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1Machine Theory

# 1Machine Theory - There was a need for a more precise...

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Machine (Automata) Theory Algorithm : A list of instructions specifying a finite sequence of operations that will give the answer to any mathematical problem of a given type. Examples: algorithms for each of the four arithmetic operations—addition, subtraction, multiplication, and division. The Euclidean algorithm: Given two integers a and b, find their greatest common divisor (the greatest integer that will divide them both without remainder). Given a and b: 1. If the numbers are equal, then each is the required result. Stop. 2. If the numbers are not equal, subtract the smaller from the larger and replace a and b by the subtrahend and remainder. Go to step 1.

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Unformatted text preview: There was a need for a more precise definition of “algorithm.” Recall the Hilbert Program (consistency, completeness, and decidability). Machine Theory (cont.) Kurt Gödel’s proof (1931): Given an axiomatic system that is comparable in complexity to, e.g., arithmetic, then it cannot be both consistent and complete. There will always be true but formally undecidable sentences in such a system. But can we at least devise an algorithm for recognizing such undecidable sentences? Alan Turing’s proof (1936): No! To prove this, Turing invented a formal machine (Turing Machine) that amounted to a precise definition of an algorithm....
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1Machine Theory - There was a need for a more precise...

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