FORMAL REASONING ABOUT SYSTEMS, SOFTWARE AND HARDWARE Using Functional, Predicates and Relations

摘要

Formal reasoning in the sense of "letting the symbols do the work" was Leibniz's dream, but making it possible and convenient for everyday practice irrespective of the availability of automated tools is due to the calculational approach that emerged from Computing Science. This tutorial provides an initiation in a formal calculational approach that covers not only the discrete world of software and digital hardware, but also the "continuous" world of analog systems and circuits. The formalism (Funmath) is free of the defects of traditional notation that hamper formal calculation, yet, by the unified way it captures the conventions from applied mathematics, it is readily adoptable by engineers. The fundamental part formalizes the equational calculation style found so convenient ever since the first exposure to high school algebra, followed by concepts supporting expression with variables (pointwise) and without (point-free). Calculation rules are derived for (ⅰ) proposition calculus, including a few techniques for fast "head" calculation; (ⅱ) sets; (ⅲ) functions, with a basic library of generic functionals that are useful throughout continuous and discrete mathematics; (ⅳ) predicate calculus, making formal calculation with quantifiers as "routine" as with derivatives and integrals in engineering mathematics. Pointwise and point-free forms are covered. Uniform principles for designing convenient operators in diverse areas of discourse are presented. Mathematical induction is formalized in a way that avoids typical errors associated with informal use. Illustrative examples are provided throughout. The applications part shows how to use the formalism in computing science, including data type definition, systems specification, imperative and functional programming, formal semantics, deriving theories of programming, and also in continuous mathematics relevant to engineering.