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As we are primarily concerned with questions of what is and what is not computable relative to some particular model of computation, we will usually base our explorations of languages on abstract automata. These are "checking machines" in which the input is a string over some speci?c alphabet. We say such a machine accepts a string if the computation on that input results in a TRUE output. We say that it recognizes a language if it accepts all and only the strings in that language.
Generally, in exploring a class of languages, we will de?ne a class of automata that recognize all and only the languages in the class-a particular sort of automaton, the peculiarities of which exactly capture the characteristics of the class of languages. We say the class of automata characterizes the class of languages. We will actually go about this both ways. Sometimes we will de?ne the class of languages ?rst, as we have in the case of the Finite Languages, and then look for a class of automata that characterize it. Other times we will specify the automata ?rst (by, for instance, modifying a previously de?ned class) and will then look for the class of languages it characterizes. We will use the same general methods no matter which way we are working.
The de?nition of the class of automata will specify the resources the machine provides along with a general algorithm for employing those resources to recognize languages in the class. The details that specialize that algorithm for a particular language are left as parameters. The only restriction on the nature of these parameters is that there must be ?nitely many of them and they must range over ?nite objects.
example of multitape turing machine
Computer has a single LIFO stack containing ?xed precision unsigned integers (so each integer is subject to over?ow problems) but which has unbounded depth (so the stack itself nev
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
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As de?ned the powerset construction builds a DFA with many states that can never be reached from Q′ 0 . Since they cannot be reached from Q′ 0 there is no path from Q′ 0 to a sta
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
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First model: Computer has a ?xed number of bits of storage. You will model this by limiting your program to a single ?xed-precision unsigned integer variable, e.g., a single one-by
Construct a PDA that accepts { x#y | x, y in {a, b}* such that x ? y and xi = yi for some i, 1 = i = min(|x|, |y|) }. For your PDA to work correctly it will need to be non-determin
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