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The Universality Problem is the dual of the emptiness problem: is L(A) = Σ∗?
It can be solved by minor variations of any one of the algorithms for Emptiness or (with a little less work) it can simply be reduced to Emptiness.
Theorem (Universality) The Universality Problem for Regular Languages is decidable.
Proof: L(A) = Σ*⇔ L(A) = ∅. As regular languages are effectively closed under complement we can simply build the DFA for the complement of L(A) and ask if it recognizes the empty language.
A finite, nonempty ordered set will be called an alphabet if its elements are symbols, or characters. A finite sequence of symbols from a given alphabet will be called a string ove
Suppose G = (N, Σ, P, S) is a reduced grammar (we can certainly reduce G if we haven't already). Our algorithm is as follows: 1. Define maxrhs(G) to be the maximum length of the
Paths leading to regions B, C and E are paths which have not yet seen aa. Those leading to region B and E end in a, with those leading to E having seen ba and those leading to B no
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
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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
This close relationship between the SL2 languages and the recognizable languages lets us use some of what we know about SL 2 to discover properties of the recognizable languages.
Find the Regular Grammar for the following Regular Expression: a(a+b)*(ab*+ba*)b.
We saw earlier that LT is not closed under concatenation. If we think in terms of the LT graphs, recognizing the concatenation of LT languages would seem to require knowing, while
The path function δ : Q × Σ* → P(Q) is the extension of δ to strings: This just says that the path labeled ε from any given state q goes only to q itself (or rather never l
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