What is the Numerical Domain of an Algorithm and How is it Determined?

[agapito]

Hi everyone. My book (Smith's "An Introduction to Goedel's Theorems") defines the numerical domain of an algorithm as the set of naturals that, when input individually to the algorithm, result in its "working", that is to terminate and output some result. In the book it is also stated that any wff of a programming language (e. g. C++) can be considered an algorithm, having a numerical domain.

I'm not clear about how an instruction such as "Go To" can be considered an algorithm, or if it is, then how does one determine its numerical domain.

Can someone please explain this? All help greatly appreciated.

 

[S.G. Janssens]

Logic is not my area of expertise, but your question made me think of Fortran 77's "computed" and "assigned" GO TO statements, https://www.fortran.com/F77_std/rjcnf0001-sh-11.html for some documentation on the (now obsolete) 77 standard.

A "computed" GO TO is then a function defined on $\{1,\ldots,n\}$ mapping to the set of labels $\{s_1,\ldots,s_n\}$.
An "assigned" GO TO is a function from the set of labels to itself.
Finally, an unconditional GO TO is a constant function defined on $\mathbb{N}$.

Maybe this helps you in your thoughts?

 

[agapito]

Logic is not my area of expertise, but your question made me think of Fortran 77's "computed" and "assigned" GO TO statements, https://www.fortran.com/F77_std/rjcnf0001-sh-11.html for some documentation on the (now obsolete) 77 standard.

A "computed" GO TO is then a function defined on $\{1,\ldots,n\}$ mapping to the set of labels $\{s_1,\ldots,s_n\}$.
An "assigned" GO TO is a function from the set of labels to itself.
Finally, an unconditional GO TO is a constant function defined on $\mathbb{N}$.

Maybe this helps you in your thoughts?

Thanks. OK in the case of GO TO we can say that the set {1,2,...n} is the algorithmic domain of the instruction. The question, then, is whether each instruction (wff) of the programming language can be similarly considered a function of some numeric set? Thanks again for your help.

 

[HOI]

It's quite possible that the author was not thinking of "Go To" since no modern language uses that.