CS 3240: Languages and Computation Context-Free Languages

A single step derivation “  ” consist of the substitution of a variable by a string according to a substitution rule in R Note that rules use single arrows “  ”, while derivations themselves use double arrows “  ” A sequence of several derivations (or none) is indicated by “  * ”  Previous example: “S  * aabbaa” L is a Context Free Language if and only if there is a context free grammar G=(V, Σ, P, S) such that L = L(G) = { w | w  Σ * and S  * w }

The language generated by a grammar

Some Remarks The language L(G) = { w | w  Σ* and S  * w } contains only strings of terminals, not variables. Notation: We can agglomerate several rules for one variable: A  B A  01 by A  B | 01 | AA A  AA What is the CFG ({S},{(,)},P, S) that produces the language of correct parentheses like (), (()), or ()(())?  Answer: S→ (S) | SS | 

Parse trees

Yield of a parse tree

From tree to derivation

From derivations to recursive inference

ambiguity

Removing ambiguity

Ambiguity and leftmost derivations

Inherent ambiguity