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When we say "solved algorithmically" we are not asking about a speci?c programming language, in fact one of the theorems in computability is that essentially all reasonable programming languages are equivalent in their power. Rather, we want to know if there is an algorithm for solving it that can be expressed in any rigorous way at all. Similarly, we are not asking about whether the problem can be solved on any particular computer, but whether it can be solved by any computing mechanism, including a human using a pencil and paper (even a limitless supply of paper).
What we need is an abstract model of computation that we can treat in a rigorous mathematical way. We'll start with the obvious model:
Here a computer receives some input (an instance of a problem), has some computing mechanism, and produces some output (the solution of that instance). We will refer to the con?guration of the computing mechanism at a given point in it's processing as its internal state. Note that in this model the computer is not a general purpose device: it solves some speci?c problem. Rather, we consider a general purpose computer and a program to both be part of a single machine. The program, in essence, specializes the computer to solve a particular problem.
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
how to convert a grammar into GNF
Claim Under the assumptions above, if there is an algorithm for checking a problem then there is an algorithm for solving the problem. Before going on, you should think a bit about
Consider a water bottle vending machine as a finite–state automaton. This machine is designed to accept coins of Rs. 2 and 5 only. It dispenses a single water bottle as soon as the
Since the signi?cance of the states represented by the nodes of these transition graphs is arbitrary, we will allow ourselves to use any ?nite set (such as {A,B,C,D,E, F,G,H} or ev
A Turing machine is a theoretical computing machine made-up by Alan Turing (1937) to serve as an idealized model for mathematical calculation. A Turing machine having of a line of
build a TM that enumerate even set of even length string over a
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
Applying the pumping lemma is not fundamentally di?erent than applying (general) su?x substitution closure or the non-counting property. The pumping lemma is a little more complica
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 langua
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