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Unformatted text preview: Computability Theory and Complexity Theory COT 6410 Computability Theory The study of what can/cannot be done via purely mechanical means. Complexity Theory The study of what can/cannot be done well via purely mechanical means. What is it that we are talking about? Solving problems algorithmically! A Problem: • Set of input data items (set of input "instances") • A set of rules or relationships between data and other values • A question to be answered or set of values to be obtained { Examples: Search a list for a key, SubsetSum, Graph Coloring } Each instance has an 'answer .' An instance’s answer is the solution of the instance  it is not the solution of the problem. A solution of the problem is a computational procedure that finds the answer of any instance A Procedure (or Program): A finite set of operations (statements) such that • Each statement is formed from a predetermined finite set of symbols and is constrained by some set of language syntax rules. • The current state of the machine model is finitely presentable. • The semantic rules of the language specify the effects of the operations on the machine’s state and An Algorithm: A procedure that • Correctly solves any instance of a given problem. • Completes execution in a finite number of steps no matter what input it receives. { Example algorithm: Linearly search a finite list for a key; If key is found, answer “Yes”; If key is not found, answer “No”; } { Example procedure: Linearly search a finite list for a key; If key is found, answer “Yes”; Procedures versus Algorithms Looking back at our approaches to “find a key in a finite list,” we see that the algorithm always halts and always reports the correct answer. In contrast, the procedure does not halt in some cases, but never lies. What this illustrates is the essential distinction between and algorithm and a procedure – algorithms always halt in some finite number of steps, whereas procedures may run on Notion of "Solvable" A problem is solvable if there exists an algorithm that solves it (provides the correct answer for each instance). The fact that a problem is solvable or, equivalently, decidable does not mean it is solved . To be solved, someone must have actually produced a correct algorithm. The distinction between solvable and solved is subtle. Solvable is an innate property – an unsolvable problem can never An Old Solvable Problem Does there exist a set of positive whole numbers, a, b, c and an n>2 such that an+bn = cn?...
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 Fall '10
 Dutton
 Computational complexity theory, polynomial time

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