Lecture 2

Nondeterministic Finite Automata (NFA)

Difference from DFA:

• A state can have zero or more outgoing arrows with the same label.
• Allows for empty steps: arrows with label λ that don't consume a symbol

Defined like DFAs, except for transition function. Consists of:

• Q finite set of states
• Σ finite input alphabet
• δ : Q × (Σ  ∪ λ)  → 2^Q transition function, 2^Q is set of all subsets of Q
• q₀ ∈ Q starting state
• F ⊂ Q of final states

Nondeterministic means that it can accept a word in more than one way.

For an L, you cannot form $\overline{L}$ by flipping final and nonfinal states.

A language L is accepted by NFA iff L is regular.

Transforming NFA into DFA

Start with the λ-closure of the initial states – i.e. the initial state plus any states you can reach with a λ transition. Then, see what states you can go to from there, and every time compute the λ-closure.

Grammars

Grammar defines a language. With grammar rules you can construct a sentence.

Consists of:

• V finite set of non-terminals (variables)
• T finite set of terminals
• S ∈ V start symbol
• P finite set of production rules x → y where
  • $x \in (V \cup T)^+$ containing at least one symbol from V
  • $y \in (V \cup T)^*$
  • for CFGs, x ∈ V

If x → y is production rule, then we have derivation step uxv ⇒ uyv for every u,v ∈ (V ∪ T)^*.

Example

Find grammar G such that L(G) = {a,b}^* {c} {b,c}^*

S → AcB
A → aA   B → bB
A → bA   B → cB
A → λ    B → λ

Backus Naur Form (BNF) - Context-free grammar

Non-terminals are indicated by < and >, and you can abbreviate multiple rules with an or.

So like:

S → a
  | b
  | λ

Right linear grammars

Right linear if all production rules are of form A → uB or A → u, where A,B are variables and u is a terminal.

It’s strictly right linear if $|u|  \leq 1$ (so $u \in (T  \cup \lambda)$)