CS 3240: Languages and Computation

Context-Free Languages

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 | AAA 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