA $$N$$ we can build a DFA $$M$$ that recognizes the same languages

• $$Q'=\mathcal{P}$$
• $$q_0'=\{q_o\}$$
• $$F'=\{R\subseteq Q:R\cap F\neq\emptyset\}$$