Abstract
This chapter presents a number of ideas that originated an evolution of programming from arts and crafts to a science. The chapter describes computer arithmetic in two stages. In the first stage, axioms are given for arithmetic operations on natural numbers, which are valid independently of their computer representation, and choices of supplementary axioms are proposed for characterizing various possible implementations. In the second part, an axiomatic definition of program execution is introduced. An axiomatic approach is indispensable for achieving program reliability. The usefulness of program proving is advocated in view of the cost of programming errors and program testing. The chapter discusses the definition of formal language. The axioms and rules of inference can be understood as the ultimate definitive specification of the meaning of the language.
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