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Recursively Enumerable Recursive Languages

1 CS 301 - Lecture 23 Recursive and Recursively Enumerable Languages Fall 2008 Review Languages and Grammars Alphabets, strings, Languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of NFA and DFA Regular Expressions Regular Grammars Properties of Regular Languages Languages that are not regular and the pumping lemma Context Free Languages Context Free Grammars Derivations: leftmost, rightmost and derivation trees Parsing and ambiguity Simplifications and Normal Forms Nondeterministic Pushdown Automata Pushdown Automata and Context Free Grammars Deterministic Pushdown Automata Pumping Lemma for context free grammars Properties of Context Free Grammars Turing Machines Definition, Accepting Languages , and Computing Functions Combining Turing Machines and Turing s Thesis Turing Machine Variations Universal Turing Machine and Linear Bounded Automata Today: Recursive and Recursively Enumerable Languages Recursively Enumerable and Recursive Languages Definition: A language is Recursively Enumerable if some Turing machine accepts it 2 For string.

Proof Using Uncountable Sets A set is uncountable if it is not countable Definition: Theorem: Let be an infinite countable set The powerset of is uncountable 2SS S Proof: Since is countable, we can write S S={s 1,s 2,s 3,…} Elements of S

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