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Applied Computer Science II Chapter 2 : Context-free languages

Applied Computer Science II Chapter 2 : Context-free languages. Prof. Dr. Luc De Raedt Institut für Informatik Albert-Ludwigs Universität Freiburg Germany. Overview. Context free grammars Pushdown Automata Equivalence of PDAs and CFGs Non-context free grammars Pumping lemma.

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Applied Computer Science II Chapter 2 : Context-free languages

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  1. Applied Computer Science IIChapter 2 : Context-free languages Prof. Dr. Luc De Raedt Institut für Informatik Albert-Ludwigs Universität Freiburg Germany

  2. Overview • Context free grammars • Pushdown Automata • Equivalence of PDAs and CFGs • Non-context free grammars • Pumping lemma

  3. Context free languages • Extend regular languages • First studied for natural languages • Often used in computer languages • Compilers • Parsers • Pushdown automata

  4. Key Concepts

  5. Definition Context free grammar

  6. Parsing • Construct meaning (parse tree)

  7. Constructing CFGs • As the union of simpler CFGs

  8. Constructing CFGs • When given a DFA

  9. Constructing CFGs • Languages consisting of “linked” strings

  10. Constructing CFGs • Strings that may contain structures that appear recursively as part of other (or the same) structures

  11. Ambiguity • If generates the same string in several wasy, then the grammar is ambiguous • E.g. • Grammar does not capture usual precedence relations • One of the main problems in natural language processing • “the boy touches the girl with the flower”

  12. Defining ambiguity • Leftmostderivation : • At every step in the derivation the leftmost variable is replaced • A string is derived ambiguously in a CFG if it has two or more different leftmost derivations • A grammar is ambiguous if it generates some string ambiguously • Some context free languages are inherently ambiguous, ie. every grammar for the language is ambiguous

  13. Chomsky Normal Form

  14. Pushdown automata • Schema of a finite automaton

  15. Pushdown automaton • Includes a stack • Push something on top of stack • Pop something from top of stack • Last in first out principle • As in cafeteria – tray

  16. An example PDA

  17. Formal definition

  18. Computation with PDAs

  19. Another example • Non determinism essential for this language

  20. Another example

  21. A CFL accepts a string if there exists a derivation of the string • Involves intermediate strings • Represent intermediate strings on PDA

  22. Substitute variables by strings • Replace top variable by string

  23. Resulting PDA

  24. A construction to substitute variable by a string

  25. Pumping lemma

  26. Overview • Context free grammars • Pushdown Automata • Equivalence of PDAs and CFGs • Non-context free grammars • Pumping lemma

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