CS 3240: Languages and Computation

1. CS 3240: Languages and Computation Context-Free Languages

2. 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 }

3. The language generated by a grammar

4. 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 | 

5. Parse trees

6. Yield of a parse tree

8. From tree to derivation

10. From derivations to recursive inference

11. ambiguity

12. Removing ambiguity

13. Ambiguity and leftmost derivations

14. Inherent ambiguity