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By Walter J Savitch

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A finite-state acceptor is neither hardware nor exactly a program but a formal mathematical construct that can model both of these. We now describe the working of this abstract mathematical device in more detail and then give a formal mathematical definition for finite-state acceptors. CONCEPTUAL DESCRIPTION A finite-state acceptor consists of a device that can exist in only a finite number of different states. There is an input alphabet associated with the device. Any string of symbols from this alphabet may be given to the machine as input.

Since the last state, , is an accepting state, the string 11010 is accepted. This machine will accept those strings of zeros and ones which contain an odd number of ones and rejects those strings of zeros and ones which contain an even number of ones. In terms of the mathematical formalism, this machine is a five-tuple (S, :E, 8, s, Y) where S = {, }, :E = {0, 1}, s =, Y ={}, and 8 is defined by the following equations: 8(, 0) = , 8(, 1) = , 8(, 0) = , and 8(, 1) = .

In describing the various grammars we will simply give the productions and specify the start symbol. All symbols other than a and b will be nonterminals. 1 and, as we have seen, L(Go) = (anbn In~ 0}. G0 can be described as follows: start symbol = S productions: S-A, S - aSb 22 CHAPTER 2. 12. 12, take G 1 = G0. 12. 13 is then as follows: start symbol = s2 productions: S2- A, S - aSb, S2 - aSb, S - ab, S2- ab G' is a nonerasing cfg with no productions of the form A -B. 12 and so G' is equivalent to Go.

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