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In historical terms, the digital computer is very, very new. The science of computing is yet newer. Compared to its older sister - mathematics - which is thousands of years old, it is hardly in the embryonic stage of development. Yet, computing science is already having a major influence on our problem-solving skills, amounting to a revolution in the art of effective reasoning.
Because of the challenges of programming (which means instructing a dumb machine how to solve each instance of a problem) and the unprecedented scale of programming problems, computing scientists have had to hone their problem-solving skills to a very fine degree. This has led to advances in logic, and to changes in the way that mathematics is practised. This book forms an introduction to problem-solving using the insights that have been gained in computing science.
Algorithms
Solutions to programming problems are formulated as so-called algorithms. An algorithm is a well-defined procedure, consisting of a number of instructions, that are executed in turn in order to solve the given problem.
Normally, an algorithm will have certain inputs; for each input, the algorithm should compute an output which is related to the input by a certain so-called input-output relation. Formulating an algorithm makes problem-solving decidedly harder, because it is necessary to formulate very clearly and precisely the procedure for solving the problem. The more general the problem, the harder it gets. The advantage, however, is a much greater understanding of the solution. The process of formulating an algorithm demands a full understanding of why the algorithm is correct.
Intended Audience:
This book aims to impart these new skills and insights to a broad audience, using an example-driven approach. It aims to demonstrate the importance of mathematical calculation, but the chosen examples are typically not mathematical; instead, they are problems that are readily understood by a lay person, with only elementary mathematical knowledge. The book also aims to challenge; most of the problems are quite difficult, at least to the untrained or poorly trained practitioner.
In historical terms, the digital computer is very, very new. The science of computing is yet newer. Compared to its older sister - mathematics - which is thousands of years old, it is hardly in the embryonic stage of development. Yet, computing science is already having a major influence on our problem-solving skills, amounting to a revolution in the art of effective reasoning.
Because of the challenges of programming (which means instructing a dumb machine how to solve each instance of a problem) and the unprecedented scale of programming problems, computing scientists have had to hone their problem-solving skills to a very fine degree. This has led to advances in logic, and to changes in the way that mathematics is practised. This book forms an introduction to problem-solving using the insights that have been gained in computing science.
Algorithms
Solutions to programming problems are formulated as so-called algorithms. An algorithm is a well-defined procedure, consisting of a number of instructions, that are executed in turn in order to solve the given problem.
Normally, an algorithm will have certain inputs; for each input, the algorithm should compute an output which is related to the input by a certain so-called input-output relation. Formulating an algorithm makes problem-solving decidedly harder, because it is necessary to formulate very clearly and precisely the procedure for solving the problem. The more general the problem, the harder it gets. The advantage, however, is a much greater understanding of the solution. The process of formulating an algorithm demands a full understanding of why the algorithm is correct.
Intended Audience:
This book aims to impart these new skills and insights to a broad audience, using an example-driven approach. It aims to demonstrate the importance of mathematical calculation, but the chosen examples are typically not mathematical; instead, they are problems that are readily understood by a lay person, with only elementary mathematical knowledge. The book also aims to challenge; most of the problems are quite difficult, at least to the untrained or poorly trained practitioner.
