Myhill-nerode theorem, Theory of Computation

Assignment Help:

This close relationship between the SL2 languages and the recognizable languages lets us use some of what we know about SL2 to discover properties of the recognizable languages. Because they are SL2 languages, the runs of an automaton A (and, equivalently, the strings of pairs licensed by G2A) will satisfy the 2-suffix substitution closure property. This means that every recognizable language L is a homomorphic image of some language L′ (over an alphabet Σ′ , say) for which

                                                             u′1σ′v′1 ∈ L′ and u′2 σ′v′2 ∈ L′⇒ u′1σ′v′2( and u′2σ′v′1) ∈ L′.

Moreover, u′1σ′v′1 ∈ L′ and u′1σ′v′2 ∈ L′⇒ u′2σ′v′2 ∈ L′

The hypothetical u′1σ′ and u′2σ′ are indistinguishable by the language. Any continuation that extends one to a string in L′ will also extend the other to a string in L′ ; any continuation that extends one to a string not in L′ will extend the other to a string not in L′.

For the SL2 language L′ the strings that are indistinguishable in this way are marked by their ?nal symbol. Things are not as clear for the recognizable language L because the homomorphism may map many symbols of Σ′ to the same symbol of Σ. So it will not generally be the case that we can easily identify the sets of strings that are indistinguishable in this way. But they will, nevertheless, exist. There will be pairs of strings u1 and u2 - namely the homomorphic images of the pairs u′1σ′ and u′2σ′-for which any continuation v, it will be the case that u1v ∈ L iff u2v ∈ L.

This equivalence between strings (in the sense of being indistinguishable by the language in this way) is the key to characterizing the recognizable languages purely in terms of the strings they contain in a way analogous to the way suffix substitution closure characterizes the SL2.


Related Discussions:- Myhill-nerode theorem

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

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

Non - sl languages, Application of the general suffix substitution closure ...

Application of the general suffix substitution closure theorem is slightly more complicated than application of the specific k-local versions. In the specific versions, all we had

# Help, #Your company has 25 licenses for a computer program, but you disco...

#Your company has 25 licenses for a computer program, but you discover that it has been copied onto 80 computers. You informed your supervisor, but he/she is not willing to take an

Chomsky-schutzenberger, The upper string r ∈ Q+ is the sequence of states v...

The upper string r ∈ Q+ is the sequence of states visited by the automaton as it scans the lower string w ∈ Σ*. We will refer to this string over Q as the run of A on w. The automa

Nfas with e-transitions, We now add an additional degree of non-determinism...

We now add an additional degree of non-determinism and allow transitions that can be taken independent of the input-ε-transitions. Here whenever the automaton is in state 1

Automata, As we are primarily concerned with questions of what is and what ...

As we are primarily concerned with questions of what is and what is not computable relative to some particular model of computation, we will usually base our explorations of langua

Fsa as generators, The SL 2 languages are speci?ed with a set of 2-factors...

The SL 2 languages are speci?ed with a set of 2-factors in Σ 2 (plus some factors in {?}Σ and some factors in Σ{?} distinguishing symbols that may occur at the beginning and en

Mapping reducibility, Can you say that B is decidable? If you somehow know...

Can you say that B is decidable? If you somehow know that A is decidable, what can you say about B?

Closure properties of recognizable languages, We got the class LT by taking...

We got the class LT by taking the class SL and closing it under Boolean operations. We have observed that LT ⊆ Recog, so certainly any Boolean combination of LT languages will also

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