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A context free grammar G = (N, Σ, P, S) is in binary form if for all productions
A we have |α| ≤ 2. In addition we say that G is in Chomsky Normaml Form (CNF) if it is in binary form and if the only sorts of production have the form
A → a (where a is a terminal symbol)
or
A → BC (where B and C are non-terminals)
We will show that every CFG G is λ- equivalent to a grammar G' that is in CNF (i.e. the only difference between G and G' is that may or may not be included. Since we know how to test for the presence of in our languages we will be able to construct equivalent grammars.
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Lemma 1 A string w ∈ Σ* is accepted by an LTk automaton iff w is the concatenation of the symbols labeling the edges of a path through the LTk transition graph of A from h?, ∅i to
For example, the question of whether a given regular language is positive (does not include the empty string) is algorithmically decidable. "Positiveness Problem". Note that
While the SL 2 languages include some surprisingly complex languages, the strictly 2-local automata are, nevertheless, quite limited. In a strong sense, they are almost memoryless
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Kleene called this the Synthesis theorem because his (and your) proof gives an effective procedure for synthesizing an automaton that recognizes the language denoted by any given r
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