Erik Jonsson School of Engineering and Computer Science FEARLESS Engineering CS 4384 – 001 Automata Theory Thursday: Context-Free Languages Sections Tuesday

EXAMINATION ANSWERS Regular Languages 2 ANSWERS

Distinguished States 3

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Context-free Languages Chapter Three 5Tuesday 2-25

Definition 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 U Σ)*) S: a start symbol (a variable)

How do we use rules? 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}.

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

Context-free 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 ) 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). 9

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12 Palindromes

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

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Parsing Expressions 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 20

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22Parse Tree

Closure Properties 23

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

Theorem 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. sq1q1 f=q n x1x1 xnxn S x 1 q 1 x 1 x 2 q 2 ··· x 1 …x n f x 1 …x n 28

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Regular Grammars 30Note: A -> a is unnecessary

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

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Simplified and Normal Forms 33