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

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

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

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Key Concepts

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Definition Context free grammar

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Parsing –Construct meaning (parse tree)

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Constructing CFGs As the union of simpler CFGs

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Constructing CFGs When given a DFA

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Constructing CFGs Languages consisting of “linked” strings

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

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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”

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Defining ambiguity Leftmost derivation : –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

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Chomsky Normal Form

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Pushdown automata Schema of a finite automaton

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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

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An example PDA

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Formal definition

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Computation with PDAs

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Another example Non determinism essential for this language

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Another example

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

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Substitute variables by strings Replace top variable by string

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Resulting PDA

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A construction to substitute variable by a string

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Pumping lemma

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

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