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CS310

CS310. Context Free Languages and Grammars Sections:2.1 page 99 September 27, 2006. Quick Review. y. z. x. Pumping Lemma

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CS310

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  1. CS310 Context Free Languages and Grammars Sections:2.1 page 99 September 27, 2006

  2. Quick Review y z x • Pumping Lemma • If A is a regular language, then there is a number p where, if s is any string in A of length at least p, then s may be divided into three pieces, s = xyz, satisfying the following conditions: • i >= 0, xyiz є L(M) • |y| > 0 • |xy| <= p s = xyz (x, z may be ε)

  3. Context Free Grammar • Another way to represent a language • Can represent more languages than FA • Produces a “Context Free Language” • Pushdown Automata: machine that recognizes a context free language • Trivia: • First used to describe human languages • Now used to parse computer languages (C, C++)

  4. Context Free Grammar • Example A -> 0A1 A -> B B -># Variables: A, B (may appear on LHS and RHS) Terminals: 0, 1, # (only appear on the RHS) Start variable: Variable on LHS of top rule Language: { 0n#1n | n >= 0 } Example: A -> 0A1 -> 00A11 -> 00#11

  5. Example • A -> 0A1 -> 00A11 -> 00#11 • derivation • write u -*> v if there is a derivation of the string v from u using the grammar, where u and v are strings of terminals and variables 0A1 -*> 00#11 • Parse Tree A A A B 0 0 0 # 1 1 1

  6. Exercise R -> XRX | S S -> aTb | bTa T -> XTX | X | ε X -> a | b Variables, terminals of G? Start variable? True or false? T -*> aba

  7. Formal Definition • A context free grammar (CFG) G is a 4-tuple (V, ∑, R, S) • V finite set of variables • ∑ finite set of terminals • R set of rules of form: • variable -> (string of variables and terminals) • S є V, start variable • The language of the grammar is: • L(G) = { w є∑* | S -*> w} • w is in ∑* and can be derived from S

  8. Example L = { w є {a, b}* | aa is a substring } Find a grammar that generates this language

  9. Example • Grammar G2 on page 101 • Show derivation for “a boy sees a flower” • Notice how this statement is non-creepy? • Show the parse tree

  10. Ambiguous Grammar • E -> E + E | E x E | E | a • Find parse tree for: a + a x a

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