Prof. Busch - LSU1 Properties of Context-Free languages

Prof. Busch - LSU2 Context-free languages are closed under: Union is context free is context-free Union

Prof. Busch - LSU3 Example Union LanguageGrammar

Prof. Busch - LSU4 In general: The grammar of the union has new start variable and additional production For context-free languages with context-free grammars and start variables

Prof. Busch - LSU5 Context-free languages are closed under: Concatenation is context free is context-free Concatenation

Prof. Busch - LSU6 Example Concatenation LanguageGrammar

Prof. Busch - LSU7 In general: The grammar of the concatenation has new start variable and additional production For context-free languages with context-free grammars and start variables

Prof. Busch - LSU8 Context-free languages are closed under: Star-operation is context freeis context-free Star Operation

Prof. Busch - LSU9 Example Language Grammar Star Operation

Prof. Busch - LSU10 In general: The grammar of the star operation has new start variable and additional production For context-free language with context-free grammar and start variable

Prof. Busch - LSU11 Negative Properties of Context-Free Languages

Prof. Busch - LSU12 Context-free languages are not closed under: intersection is context free not necessarily context-free Intersection

Prof. Busch - LSU13 Example Context-free: NOT context-free Intersection

Prof. Busch - LSU14 Context-free languages are not closed under: complement is context freenot necessarily context-free Complement

Prof. Busch - LSU15 NOT context-free Example Context-free: Complement

Prof. Busch - LSU16 Intersection of Context-free languages and Regular Languages

Prof. Busch - LSU17 The intersection of a context-free language and a regular language is a context-free language context free regular context-free

Prof. Busch - LSU18 for NPDA DFA Construct a new NPDA machine that accepts Machine context-free regular simulates in parallel and

Prof. Busch - LSU19 transition NPDADFA transition NPDA

Prof. Busch - LSU20 transition NPDADFA transition NPDA

Prof. Busch - LSU21 initial state NPDADFA Initial state NPDA

Prof. Busch - LSU22 final state final states NPDADFA final states NPDA

Prof. Busch - LSU23 Example: NPDA context-free

Prof. Busch - LSU24 DFA regular

Prof. Busch - LSU25 Automaton for: NPDA context-free

Prof. Busch - LSU26 simulates in parallel and accepts stringif and only if accepts string and accepts string In General:

Prof. Busch - LSU27 Therefore: is NPDA is context-free

Prof. Busch - LSU28 Applications of Regular Closure

Prof. Busch - LSU29 The intersection of a context-free language and a regular language is a context-free language context free regular context-free Regular Closure

Prof. Busch - LSU30 An Application of Regular Closure Prove that: is context-free

Prof. Busch - LSU31 We know: is context-free

Prof. Busch - LSU32 is regular We also know:

Prof. Busch - LSU33 regularcontext-free is context-free (regular closure)

Prof. Busch - LSU34 Another Application of Regular Closure Prove that: is not context-free

Prof. Busch - LSU35 context-freeregularcontext-free If is context-free Then Impossible!!! Therefore, is not context free (regular closure)