Discrete Mathematics

Read e-book online Algebraic theory of automata and languages PDF

By Masami Ito

ISBN-10: 9810247273

ISBN-13: 9789810247270

ISBN-10: 9812562680

ISBN-13: 9789812562685

The idea of formal languages and the idea of automata have been either initiated within the overdue Fifties, explains Ito (Kyoto Sangyo U., Japan), and the 2 fields have given that constructed into vital theoretical foundations of computing device technological know-how. He appears to be like on the from the algebraic viewpoint, starting with the algebraic constitution of automata, and in part ordered units of automata as one of those international concept. Then he delves into grammars, languages, and operations on languages. To finish, he introduces directable automata as a different case.

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To solve this equation, we multiply both sides by a , the multiplicative inverse of a, to obtain χ = a~ b. So why can't we always solve Ax. — b when A is singular? Let's look at an example. 6 (Solving A x = b ) . Solve the equation ' 1 2 3 1 2 3 1 " 2 3 Xl = . X 3 ' 0 " 0 0 Solution. Let's take an alternative approach to solving this problem. 8) 1 0 3 . So we have 1 0 0 0 2 0 0 0 3 " 1 2 3 1 2 3 " 1 1 1 1 1 1 1 " Xl 2 3 j . 8) gives 1 1 1 1 1 1 1" 1 1 " 1 0 0 Xl *2 . X 3 J 0 0 0 1 2 " 0" 0 0 1 3 0 ' 0 " 0 0 X\ + x 2 Xl +X + X3 Xl + X + X3 2 2 It is easy to see that χ = x = χ = 0 or χ = x = 1 and x = - 2 are solutions to this equation.

Then we would dot ν with the vector c = [±,0,0, i ] . So we would compute three inner products: a v , b v , and c v . We store these row vectors in a matrix A and then consider the product Av as a 3-vector containing our three dot products. T T a " ] 4 1 6 Av - 1 2 1 1 4 1 3 4 0 0 1 3 T 100 1 " 4 1 6 1 2 - T 200 250 L 150 ~ 1 ( 1 0 0 ) + ± ( 2 0 0 ) + ± ( 2 5 0 ) + 1(150) 175 £100) + | ( 2 0 0 ) + ±(250) + 1(150) ±(100) + 0 · 200 + 0 • 250 + §(150) 125 Of course, we might want to apply our weighted averages to other columns in addition to v .

After quantization, it is impossible to exactly retrieve the original image . 3, but nearly all the values of the quantized transformation are different than the corresponding values in the transformation. The final step before transmission is to encode the quantized transformation. That is, instead of using 8 bits to store each integer, we will try to group together like integers and possibly use a smaller number of bits to store those values that occur with greater frequency. Since the quantized wavelet transformation contains a large number of zeros (black pixels), we expect the encoded transform to require fewer bits.

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Algebraic theory of automata and languages by Masami Ito


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