order are considered to be identical states. This view actually mirrors the behavior of some concrete implementations, such as linked lists with hash cons. Instead of create(), a functional-style definition of an abstract stack may assume the existence of a special stack state, the empty stack, designated by a special symbol like Λ or "()"; or define a bottom() operation that takes no arguments and returns this special stack state. Note that the axioms imply that push(Λ, x) ≠ Λ. In a functional-style definition of a stack one does not need an empty predicate: instead, one can test whether a stack is empty by testing whether it is equal to Λ. Note that these axioms do not define the effect of top(s) or pop(s), unless s is a stack state returned by a push. Since push leaves the stack non-empty, those two operations are undefined (hence invalid) when s = Λ. On the other hand, the axioms (and the lack of side effects) imply that push(s, x) = push(t, y) if and only if x = y and s = t. As in some other branches of mathematics, it is customary to assume also that the stack states are only those whose existence can be proved from the axioms in a finite number of steps. In the abstract stack example above, this rule means that every stack is a finite sequence of values, that becomes the empty stack (Λ) after a finite number of pops. By themselves, the axioms above do not exclude the existence of infinite stacks (that can be popped forever, each time yielding a different state) or circular stacks (that return to the same state after a finite number of pops). In particular, they do not exclude states s such that pop(s) = s or push(s, x) = s for some x. However, since one cannot obtain such stack states with the given operations, they are assumed "not to exist". Whether to include complexity Aside from the behavior in terms of axioms, it is also possible to include, in the definition of an ADT operation, their algorithmic complexity. Alexander Stepanov, designer of the C++ Standard Template Library, included complexity guarantees in the STL specification, arguing: Advantages of abstract data typing Encapsulation Abstraction provides a promise that any implementation of the ADT has certain properties and abilities; knowing these is all that is required to make use of