Erik Jonsson School of Engineering and Computer Science

1. Erik Jonsson School of Engineering and Computer Science CS 4384– 001 Automata Theory http://www.utdallas.edu/~pervin Thursday: Context-Free Languages Sections 3.1-3.3 Tuesday2-25-14 FEARLESS Engineering

2. EXAMINATION ANSWERS Regular Languages ANSWERS

3. Distinguished States

4. Distinguished States

5. Context-free Languages Chapter Three Tuesday 2-25

6. A context-free grammar (CFG) G is a quadruple (V, Σ, R, S) where • V: a set of non-terminal symbols (Variables) • Σ: a set of terminals (V ∩ Σ = Ǿ) (Alphabet) • R: a set of rules (R: V → (V UΣ)*) • S: a start symbol (a variable) Definition

7. If R: A → B, then xAyxBy and we say that xBy is derived from xAy using rule R. • If s ··· t, then we write s * t. • A string x in Σ* is generated by G=(V,Σ,R,S) if S * x. • L(G) = { x in Σ* | S * x}. How do we use rules?

8. G = ({S}, {0,1}. {S → 0S1 | ε }, S) • ε in L(G) because S ε . • 01 in L(G) because S 0S1 01. • 0011 in L(G) because S 0S1 00S11 0011. • 0 1 in L(G) because S * 0 1 . • L(G) = {0 1 | n > 0} n n n n n n Example

9. A context-free grammar is “context-free’’ because we can use a rule R no matter what the context. E.g., (M&S P.89, Def. 3.1.2) from the rule R: A -> w, we can derive from the string uAv in one step the string uwv, no matter what the context (that is, for any strings u and v). Context-free

12. Palindromes

13. Any positive even number of a’s with any number of b’s anywhere

20. G = (V, ∑, R, E) V = {E, T, F} ∑= {x, 1, 2, +, *, (, )} R = { E -> E + T | T; T -> T * F | F; F -> (E) | x1 | x2 } Expressions, Terms, Factors Parsing Expressions

22. Parse Tree

23. Closure Properties

27. For every regular language L, there exists a CFG G (in fact a Regular Grammar) such that L=L(G). Theorem

28. For every regular language L, there exists a CFG G such that L=L(G). Proof. Let L=L(M) for a DFA M=(Q, Σ, δ, s, F). Construct a CFG G=(V, Σ, R, S) as follows. V = Q, Σ = Σ, R = { q → ap | δ(q, a) = p } U { f → ε | f in F}, S = q0. Theorem x1 xn S x1q1 x1x2q2 ··· x1…xnf x1…xn f=qn s q1

30. Regular Grammars Note: A -> a is unnecessary

31. What's wrong with (ab)*a*?

33. Simplified and Normal Forms