Let $\Gamma$ be a finite set of alphabets, $M$ be a single datatype of memory states. A memory state is a function $m : \Z \to \Gamma$. Any Turing machine $T$ can be regarded as computing a certain partial function $f_T : M \rightharpoonup M$ in the way: $f_T(m) = m'$ if the execution of $T$ with initial state $m$, eventually halts yielding the final memory state $m'$.

The model $T_2$ consisting of the single datatype $\N$ together with all Turing-computable partial functions $\N \rightharpoonup \N$. This model needs some convention for representing natural numbers via memory states.

Also known as Kleene's first model.

The model $T_3$ conceptually have a read-only input tape, a write-only output tape, and a working tape that permits both reading and writing. The input and output tapes are functions $d : \N \to \Gamma$, $D$ is the set of all such total functions. Thus, the model consisting of the single datatype $D$ and all machine-computable partial functions $f : D \rightharpoonup D$.