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Understanding Context-free Grammars: Rules and Language Generation

Learn about context-free grammars (CFGs) and how they generate languages through production rules. See examples and understand how to work with CFGs.

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Understanding Context-free Grammars: Rules and Language Generation

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  1. [Section 6.1] Context-free Grammars Example : S  Shortened notation : S  aSa S  | aSa | bSb S  bSb Which strings can be generated from S ?

  2. [Section 6.1] Context-free Grammars • Def: A context-free grammar (CFG) is a 4-tuple G = (V,,S,P) where • V is a finite set of variables (nonterminal symbols) •  is a finite set of terminal symbols (terminals) • S 2 V is the start symbol • P is the finite set of grammar rules (productions) of the form A , where A 2 V and 2 (V [)* • (V and  are disjoint.) • For example from previous slide :

  3. [Section 6.1] Context-free Grammars A derivation of the string “abaaba” : Def : Let G=(V,,S,P) be a CFG. For ,2 (V [)* we write )G if there is a production C  in P such that  = 1C2 and  = 12. We say that  is derived from . If 1, …, n2 (V [)* and k)k+1 for k=1,…,n-1, we write 1)G*n. Language generated by G, denoted L(G), is { x 2* | ______ }

  4. [Section 6.1] Context-free Grammars Examples : { akbk | k ¸ 0 } { x 2 {a,b}* | x has at least 2 b’s } { x 2 {a,b}* | |x| is even }

  5. [Section 6.1] Context-free Grammars Example : S  | abS | baS | aSb | bSa | Sab | Sba What is the language generated by this grammar ? (We’ve seen this before…)

  6. [Section 6.1] Context-free Grammars Our G : S  | abS | baS | aSb | bSa | Sab | Sba | SS Recall that we proved (using structural induction) that every string generated by G contains the same number of a’s and b’s. Moreover, we can generate every string with equal number of a’s and b’s by the rules of G.

  7. [Section 6.2] Closure Properties of CFL’s • Given are two CFG’s G1 = (V1,,S1,P1) and G2 = (V2,,S2,P2). • Give a CFG G such that L(G) = L(G1) [ L(G2). • Give a CFG G such that L(G) = L(G1)L(G2). • Give a CFG G such that L(G) = L(G1)*. • Thus, context-free languages (CFL’s) are closed under union, concatenation, and Kleene’s star.

  8. [Section 6.4] Derivation Trees & Ambiguity Consider G: S  S + S | S * S | (S) | x Describe the language generated by G: Give all derivations of x + x * x :

  9. [Section 6.4] Derivation Trees We will draw derivation trees of x + x * x : A leftmost derivation is a derivation in which each derivation step uses a rule for the leftmost nonterminal symbol. Derivation trees are in 1-1 correspondence with leftmost derivations.

  10. [Section 6.4] Ambiguity A CFG G is ambiguous iff there exists a string x 2 L(G) such that x has more than one distinct derivation trees (or, equivalently, more than one leftmost derivation). Is S  S + S | S * S | (S) | x ambiguous ? Can you give an unambiguous grammar for mathematical expressions ?

  11. [Section 6.4] Ambiguity in Programming Languages Consider the following production rule : <statement>  if ( <expression> ) <statement> | if ( <expression> ) <statement> else <statement> | <otherstatement> What are the terminals and nonterminals in the above ? How to fix the problem ?

  12. [Section 6.5] CFL’s vs. Regular Languages Is every context-free language regular ? Is every regular language context-free ? (Recall the definition of regular languages.)

  13. [Section 6.6] Parsing & Normal Forms How to determine if a string is in a language generated by a CFG ? (Algorithms that do this are called parsers.) It helps to have the grammar in a normal form (more restrictions than the original definition of CFG’s). Chomsky normal form : Every rule is of the form A  or A  A1A2 where A,A1,A22 V and 2. Example : S  S + S | S * S | (S) | x Convert this grammar into Chomsky normal form.

  14. [Section 6.6] Parsing & Normal Forms How to determine if a string is in a language generated by a CFG ? (Algorithms that do this are called parsers.) It helps to have the grammar in a normal form (more restrictions than the original definition of CFG’s). Chomsky normal form : Every rule is of the form A  or A  A1A2 where A,A1,A22 V and 2. Example : S  S + S | S * S | (S) | x Convert this grammar into Chomsky normal form.

  15. [Section 6.6] Parsing & Normal Forms Greibach normal form : Every rule is of the form A  where 2 and 2 V*. Example : S  S + S | S * S | (S) | x Convert this grammar into Greibach normal form.

  16. Machines for CFG ? Deterministic or nondeterministic ? Should be able to recognize { akbk | k ¸ 0 }, { w 2 {a,b}* | w is a palindrome }, but not recognize { ww | w 2 {a,b}* }.

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