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Invariance 1: not [ 3 a, b,t (a, t) E elems and (b, t) E elems and a=Fb] This means that this property is an invariance and must always be true about a stack. A second invariance must hold that c, the current time-stamp, be greater than all the time-stamps in the stack. invariance 2:\::1 (a, t) E elems c > t Finally, a last invariance should ensure that a stack may not have more than max objects. invariance 3: I elems I ~ max Now we specify each operation by giving its pre (condition) and post (condition) which must be true before and after the operation respectively.

In principle, such data structures need not be in the interface section, but translators for such languages need the extra information for efficient compilation. (Modula-2 allows a somewhat restricted form of data abstraction known as opaque types. ) (3) Since the implementation of an ADT is hidden from the applications, different strategies can be used in the implementation without affecting the programs using that ADT. One particular advantage is that trade-offs between time and space can be made.

Each operation is primarily a function which, when applied to an object of a given type (or several objects), returns an object of a specified type. For instance, the arithmetic operation'+' is a function '+': integer* integer~ integer which when given two integers returns another integer. A special operator,':=', (that is, the assignment operation) is a function with two arguments of the same type which returns no value but changes the state of the program by copying the second argument into the first argument.

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