Applying the pumping lemma, Theory of Computation

Assignment Help:

Applying the pumping lemma is not fundamentally di?erent than applying (general) su?x substitution closure or the non-counting property. The pumping lemma is a little more complicated-rather than just the single universal quanti?er ("for all languages L") and single existential quanti?er ("there exists n"), we have a nest of alternating quanti?ers (denoting "for all" as ∀ and "there exists" as ∃):

(∀L)[L regular ⇒

(∃n)[

(∀x)[x ∈ L and |x| ≥ n ⇒

(∃u, v,w)[x = uvw and

|uv| ≤ n and

|v| ≥ 1 and

(∀i ≥ 0)[uviw ∈ L]]]]].

Just as with the lemmas for the local languages, we will approach this as an adversary game. Our proof will consist of a strategy for showing that L fails to satisfy the pumping lemma. Our choices are the "for all"s; the "there exists"s are our adversary's choices. There are just a few more rounds in this game than there were in the lemmas for the local languages. The key things are being clear about which are our choices and which are the adversary's and making sure that our strategy accounts for every legal choice the adversary
might make.

The game starts with our choice of the L we wish to prove to be non regular. Our adversary then chooses some n, we choose a string x ∈ L of length at least n, etc. We win if, at the end of this process, we can choose i such that uviw ∈ L. Of course, our strategy at each step will depend on the choices our adversary has made.

What we end up with is a proof by contradiction. For instance:

To show that Lab = {ajbj| j ≥ 0} is not regular.


Related Discussions:- Applying the pumping lemma

Give a strictly 2-local automaton, Let L 3 = {a i bc j | i, j ≥ 0}. Give ...

Let L 3 = {a i bc j | i, j ≥ 0}. Give a strictly 2-local automaton that recognizes L 3 . Use the construction of the proof to extend the automaton to one that recognizes L 3 . Gi

Turing machine, design a turing machine that accepts the language which con...

design a turing machine that accepts the language which consists of even number of zero''s and even number of one''s?

Automata answer, build a TM that enumerate even set of even length string o...

build a TM that enumerate even set of even length string over a

Transition and path functions, When an FSA is deterministic the set of trip...

When an FSA is deterministic the set of triples encoding its edges represents a relation that is functional in its ?rst and third components: for every q and σ there is exactly one

Automata, automata of atm machine

automata of atm machine

Programming languages, Different types of applications and numerous program...

Different types of applications and numerous programming languages have been developed to make easy the task of writing programs. The assortment of programming languages shows, dif

Kleene closure, One might assume that non-closure under concatenation would...

One might assume that non-closure under concatenation would imply non closure under both Kleene- and positive closure, since the concatenation of a language with itself is included

Union, Intuitively, closure of SL 2 under intersection is reasonably easy ...

Intuitively, closure of SL 2 under intersection is reasonably easy to see, particularly if one considers the Myhill graphs of the automata. Any path through both graphs will be a

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