Properties of ContextFree languages Costas Busch LSU 1

Properties of Context-Free languages Costas Busch - LSU 1

Union Context-free languages are closed under: Union is context free is context-free Costas Busch - LSU 2

Example Language Grammar Union Costas Busch - LSU 3

In general: For context-free languages with context-free grammars and start variables The grammar of the union has new start variable and additional production Costas Busch - LSU 4

Concatenation Context-free languages Concatenation are closed under: is context free is context-free Costas Busch - LSU 5

Example Language Grammar Concatenation Costas Busch - LSU 6

In general: For context-free languages with context-free grammars and start variables The grammar of the concatenation has new start variable and additional production Costas Busch - LSU 7

Star Operation Context-free languages Star-operation are closed under: is context free is context-free Costas Busch - LSU 8

Example Language Grammar Star Operation Costas Busch - LSU 9

In general: For context-free language with context-free grammar and start variable The grammar of the star operation has new start variable and additional production Costas Busch - LSU 10

Negative Properties of Context-Free Languages Costas Busch - LSU 11

Intersection Context-free languages intersection are not closed under: is context free Costas Busch - LSU not necessarily context-free 12

Example Context-free: Intersection NOT context-free Costas Busch - LSU 13

Complement Context-free languages complement are not closed under: is context free not necessarily context-free Costas Busch - LSU 14

Example Context-free: Complement NOT context-free Costas Busch - LSU 15

Intersection of Context-free languages and Regular Languages Costas Busch - LSU 16

The intersection of a context-free language and a regular language is a context-free language context free regular context-free Costas Busch - LSU 17

Machine NPDA for DFA for regular context-free Construct a new NPDA machine that accepts simulates in parallel Costas Busch - LSU and 18

NPDA DFA transition NPDA transition Costas Busch - LSU 19

NPDA DFA transition NPDA transition Costas Busch - LSU 20

NPDA DFA initial state NPDA Initial state Costas Busch - LSU 21

NPDA DFA final states NPDA final states Costas Busch - LSU 22

Example: context-free NPDA Costas Busch - LSU 23

regular DFA Costas Busch - LSU 24

context-free Automaton for: NPDA Costas Busch - LSU 25

In General: simulates in parallel accepts string and if and only if accepts string and accepts string Costas Busch - LSU 26

Therefore: is NPDA is context-free Costas Busch - LSU 27

Applications of Regular Closure Costas Busch - LSU 28

The intersection of a context-free language and a regular language is a context-free language Regular Closure context free regular context-free Costas Busch - LSU 29

An Application of Regular Closure Prove that: is context-free Costas Busch - LSU 30

We know: is context-free Costas Busch - LSU 31

We also know: is regular Costas Busch - LSU 32

context-free regular (regular closure) context-free is context-free Costas Busch - LSU 33

Another Application of Regular Closure Prove that: is not context-free Costas Busch - LSU 34

is context-free If (regular closure) Then context-free regular context-free Impossible!!! Therefore, is not context free Costas Busch - LSU 35