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

A composable-reset DFA (CR-DFA) is a five-tuple, Question 2 (10 pt): In thi...

Question 2 (10 pt): In this question we look at an extension to DFAs. A composable-reset DFA (CR-DFA) is a five-tuple, (Q,S,d,q0,F) where: – Q is the set of states, – S is the alph

Shell script, shell script to print table in given range

shell script to print table in given range

Complement - operations on languages, The fact that SL 2 is closed under i...

The fact that SL 2 is closed under intersection but not under union implies that it is not closed under complement since, by DeMorgan's Theorem L 1 ∩ L 2 = We know that

Ogdens lemma, proof ogdens lemma .with example i am not able to undestand ...

proof ogdens lemma .with example i am not able to undestand the meaning of distinguished position .

Generalization of the interpretation of local automata, The generalization ...

The generalization of the interpretation of strictly local automata as generators is similar, in some respects, to the generalization of Myhill graphs. Again, the set of possible s

Local myhill graphs, Myhill graphs also generalize to the SLk case. The k-f...

Myhill graphs also generalize to the SLk case. The k-factors, however, cannot simply denote edges. Rather the string σ 1 σ 2 ....... σ k-1 σ k asserts, in essence, that if we hav

Construct a regular expression, Given any NFA A, we will construct a regula...

Given any NFA A, we will construct a regular expression denoting L(A) by means of an expression graph, a generalization of NFA transition graphs in which the edges are labeled with

Myhill-nerode theorem, The Myhill-Nerode Theorem provided us with an algori...

The Myhill-Nerode Theorem provided us with an algorithm for minimizing DFAs. Moreover, the DFA the algorithm produces is unique up to isomorphism: every minimal DFA that recognizes

Myhill graphs, Another way of representing a strictly 2-local automaton is ...

Another way of representing a strictly 2-local automaton is with a Myhill graph. These are directed graphs in which the vertices are labeled with symbols from the input alphabet of

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