By Michael A. Arbib, A. J. Kfoury, Robert N. Moll

Computer technology seeks to supply a systematic foundation for the learn of tell a tion processing, the answer of difficulties by means of algorithms, and the layout and programming of pcs. The final 40 years have visible expanding sophistication within the technological know-how, within the microelectronics which has made machines of outstanding complexity economically possible, within the advances in programming method which permit giant courses to be designed with expanding pace and lowered errors, and within the improvement of mathematical strategies to permit the rigorous specification of software, strategy, and computing device. the current quantity is one in all a sequence, The AKM sequence in Theoretical desktop technology, designed to make key mathe matical advancements in computing device technology quite simply available to lower than graduate and starting graduate scholars. in particular, this quantity takes readers with very little mathematical historical past past highschool algebra, and provides them a style of a couple of issues in theoretical laptop technology whereas laying the mathematical starting place for the later, extra distinct, learn of such issues as formal language concept, computability conception, programming language semantics, and the research of application verification and correctness. bankruptcy 1 introduces the elemental recommendations of set thought, with specified emphasis on capabilities and family, utilizing an easy set of rules to supply motivation. bankruptcy 2 provides the proposal of inductive evidence and offers the reader an exceptional grab on some of the most very important notions of desktop technological know-how: the recursive definition of features and information structures.

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In the last section, we studied induction on N. {O} was the basis, and the passage from n to n + 1 was the induction step. Here we consider ND with the set D of digits as the basis. The passage from any string w to each of the strings wd, one for each digit in D, is the induction step. " 9 Recursive Definition of Addition. Having already defined the successor function, we may define m + n by induction on n as follows: Basis Step: m + 0 = m. Induction Step: Given that m + n is already defined, we set m +

M * (n + p) = m * n + m * p for all m, n, and p in N. This calls for yet another general definition: 13 Definition. A quintuple (S, +, ,,0, 1) is called a semiring with additive identity 0 and multiplicative identity 1 if 1. (S, +, 0) is a commutative monoid; 2. (S,·, 1) is a monoid; 3. distributes over + : a· (b + c) = a . b + a . c for all a, b, cin S. One reason for the interest of this concept to computer scientists is given by the following example. Another is given in Exercise 6. 2. 14 Observation.

1 Induction on the Natural Numbers In the course of this book a variety of proof techniques will be used to establish theorems and propositions. In this chapter we introduce one particular proof technique, proof by induction. Induction is especially important for computer scientists because it can serve as a principle of computation as well as a method of proof. Programmes defined using recursion are really doing induction "backwards" during execution. Proof by induction is a powerful method of proving that a property holds for all natural numbers.