§16§In ordinary mathematics, an equation can be written down which is syntactically correct, but for which no solution exists. For example, consider the equation x = x + 1 defined over the real numbers; there is no value of x which satisfies it. Similarly it is possible to specify objects using the formal specification language Z [3,4], which can not possibly exist. Such specifications are called inconsistent and can arise in a number of ways. Example 1 The following Z specification of a functionf, from integers to integers "f x : ~ 1 x ~ O〓 fx = x + 1 (i) "f x : ~ 1 x ~ O〓 fx = x + 2 (ii) is inconsistent, because axiom (i) gives f 0 = 1, while axiom (ii) gives f 0 = 2. This contradicts the fact that f was declared as a function, that is, f must have a unique result when applied to an argument. Hence no suchfexists. Furthermore, iff 0 = 1 andfO = 2 then 1 = 2 can be deduced! From 1 = 2 anything can be deduced, thus showing the danger of an inconsistent specification. Note that all exa §16§mples and proofs start with the word Example or Proof and end with the symbol.1. §04§Theoretical Foundations.- On Recursive Free Types in Z.- On Free Type Definitions in Z.- Z and Hoare Logics.- W: A Logic for Z.- Scope of Use.- The Use of Z.- Extending the Useful Application Domain for Formal Methods.- Domains of Application for Formal Methods.- Z-, an Executable Subset of Z.- Special Applications.- Engineering Human-Error Tolerant Software.- Techniques for Partial Specification and Specification of Switching Systems.- Tools.- Z and Eves.- zedB: A Proof Tool for Z Built on B.- Structured Methods and Object-Oriented Approaches.- A Method for the Specification of Relational Database Applications.- Structured Analysis' A Draft Method for Writing Z Specifications.- Enhancing the Structure of Z Specifications.- ZOOM Workshop Report.- Bibliography etc.- Select Z Bibliography and Frequently Asked Questions.- Author Index.
