closure properties of languages table

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Closure Properties of Decidable Languages

Closure Properties of Decidable Languages Decidable languages are closed under ∪ ° * ∩ and complement Example: Closure under ∪ Need to show that union of 2 decidable L’s is also decidable Let M1 be a decider for L1 and M2 a decider for L2 A decider M for L1 ∪ L2: On input w:

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Closure Properties of Regular Languages

Recall a closure property is a statement that a certain operation on languages when applied to languages in a class (e g the regular languages) produces a result that is also in that class For regular languages we can use any of its representations to prove a closure property

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Closure Properties of Regular Languages

Formally: = Σ* - L L Σ* Given a language L ⊆ Σ* the complement of that language (denoted L) is the language of all strings in Σ* that aren\'t in L Formally: = Σ* - L L Σ* As we saw a few minutes ago a regular language is a language recognized by some DFA (or NFA) Question: If L is a regular language is L necessarily a regular language?

How do you find a smallest closure property?
You can’t find a “smallest.” A 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. Why Closure Properties?
How do you close a regular language?
Regular languages are closed under the following operations: Kleen Closure: RS is a regular expression whose language is L, M. R* is a regular expression whose language is L*. Positive closure: RS is a regular expression whose language is L, M. is a regular expression whose language is .
What are closure properties on regular languages?
Closure properties on regular languages are defined as certain operations on regular language that are guaranteed to produce regular language. Closure refers to some operation on a language, resulting in a new language that is of the same “type” as originally operated on i.e., regular. Regular languages are closed under the following operations:
Are regular languages closed under difference?
No, regular languages are not closed under difference. If L1,L2 and L1-L2 may not necessarily be a regular language. Q2. How can closure properties be proven for regular languages?

Closure Properties

Recall a closure property is a statement that a certain operation on languages, when applied to languages in a class (e.g., the regular languages), produces a result that is also in that class. For regular languages, we can use any of its representations to prove a closure property. infolab.stanford.edu

Closure Under Union

If L and M are regular languages, so is L  M. Proof: Let L and M be the languages of regular expressions R and S, respectively. Then R+S is a regular expression whose language is L  M. infolab.stanford.edu

Closure Under Concatenation and Kleene Closure

Same idea: RS is a regular expression whose language is LM. R* is a regular expression whose language is L*. infolab.stanford.edu

Closure Under Intersection

If L and M are regular languages, then so is L  M. Proof: Let A and B be DFA’s whose languages are L and M, respectively. Construct C, the product automaton of A and B. Make the final states of C be the pairs consisting of final states of both A and B. infolab.stanford.edu

Closure Under Difference

If L and M are regular languages, then so is L – M = strings in L but not M. Proof: Let A and B be DFA’s whose languages are L and M, respectively. Construct C, the product automaton of A and B. Make the final states of C be the pairs where A-state is final but B-state is not. infolab.stanford.edu

Closure Under Complementation

The complement of a language L (with respect to an alphabet Σ such that Σ* contains L) is Σ* – L. Since Σ* is surely regular, the complement of a regular language is always regular. infolab.stanford.edu

Induction: If E is

F+G, then ER = FR + GR. FG, then ER = GRFR F*, then ER = (FR)*. infolab.stanford.edu

Closure Under Homomorphism

If L is a regular language, and h is a homomorphism on its alphabet, then h(L) = {h(w) w is in L} is also a regular language. Proof: Let E be a regular expression for L. Apply h to each symbol in E. Language of resulting RE is h(L). infolab.stanford.edu

Example – Continued

abε* + ε(ab)* can be simplified. ε* = ε, so abε* = abε. ε is the identity under concatenation.  That is, εE = Eε = E for any RE E. Thus, abε* + ε(ab)* = abε + ε(ab)* = ab + (ab)*. Finally, L(ab) is contained in L((ab)*), so a RE for h(L) is (ab)*. infolab.stanford.edu

Proof – (2)

The transitions for B are computed by applying h to an input symbol a and seeing where A would go on sequence of input symbols h(a). Formally, δB(q, a) = δA(q, h(a)). infolab.stanford.edu

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