# Refinement Calculus: A Systematic Introduction by Ralph-Johan Back

By Ralph-Johan Back

Much present study in laptop technological know-how is worried with questions: is a software right? and the way do we increase an accurate application keeping correctness? This latter query is named the refinement of courses and the aim of this ebook is to contemplate those questions in a proper environment. in reality, correctness seems to be a different case of refinement and so the point of interest is on refinement. even though an affordable historical past wisdom is believed from arithmetic and CS, the booklet is a self-contained advent appropriate for graduate scholars and researchers coming to this topic for the 1st time. there are many workouts supplied of various levels of challenge.

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This is the extensionality principle for functions generalized to partial orderings. We say that the poset A ~ B is a pointwise extension of the poset B. 4. 42 2. 4. Pointwise extension ordering The function indicated by the dashed arrows is here greater than the function indicated by solid lines. We have the same result for the pointwise extension to function spaces as for Cartesian products. 5 Let B be a poset and A any set. Then the pointwise extension of B to A --+ B is also a poset. Furthermore, this poset is a lattice (complete, bounded, distributive, Boolean lattice) if B is a lattice (complete, bounded, distributive, Boolean lattice).

This book incorporated his extensions to the refinement calculus and contained a number of case studies that described how to derive programs from specifications in the refinement calculus framework. This established the applicability of the calculus to practical programming problems. The book reformulates the original refinement calculus in a notation that is influenced by the Z specification style [133]. Another driving force for the further development of the calculus was the need to reengineer its mathematical basis.

A set of elements B need not have a greatest lower bound. However, if there is a greatest lower bound, then it is unique. The greatest lower bound of an arbitrary set B is characterized by the following two properties: bE B (Vb E B • a ~ b) => => nB ~ b , a ~ nB , (general n elimination) (general n introduction) The first property states that nB is a lower bound, and the second that it is the greatest of all lower bounds. The names for these properties reflect their intended use. The first property is used when we need to go from the meet to an element greater than the meet.