Turing computable

A function is Turing computable if the function’s value can becomputed with a Turing machine.

More specifically, let $D$ be a set of words in a given alphabet andlet $f$ be a function which maps elements of $D$ to words on thesame alphabet. We say that $f$ is Turing computable if thereexists a Turing machine such that

• If one starts the Turing machine with a word $w\in D$ as theinitial content of the tape, the computation will halt.

• When the computation halts, the tape will read $f(w)$.

Formally, let $\mathrm{\Sigma }$ be an alphabet and $f:{\mathrm{\Sigma }}^{*}\to {\mathrm{\Sigma }}^{*}$ on words over $\mathrm{\Sigma }$. Then $f$ is said to be Turing-computable if there is a Turing machine $T$ over $\mathrm{\Sigma }$ (its input alphabet), as defined in this entry (http://planetmath.org/FormalDefinitionOfATuringMachine), such that for any $w\in {\mathrm{\Sigma }}^{*}$,

$$(s,{\tau }_w,1){\to }^{*}(h,{\tau }_{f(w)},m)$$

for some $m$. Here, $h$ is a halt state (either an accept or a reject state), and ${\tau }_w$ for any word $w$ is defined as the tape description such that the content of the $i$-th square is the $i$-th letter of $w$, and blank everywhere else.

Because of the fact that all types of Turing machines (deterministic,non-deterministic, single head, multiple head, etc.) all have the samecomputational power, it does not matter which type of Turing machineone uses in the definition.

It is not hard to find examples of Turing computable functions —because Turing machines provide an idealized model for the operationof the digital computer, any function which can be evaluated by acomputer provides an example.