Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

Example Grammar :

Definition: Context-Free Grammars Grammar Productions of the form: String of variables and terminals VariablesTerminal symbols Start variable Variable

Leftmost derivation: Rightmost derivation:

Derivation Tree

yield Derivation Tree

Simplifications of Context-Free Grammars

Example

Normal Forms for Context-free Grammars

Chomsky Normal Form Each productions has form: variable or terminal

Introduce variables for terminals:

Introduce intermediate variable:

Final grammar in Chomsky Normal Form: Initial grammar

Then, for every symbol : In productions: replace with Add production New variable:

Replace any production with New intermediate variables:

Greinbach Normal Form All productions have form: symbolvariables

Pushdown Automata PDAs

Pushdown Automaton -- PDA Input String Stack States

The States Input symbol Pop symbol Push symbol

top input stack Replace

Push top input stack

Pop top input stack

No Change top input stack

Pop top input stack Empty Stack empty The automaton HALTS No possible transition after

Pop top input stack A Possible Transition

Non-Determinism PDAs are non-deterministic Allowed non-deterministic transitions

Formalities for PDAs

Transition function:

Formal Definition Pushdown Automaton (PDA) States Input alphabet Stack alphabet Transition function Final states Stack start symbol Initial state

Instantaneous Description Current state Remaining input Current stack contents

A computation:

Deterministic PDAs - DPDAs

Deterministic PDA: DPDA Allowed transitions: (deterministic choices)

Allowed transitions: (deterministic choices)

Not allowed: (non deterministic choices)

PDAs Have More Power than DPDAs

Positive Properties of Context-Free languages

Context-free languages are closed under: Union is context free is context-free Union

Context-free languages are closed under: Concatenation is context free is context-free Concatenation

Context-free languages are closed under: Star-operation is context freeis context-free Star Operation

Negative Properties of Context-Free Languages

Context-free languages are not closed under: intersection is context free not necessarily context-free Intersection

Context-free languages are not closed under: complement is context freenot necessarily context-free Complement

Intersection of Context-free languages and Regular Languages

The intersection of a context-free language and a regular language is a context-free language context free regular context-free

An Application of Regular Closure Prove that: is context-free

Decidable Properties of Context-Free Languages

Membership Question: for context-free grammar find if string Membership Algorithms: Parsers Exhaustive search parser CYK parsing algorithm

Empty Language Question: for context-free grammar find if Algorithm: 1.Remove useless variables 2.Check if start variable is useless

Infinite Language Question: for context-free grammar find if is infinite Algorithm: 1. Remove useless variables 3. Create dependency graph for variables 4. If there is a loop in the dependency graph then the language is infinite 2. Remove unit and productions