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Fall 2003Costas Busch - RPI1 Properties of Context-Free languages
Fall 2003Costas Busch - RPI2 Context-free languages are closed under: Union is context free is context-free Union
Fall 2003Costas Busch - RPI3 Example Union LanguageGrammar
Fall 2003Costas Busch - RPI4 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
Fall 2003Costas Busch - RPI5 Context-free languages are closed under: Concatenation is context free is context-free Concatenation
Fall 2003Costas Busch - RPI6 Example Concatenation LanguageGrammar
Fall 2003Costas Busch - RPI7 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
Fall 2003Costas Busch - RPI8 Context-free languages are closed under: Star-operation is context freeis context-free Star Operation
Fall 2003Costas Busch - RPI9 Example Language Grammar Star Operation
Fall 2003Costas Busch - RPI10 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
Fall 2003Costas Busch - RPI11 Negative Properties of Context-Free Languages
Fall 2003Costas Busch - RPI12 Context-free languages are not closed under: intersection is context free not necessarily context-free Intersection
Fall 2003Costas Busch - RPI13 Example Context-free: NOT context-free Intersection
Fall 2003Costas Busch - RPI14 Context-free languages are not closed under: complement is context freenot necessarily context-free Complement
Fall 2003Costas Busch - RPI15 NOT context-free Example Context-free: Complement
Fall 2003Costas Busch - RPI16 Intersection of Context-free languages and Regular Languages
Fall 2003Costas Busch - RPI17 The intersection of a context-free language and a regular language is a context-free language context free regular context-free
Fall 2003Costas Busch - RPI18 for NPDA DFA Construct a new NPDA machine that accepts Machine context-free regular simulates in parallel and
Fall 2003Costas Busch - RPI19 transition NPDADFA transition NPDA
Fall 2003Costas Busch - RPI20 transition NPDADFA transition NPDA
Fall 2003Costas Busch - RPI21 initial state NPDADFA Initial state NPDA
Fall 2003Costas Busch - RPI22 final state final states NPDADFA final states NPDA
Fall 2003Costas Busch - RPI23 Example: NPDA context-free
Fall 2003Costas Busch - RPI24 DFA regular
Fall 2003Costas Busch - RPI25 Automaton for: NPDA context-free
Fall 2003Costas Busch - RPI26 simulates in parallel and accepts stringif and only if accepts string and accepts string In General:
Fall 2003Costas Busch - RPI27 Therefore: is NPDA is context-free
Fall 2003Costas Busch - RPI28 Applications of Regular Closure
Fall 2003Costas Busch - RPI29 The intersection of a context-free language and a regular language is a context-free language context free regular context-free Regular Closure
Fall 2003Costas Busch - RPI30 An Application of Regular Closure Prove that: is context-free
Fall 2003Costas Busch - RPI31 We know: is context-free
Fall 2003Costas Busch - RPI32 is regular We also know:
Fall 2003Costas Busch - RPI33 regularcontext-free is context-free (regular closure)
Fall 2003Costas Busch - RPI34 Another Application of Regular Closure Prove that: is not context-free
Fall 2003Costas Busch - RPI35 context-freeregularcontext-free If is context-free Then Impossible!!! Therefore, is not context free (regular closure)
PDAs Accept Context-Free Languages
CS2303-THEORY OF COMPUTATION Closure Properties of Regular Languages
6.2 Closure Properties of CFL's
Fall 2006Costas Busch - RPI1 Reductions. Fall 2006Costas Busch - RPI2 Problem is reduced to problem If we can solve problem then we can solve problem.
Fall 2006Costas Busch - RPI1 Time Complexity. Fall 2006Costas Busch - RPI2 Consider a deterministic Turing Machine which decides a language.
Fall 2006Costas Busch - RPI1 A Universal Turing Machine.
Fall 2006Costas Busch - RPI1 Turing Machines. Fall 2006Costas Busch - RPI2 The Language Hierarchy Regular Languages Context-Free Languages ? ?
Linear Bounded Automata LBAs
Fall 2006Costas Buch - RPI1 Simplifications of Context-Free Grammars.
Fall 2006Costas Busch - RPI1 Decidable Languages.
Properties of Regular Languages
Lecture 8 From NFA to Regular Language. Induction on k= # of states other than initial and final states K=0 a a* b a c d c*a(d+bc*a)*
Automata and Formal Languages Tim Sheard 1 Lecture 9 Reasoning with DFAs.
NFAs Sipser 1.2 (pages 47–54). CS 311 Fall Recall… Last time we showed that the class of regular languages is closed under: –Complement –Union.
Lecture 3UofH - COSC Dr. Verma 1 COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 3.
Prof. Busch - LSU1 Properties of Context-Free languages.
Costas Busch - RPI1 Single Final State for NFAs. Costas Busch - RPI2 Any NFA can be converted to an equivalent NFA with a single final state.
CS 310 – Fall 2006 Pacific University CS310 Finite Automata Sections:1.1 page 44 September 8, 2006.
Costas Busch - RPI1 Pushdown Automata PDAs. Costas Busch - RPI2 Pushdown Automaton -- PDA Input String Stack States.
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