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 Tuesday: Context-Free Languages Sections 3.4-3.5 Thursday2-27-14 FEARLESS Engineering

2. Distinguished States

3. Distinguished States

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

5. 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

8. Palindromes

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

14. 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

16. Parse Tree

17. Closure Properties

21. 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

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

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

25. Homework 9 Answer 1

26. Homework 10 Answer 2

27. Simplified and Normal Forms

32. Chomsky Normal Form

33. Extended Example

38. Converting to Chomsky Normal Form

41. Note that we don't need both Y1 and Y3, or both Y2 and Y4 so we could simplify slightly.

42. The empty string

43. Other Simplifications Useful variables and productions Left-recursion

45. Useless Example

47. Left-recursion

49. Pushdown Automata Many similar definitions