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lect6-DecisionPropertiesOfRL

# lect6-DecisionPropertiesOfRL - Languages ThePumpingLemma...

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1 Decision Properties of Regular  Languages General Discussion of “Properties” The Pumping Lemma Membership, Emptiness, Etc.

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2 Properties of Language Classes language class   is a set of  languages. We have one example: the regular  languages. We’ll see many more in this class. Language classes have two important  kinds of properties: 1. Decision properties. 2. Closure properties.
3 Representation of Languages Representations can be formal or informal. Example  (formal): represent a language by a  RE or DFA defining it. Example : (informal): a logical or prose  statement about its strings: {0 n 1 n  | n is a nonnegative integer} “The set of strings consisting of some number of  0’s followed by the same number of 1’s.”

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4 Decision Properties decision property   for a class of  languages is an algorithm that takes a  formal description of a language (e.g., a  DFA) and tells whether or not some  property holds. Example : Is language L empty?
5 Subtle Point : Representation  Matters You might imagine that the language is  described informally, so if my  description is “the empty language” then  yes, otherwise no. But the representation is a DFA (or a  RE that you will convert to a DFA). Can you tell if L(A) =   for DFA A?

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6 Why Decision Properties? When we talked about protocols  represented as DFA’s, we noted that  important properties of a good protocol  were related to the language of the DFA. Example : “Does the protocol terminate?”  = “Is the language finite?” Example : “Can the protocol fail?” = “Is  the language nonempty?”
7 Why Decision Properties – (2) We might want a “smallest”  representation for a language, e.g., a  minimum-state DFA or a shortest RE. If you can’t decide “Are these two  languages the same?” I.e., do two DFA’s define the same  language? You can’t find a “smallest.”

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8 Closure Properties closure property   of a language class  says that given languages in the class,  an operator (e.g., union) produces  another language in the same class. Example : the regular languages are  obviously closed under union,  concatenation, and (Kleene) closure. Use the RE representation of languages.
9 Why Closure Properties? 1. Helps construct representations. 2. Helps show (informally described)  languages not to be in the class.

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10 Example : Use of Closure Property We can easily prove L 1  = {0 n 1 n  | n >  0} is  not a regular language.
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