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An abstract data type specifies values and operations independently of their storage. A FIFO queue has ordered sequences as its abstract values; an array and a linked list can represent those same values. A representation invariant identifies valid storage states, and an abstraction function says what each valid state means.^[ref]

An abstraction function maps a valid concrete representation to its abstract value. Write $R_{\mathrm{valid}}=\{r:I(r)\}$ and $\alpha :R_{\mathrm{valid}}\to Q$. Different storage states can denote the same abstract value. Correct implementation requires both preservation of the representation invariant and agreement with the specified operation after applying the abstraction function; preserving a length bound alone would not prove first-in, first-out behavior.

The representation invariant answers "is this storage state allowed?" The abstraction function answers "which value does it represent?" Thus $I(r)$ is a truth value, whereas $\alpha (r)$ is a value of the abstract data type. Treating $\alpha$ as a partial function on all storage states, or as a function defined on $R_{\mathrm{valid}}$ only, expresses the same restriction. The abstraction function may forget storage details, and need not be executable code: its purpose is to connect an implementation to its formal specification.^[ref]

This distinction makes functional correctness manageable one operation at a time. The unreduced rational representation below separates valid storage from mathematical meaning; correct operations through an abstraction function then checks whether an implementation performs the promised addition. For an example with hidden mutable storage and failure cases, the bounded queue with encapsulation supplies a different abstraction function, reading an occupied prefix of an array as a sequence.

Representation exposure gives outside code access that can modify storage on which the representation invariant or promised behavior depends. Returning the internal array exposes a route around the API; merely hiding the field name does not close it. Incoming shared mutable storage can create the same problem. Copies or immutable values can prevent such modification, but their costs and sharing rules belong in the design.^[ref]

The bounded queue with encapsulation deliberately stores immutable labels. A FIFO queue of mutable objects needs a separate promise about whether changing a returned object changes stored content; protecting the queue's ordering and size does not automatically make every stored object immutable.

A representation invariant $I(r)$ specifies which concrete storage states $r$ are valid at the API boundary. Construction must establish it, and each completed operation must preserve it. Internal steps may temporarily violate it only while that state is inaccessible to observers; callbacks or concurrent access would require further reasoning.^[ref]

The short proof pattern is mathematical induction on the number of completed calls: the initial state satisfies $I$; every permitted transition from an $I$-state ends in an $I$-state; hence every state observable through the API satisfies $I$. An unrestricted external write falls outside this argument.