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Context Free Grammars & Parsing

Context Free Grammars & Parsing. CPSC 388 Fall 2001 Ellen Walker Hiram College. Syntax Analysis. Program structure is sequence of tokens Structure is too complex to be regular Parentheses & braces are nested If-else need to match up Structure is context free. Context Free Grammar.

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Context Free Grammars & Parsing

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  1. Context Free Grammars & Parsing CPSC 388 Fall 2001 Ellen Walker Hiram College

  2. Syntax Analysis • Program structure is sequence of tokens • Structure is too complex to be regular • Parentheses & braces are nested • If-else need to match up • Structure is context free

  3. Context Free Grammar • Rules more general than RG: • Right side can be any sequence of terminals, non-terminals • Left side must be a single non-terminal • S -> abScTdef (context free) • AS -> bSAc (not CF) • Every Regular Grammar is also a Context Free Grammar

  4. Context Free Language • The language of a CFG is the set of all strings that can be generated by repeatedly applying rules to the start symbol. • The sequence of rule applications is called a derivation

  5. Example CFG • This grammar generates a*b* where the number of a’s = number of b’s S -> aSb | e • Derivation of aaabbb: S -> aSb -> aaSbb -> aaaSbbb -> aaabbb

  6. Same # a’s as b’s • Grammar • S -> aSb | bSa | abS | baS |  • Derivation • “aabbabab” • S-> aSb -> aabSb -> aabbaSb ->aabbabaSb -> aabbabab

  7. Formal Description of a CFG • A set of terminals (e.g. {a,b}) • A set of non-terminals (e.g. {S}) • A start non-terminal (e.g. S) • A set of rules with a single nonterminal on the left side.

  8. Notation • In formal (theory) CFL’s, non-terminals are usually capital letters E -> E O E • In programming languages & compiler CFL’s non-terminals are usually names, sometimes in angle brackets (BNF) <expr> -> <expr> <op> <expr> • Terminals can be single or multiple-character symbols, or token types

  9. Parse Tree • S (start symbol) is the root • String is the leaves • Intermediate nodes are non-terminals

  10. Parse Tree Example S-> aSb -> aabSb -> aabbaSb ->aabbab

  11. Another Grammar • S -> abScB | e • B-> bB | b • Strings to generate: • abcb • ababccb • Give both derivation and parse tree • Find a string of length 5 that is accepted • Find a string with 2 c’s that is accepted

  12. Grammar for Expression • E => E O E • E => ( E ) • E => a • E => b • O => + • O => - • O => *

  13. Example Derivation • E-> E O E • -> ( E ) O E • -> ( E O E ) O E • -> ( a O E ) O E • -> ( a + E ) O E • -> ( a + b ) O E • -> ( a + b ) * E • -> ( a + b) * a

  14. More CFL’s • Palindromes over {a,b,c} • Strings with twice as many a’s as b’s • Strings with 3 more a’s than b’s (hint: create a non-terminal that generates equal #’s of a’s and b’s, then add 3 more a’s)

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