5. Equality and Function Application Rules

26 the descriptive power of RMIN and the simplicity of its semantics, they provide a powerful incentive for further investigation of RMIN in combination with ACL as the basis of a formal software development method. Notes 1. In fact it is only when using a typed theory that there is a problem with undeened terms. In untyped axiomatic set theory the application of a function outside its domain could denote the universal set (if there is one) or the empty set. For examples of such treatments see 12] and 27]. 2. In Principia Mathematica the ordered pairs in relations are the opposite way round to those in Z and B. This has an advantage when it comes to choosing meaningful names for relations, in that when naming a relation we do not have to know whether we wish to use it in innx (relational) or preex (functional) form. For example the relation wrote might be used as Scott wrote Waverly or wrote Waverly = Scott. 3. Unless its type constraint limits it to membership of a singleton or empty set. 4. This is a little diierent from the usual approach in Z, which distinguishes a type, which is a syntactic entity, from its carrier set. 5. EMIN is not considered in detail, since as noted above its existential equality can be deened as the identity relation in RMIN. 6. A set S is said to be decidable if there exists a procedure to test whether an element e is in S and this procedure will give the correct result for any element. A set S is semi-decidable if the best we can manage is a test which is guaranteed to give the correct result if e is in S, but which may not terminate if e is not in S. 7. Some further explanation of this method of expressing deenitive non-termination when an operation is invoked outside its precondition will be given in the next section. A logic covering undeenedness in program proofs. (existential equality, minimal denotation); RMIN (referential equality, minimal de-notation), and XMAX (either equality, maximal denotation). Since the existential equality of EMIN can be deened under RMIN as the identity relation, EMIN is less fully investigated than the other approaches, but receives an honourable mention as the semantics used in Principia Mathematica's theory of descriptive functions (the \Theory of Descriptions"). The B proof system is consistent with the …