Refinement Calculus: A Systematic Introduction by Ralph-Johan Back

By Ralph-Johan Back

Much present examine in laptop technological know-how is anxious with questions: is a software right? and the way will we enhance an accurate software retaining correctness? This latter query is called the refinement of courses and the aim of this e-book is to think about those questions in a proper surroundings. in reality, correctness seems to be a distinct case of refinement and so the point of interest is on refinement. even though an inexpensive heritage wisdom is thought 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 various workouts supplied of various levels of challenge.

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It is easy to see that A U --, A = D and A n --, A = 0, so the requirements on the complement are met. The natural numbers do not form a Boolean lattice. Even if we add the limit ordinal w to Nat, so that we get a complete distributive lattice, we still do not have a Boolean lattice. Atomic Lattices Let A be a lattice with a least element. 1 v b=a. Thus, an atom is an element that is one step above the bottom element in a Hasse diagram. A lattice A is said to be atomic if for each element x in A, we have that x = u{a E A I a is an atom /\ a I;; x} .

D gm :: Sm fi = {gd; S\ u ... U {gm}; Sm This statement will again cause a breach of the contract if none of the assertions holds in the state. If at least one of the assertions holds, then our agent gets to choose the alternative to continue with. This construct can be interpreted as a context-sensitive menu choice: the alternatives from which our agent chooses are items on a menu, and the assertions determine which of these items are enabled in a particular state. Recursive Contracts We can make the language of contracts even more interesting from a programming point of view by permitting recursive statements of contracts.

We construct the program and its specification hand in hand, making the program and its specification more precise as part of the program construction process. A variant of explorative programming arises when the postcondition q is given, and the task is to find an interesting statement S and a precondition p such that p ft S ~ q holds. This leads to a form of backward program derivation, where we try to derive from the postcondition a program that will achieve this postcondition. Another variant is that the precondition p is given, and we try to derive an interesting statement S and a postcondition q such that p ft S ~ q holds.

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