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The original assumption, that a TM does exist that can determine whether any particular program will run to completion when presented with any arbitrary input data, must be incorrect. That assumption led to the contradictory state illustrated by NotH. Therefore, computer scientists conclude that there can be no one algorithm that can determine whether any particular program will run to completion, or fail to run to completion, for every possible set of inputs. It would be very nice to have a program to which we could submit new code for a quick determination as to whether it would run to completion given any particular set of inputs. Alas, Turing proved that this cannot be. One can and should write test programs, but one will never succeed in writing one program which can test every program. The halting problem is one of the provably unsolvable problems in computing (Turing, Alan, On computable Numbers with an Application to the Entscheidungsproblem , Proceedings of the London Mathematical Society, 2:230 265, 1936). No one algorithm will ever be written to prove the correct or incorrect execution of every possible program when presented with any particular set of inputs. While no such algorithm can be successful, knowing that allows computer scientists to focus on problems for which there are solutions. SUMMARY An algorithm is a specific procedure for accomplishing some job. Much of computer science has to do with finding or creating better algorithms for solving computational problems. We usually describe computational algorithms using pseudocode, and we characterize the performance of algorithms using the term order of growth or theta. The order of growth of an algorithm tells us, in a simplified way, how the running time of the algorithm will vary with problems of different sizes. We provided examples of algorithms whose orders of growth were (lg n), n, n(lg n), n2, 2n and n!. Algorithm development should be considered an important part of computing technology. In fact, a better algorithm for an important task may be much more impactful than any foreseeable near-term improvement in computing hardware speed. The Turing machine is a formal mathematical model of computation, and the Church Turing thesis maintains that any algorithmic procedure to manipulate symbols can be conducted by some Turing machine. We gave example Turing machines to perform the simple binary operations of complementing and incrementing a binary number. Some problems in computing are provably unsolvable. For instance, Turing proved that it is impossible to write one computer program that can inspect any other program and verify that the program in question will, or will not, run to completion, given any specific set of inputs. While the Holy Grail of an algorithm to prove the correctness of programs has been proven to be only a phantom in the dreams of computer scientists, computer scientists at least know that is so, and can work instead on practical test plans for real programs. REVIEW QUESTIONS 2.1 Write pseudocode for an algorithm for finding the square root of a number. 2.2 Write pseudocode for finding the mean of a set of numbers. 2.3 Count the primitive operations in your algorithm to find the mean. What is the order of growth of your mean algorithm 2.4 Write pseudocode for finding the median of a set of numbers. 2.5 What is the order of growth of your algorithm to find the median 2.6 Suppose that your algorithm to find the mean is (n), and that your algorithm to find the median is (n lg n), what will be the execution speed ratio between your algorithm for the mean and your algorithm for the median when the number of values is 1,000,000 2.7 A sort routine which is easy to program is the bubble sort. The program simply scans all of the elements to be sorted repeatedly. On each pass, the program compares each element with the one next to it, and reorders the two, if they are in inverse order. For instance, to sort the following list: 67314
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