Erik Jonsson School of Engineering and Computer Science

1. Erik Jonsson School of Engineering and Computer Science CS 5349– 001 CS 4384 - 001 Automata Theory http://www.utdallas.edu/~pervin Thursday: Sections 3.1-3.3 Tuesday10-07-14 FEARLESS Engineering

2. EXAMINATION I 2 October 2014

11. We play against an opponent. Our goal is to win the game by establishing a contradiction to the PL, while the opponent tries to foil us. There are four moves in the game. • The opponent picks p. • Given p we pick a string s(p) in L of length ≥ p. • The opponent chooses the decomposition xyz subject to |xy| ≤ p, |y| ≥ 1. We have to assume that the opponent makes the choice that will make it harder for us to win the game. • We pick i so that the pumped string is not in L. The Pumping Lemma Game Don't forget! It will be on the comprehensive final exam!

12. Context-free Languages Chapter Three

13. 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 Du, Lecture #11

14. If R: A → B, then xAy xBy 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?

15. 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

16. 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

26. For every regular language L, there exists a CFG G such that L=L(G). Theorem