Specification and Implementation of Abstract Data Types

Data types in distributive categories

“…Iu Sets, by a similar analysis to that above, a type queue in our sense yields a type queue in the sense of [1, p.306]. In fact, the only difference between our notion and that of [1] is that their pop and reacf are defined on empty queues (resulting, respectively, in a queue and a value). As in our description of type stack, in Sets our arrow make corresponds to two functions of [1], namely new and make.…”

“…Further, an arrow 1 + XS -• S is a pair of arrows 1 -+ S, XS -» 5 . In [1] the first of these is called new, and the second push. Now our equation push opop = I s implies that if pop(s) = error then s = new, and pop opush = II-J-JVS implies that pop(new) = error.…”

“…An arrow into a product is a pair of arrows, and so pop, restricted to non-new stacks, can be regarded as a pair of arrows, pop: S -> 5 , and read: S -» X (this is the notation of [1]). In our description empty (of [1]) does not occur as basic data, but it is definable as empty = (true + false) o pop. The equations of [1] are all consequences of the fact that our pop and pusii are inverse.…”

“…In our description empty (of [1]) does not occur as basic data, but it is definable as empty = (true + false) o pop. The equations of [1] are all consequences of the fact that our pop and pusii are inverse. The notion of type stack in [1] is however weaker than ours since there is no requirement in [1] that push reverse the effect of pop.…”

A mathematical approach to nondeterminism in data types

“…In order not to be forced to specify all details of an implementation, we have to admit nondeterminism on the formal level (cf. [2,Sect. 4.41).…”