Class of local languages is not closed under union, Theory of Computation

Assignment Help:

Both L1 and L2 are SL2. (You should verify this by thinking about what the automata look like.)

2360_Class of local languages is not closed under union.png

We claim that L1 ∪ L2 ∈ SL2. To see this, suppose, by way of contradiction, that it was SL2. Then it would satisfy Suffx Substitution Closure. But abb ∈ L1 ∪ L2, since it is in L1, and bba ∈ L1 ∪ L2, since it is in L2, and, by Suffx Substitution Closure, this implies that aba ∈ L1 ∪ L2, which is not the case. Hence, L1 ∪ L2 is not SL2.

It's important to not misunderstand the weight of this result. It does not show that every union of two SL2 languages is not SL2-after all, the union of any SL2 language with itself must, necessarily, be SL2. It is easy to come up with pairs of distinct SL2 languages which yield an SL2 union as well,

1737_Class of local languages is not closed under union1.png

What it does say is that one can't count on the union of SL2 languages being SL2. This is in contrast to the closure result for intersection, which lets us build up languages out of simpler SL2 languages using intersection with con?dence that the result will be SL2. Showing that our desired language is the union of some simple SL2 languages doesn't help us. The result may or may not be SL.


Related Discussions:- Class of local languages is not closed under union

Mealy machine, Construct a Mealy machine that can output EVEN or ODD Accord...

Construct a Mealy machine that can output EVEN or ODD According to the total no. of 1''s encountered is even or odd.

Differentiate between dfa and nfa, Differentiate between DFA and NFA. Conve...

Differentiate between DFA and NFA. Convert the following Regular Expression into DFA. (0+1)*(01*+10*)*(0+1)*. Also write a regular grammar for this DFA.

Deterministic finite automata, conversion from nfa to dfa 0 | 1 ____...

conversion from nfa to dfa 0 | 1 ___________________ p |{q,s}|{q} *q|{r} |{q,r} r |(s) |{p} *s|null |{p}

Qbasic, Ask question #Minimum 100 words accepte

Ask question #Minimum 100 words accepte

Suffix substitution , Exercise Show, using Suffix Substitution Closure, tha...

Exercise Show, using Suffix Substitution Closure, that L 3 . L 3 ∈ SL 2 . Explain how it can be the case that L 3 . L 3 ∈ SL 2 , while L 3 . L 3 ⊆ L + 3 and L + 3 ∈ SL

Computation and languages, When we study computability we are studying prob...

When we study computability we are studying problems in an abstract sense. For example, addition is the problem of, having been given two numbers, returning a third number that is

Numerical integration, what problems are tackled under numerical integratio...

what problems are tackled under numerical integration

Designing finite automata, a finite automata accepting strings over {a,b} e...

a finite automata accepting strings over {a,b} ending in abbbba

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd