Non-regular languages, Theory of Computation

Suppose A = (Q,Σ, T, q0, F) is a DFA and that Q = {q0, q1, . . . , qn-1} includes n states. Thinking of the automaton in terms of its transition graph, a string x is recognized by the automaton iff there is a path through the graph from q0 to some qf ∈ F that is labeled x, i.e., if δ(q0, x) ∈ F. Suppose x ∈ L(A) and |x| = l. Then there is a path l edges long from q0 to qf . Since the path traverses l edges, it must visit l + 1 states.

756_Non-Regular Languages.png

Suppose, now, that l ≥ n. Then the path must visit at least n+1 states. But there are only n states in Q; thus, the path must visit at least one state at least twice. (This is an application of the pigeon hole principle: If one places k objects into n bins, where k > n, then at least one bin must contain at least two objects.)

1213_Non-Regular Languages1.png

Thus, whenever |x| ≥ n the path labeled w will have a cycle. We can break the path into three segments: x = uvw, where

• there is a path (perhaps empty) from q0 to p labeled u (i.e., δ(q0, u) = p),

• there is a (non-empty) path from p to p (a cycle) labeled v (i.e., δ(p, v) = p),

• there is a path (again, possibly empty) from p to qf labeled w (i.e., δ(p,w) = qf ).

But if there is a path from q0 to p labeled u and one from p to qf labeled w then there is a path from q0 to qf labeled uw in which we do not take the loop labeled v, which is to say uw ∈ L(A). Formally

δ(q0, uvvw) = δ(δ(q0, u), w) =  δ(p, w) = qf =  F

Similarly, we can take the v loop more than once:

δ(q0, uvvw) = δ(δ(δ(δ(q0, u), v), v),w)
= δ(δ(δ(p, v), v),w)

= δ(δ(p, v),w)

= δ(p,w) = qf ∈ F.

In fact, we can take it as many times as we like. Thus, uvi

w ∈ L(A) for all i.

This implies, then, that if the language recognized by a DFA with n states includes a string of length at least n then it contains in?nitely many closely related strings as well. We can strengthen this by noting (as a consequence of the pigeon hole principle again) that the length of the path from q0 to the ?rst time a state repeats (i.e., the second occurrence of p) must be no greater than n. Thus |uv| ≤ n.

Posted Date: 3/21/2013 1:37:55 AM | Location : United States







Related Discussions:- Non-regular languages, Assignment Help, Ask Question on Non-regular languages, Get Answer, Expert's Help, Non-regular languages Discussions

Write discussion on Non-regular languages
Your posts are moderated
Related Questions
Distinguish between Mealy and Moore Machine? Construct a Mealy machine that can output EVEN or ODD According to the total no. of 1's encountered is even or odd.

shell script to print table in given range

Sketch an algorithm for the universal recognition problem for SL 2 . This takes an automaton and a string and returns TRUE if the string is accepted by the automaton, FALSE otherwi

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 complica

De?nition (Instantaneous Description) (for both DFAs and NFAs) An instantaneous description of A = (Q,Σ, δ, q 0 , F) , either a DFA or an NFA, is a pair h q ,w i ∈ Q×Σ*, where

The fundamental idea of strictly local languages is that they are speci?ed solely in terms of the blocks of consecutive symbols that occur in a word. We'll start by considering lan

Explain Theory of Computation ,Overview of DFA,NFA, CFG, PDA, Turing Machine, Regular Language, Context Free Language, Pumping Lemma, Context Sensitive Language, Chomsky Normal For

turing machine for prime numbers

The language accepted by a NFA A = (Q,Σ, δ, q 0 , F) is NFAs correspond to a kind of parallelism in the automata. We can think of the same basic model of automaton: an inpu

Ask queyystion #Minimum 100 words accepted#