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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 over the alphabet. A string that consists of a sequence a1, a2, . . . , an of symbols will be denoted by the juxtaposition a1a2 an. Strings that have zero symbols, called empty strings, will be denoted by .
{0, 1} is a binary alphabet, and {1} is a unary alphabet. 11 is a binary string over the alphabet {0, 1}, and a unary string over the alphabet {1}.
11 is a string of length 2, |ε| = 0, and |01| + |1| = 3.
Example-The string consisting of a sequence αβ followed by a sequence β is denoted αβ. The string αβ is called the concatenation of α and β. The notation αi is used for the string obtained by concatenating i copies of the string α.
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
One might assume that non-closure under concatenation would imply non closure under both Kleene- and positive closure, since the concatenation of a language with itself is included
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.
Intuitively, closure of SL 2 under intersection is reasonably easy to see, particularly if one considers the Myhill graphs of the automata. Any path through both graphs will be a
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
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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
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
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We will assume that the string has been augmented by marking the beginning and the end with the symbols ‘?' and ‘?' respectively and that these symbols do not occur in the input al
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