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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 be recognizable. But what about the class of recognizable languages as a whole? Are Boolean combinations of recognizable (not just LT) languages also recognizable. In answering we can use the same methodology we use to show that any language is recognizable: consider what we need to keep track of in scanning a string in order to determine if it belongs to the language or not and then use that information to build our state set.
Suppose, then, that L = L1 ∩ L2, where L1 and L2 are both recognizable. A string w will be in L iff it is in both L1 and L2. Since they are recognizable there exist DFAs A1 and A2 for which L1 = L(A1) and L2 = L(A2). We can tell if the string is in L1 or L2 simply by keeping track of the state of the corresponding automaton. We can tell if it is in both by keeping track of both states simultaneously.
It is not hard to see that ε-transitions do not add to the accepting power of the model. The underlying idea is that whenever an ID (q, σ v) directly computes another (p, v) via
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
The universe of strings is a very useful medium for the representation of information as long as there exists a function that provides the interpretation for the information carrie
What is the purpose of GDTR?
Let G be a graph with n > 2 vertices with (n2 - 3n + 4)/2 edges. Prove that G is connected.
The Recognition Problem for a class of languages is the question of whether a given string is a member of a given language. An instance consists of a string and a (?nite) speci?cat
Exercise: Give a construction that converts a strictly 2-local automaton for a language L into one that recognizes the language L r . Justify the correctness of your construction.
Let L 3 = {a i bc j | i, j ≥ 0}. Give a strictly 2-local automaton that recognizes L 3 . Use the construction of the proof to extend the automaton to one that recognizes L 3 . Gi
Computer has a single unbounded precision counter which you can only increment, decrement and test for zero. (You may assume that it is initially zero or you may include an explici
Computer has a single FIFO queue of ?xed precision unsigned integers with the length of the queue unbounded. You can use access methods similar to those in the third model. In this
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