You are here
Homesearch problem
Primary tabs
search problem
If $R$ is a binary relation such that $\operatorname{field}(R)\subseteq\Gamma^{+}$ and $T$ is a Turing machine, then $T$ calculates $f$ if:

If $x$ is such that there is some $y$ such that $R(x,y)$ then $T$ accepts $x$ with output $z$ such that $R(x,z)$ (there may be multiple $y$, and $T$ need only find one of them)

If $x$ is such that there is no $y$ such that $R(x,y)$ then $T$ rejects $x$
Note that the graph of a partial function is a binary relation, and if $T$ calculates a partial function then there is at most one possible output.
A relation $R$ can be viewed as a search problem, and a Turing machine which calculates $R$ is also said to solve it. Every search problem has a corresponding decision problem, namely $L(R)=\{x\mid\exists yR(x,y)\}$.
This definition may be generalized to $n$ary relations using any suitable encoding which allows multiple strings to be compressed into one string (for instance by listing them consecutively with a delimiter).
Mathematics Subject Classification
68Q25 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff