Notes-SecC - COT 4210 Section C Spring 2001 Theorem 2...

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COT 4210 Section C Spring 2001 Theorem 2 established that NFAs are no more “powerful” than DFAs in terms of the family of languages they can recognize. However, NFAs are more compact and succinct than DFAs because of the exponential relationship between the number of states required for a DFA to recognize the same language as an NFA. We will return to this issue later, when we study the algorithm for minimizing DFAs. Equivalence of NFAs to Right-linear Grammars To establish the first equivalence suggested by the Chomsky Hierarchy, we now show that the Regular languages (defined by DFAs and NFAs) is exactly the same family of languages that can be defined by Type-3 or Right-linear grammars. This is the subject of our next theorem. Theorem 3. Let M = (Q, Σ , δ ,Q 0 , A) be an arbitrary NFA, then there is a Right-linear grammar G = (N, Σ , P, S), such that L(G) = L(M). Conversely, if G is an arbitrary RLG, then one can construct an NFA, M, for which L(M) = L(G). Proof. Let M be given. We construct G from M essentially by (a) introducing a nonterminal for each state of M, and (b) introducing one production for each transition of M. Formally, we define N = Q {S}, where S denotes the start symbol of G and is distinct from all symbols denoting states of M. P is then defined to be the union of three sets of rules denoted P[1] , P[2], and P[3]. P[1] = { S q | q Q 0 }, P[2] = { q →σ q’ | q’ δ M (q, σ ) for some σ∈ Σ { Λ } }, and P[3] = { q λ | q A } To show that L(G) = L(M), we have to show that x L(G) if and only if x L(M). Suppose that x L(M). Then there is some accepting computation of x, that is there is some sequence of configurations: (q 0 , y 0 ) M (q 1 , y 1 ) M M (q m , y m ) = (q m , λ ), where q 0 Q 0 , y 0 = x, and q m A. Then in G we have the following derivation: S G q 0 G π y 0 q m G y 0 = x. The first step of the derivation applies a rule in P[1] to rewrite S as the particular initial state (q 0 ) of M that determines an accepting computation. Each rule of π corresponds to a move of M in the accepting computation – whatever input, σ , is consumed by M on any given move, σ will be produced as output by the corresponding rule in P[2]; if σ = Λ , then no input is consumed by M and nothing is written to the sentential form by G. Thus, if M consumes y 0 , G will generate y 0 by mimicking the same transitions, but opposite in the IO sense. Finally, the last rule of the derivation is a rule in P[3] – these rules permit the derivation in G to terminate if and only if M is in an accept state. Using a similar argument it is easily shown that any string x generated by G can also be accepted by M. Thus M and G are equivalent specifications for the same language. The converse result is established in a similar fashion, but there are some details that are different. So, let us be given an arbitrary RLG, G = (N, Σ , P, S). To construct M we will introduce a state for each nonterminal, analogous to our previous construction of G from M.
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This note was uploaded on 06/09/2011 for the course COT 4210 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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Notes-SecC - COT 4210 Section C Spring 2001 Theorem 2...

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