|
|
|
@ -14882,22 +14882,21 @@ headings. |
|
|
|
% might be considered as a significant advantage of $\LLL$ over $\LLL'$. |
|
|
|
|
|
|
|
If the `operations' we are asking about are ordinary first-order |
|
|
|
functions --- that is, both their inputs and outputs are of ground |
|
|
|
type (strings, arbitrary-size integers etc.)\ --- then the situation |
|
|
|
is easily summarised. At such types, all reasonable languages give |
|
|
|
rise to the same class of programmable functions, namely the |
|
|
|
Church-Turing computable ones. As for complexity, the runtime of a |
|
|
|
program is typically analysed with respect to some cost model for |
|
|
|
basic instructions (e.g.\ one unit of time per array access). |
|
|
|
Although the realism of such cost models in the asymptotic limit can |
|
|
|
be questioned (see, e.g., \citep[Section~2.6]{Knuth97}), it is broadly |
|
|
|
taken as read that such models are equally applicable whatever |
|
|
|
programming language we are working with, and moreover that all |
|
|
|
respectable languages can represent all algorithms of interest; thus, |
|
|
|
one does not expect the best achievable asymptotic run-time for a |
|
|
|
typical algorithm (say in number theory or graph theory) to be |
|
|
|
sensitive to the choice of programming language, except perhaps in |
|
|
|
marginal cases. |
|
|
|
functions, that is both their inputs and outputs are of ground type |
|
|
|
(strings, arbitrary-size integers etc), then the situation is easily |
|
|
|
summarised. At such types, all reasonable languages give rise to the |
|
|
|
same class of programmable functions, namely the Church-Turing |
|
|
|
computable ones. As for complexity, the runtime of a program is |
|
|
|
typically analysed with respect to some cost model for basic |
|
|
|
instructions (e.g.\ one unit of time per array access). Although the |
|
|
|
realism of such cost models in the asymptotic limit can be questioned |
|
|
|
(see, e.g., \citet[Section~2.6]{Knuth97}), it is broadly taken as read |
|
|
|
that such models are equally applicable whatever programming language |
|
|
|
we are working with, and moreover that all respectable languages can |
|
|
|
represent all algorithms of interest; thus, one does not expect the |
|
|
|
best achievable asymptotic run-time for a typical algorithm (say in |
|
|
|
number theory or graph theory) to be sensitive to the choice of |
|
|
|
programming language, except perhaps in marginal cases. |
|
|
|
|
|
|
|
The situation changes radically, however, if we consider |
|
|
|
\emph{higher-order} operations: programmable operations whose inputs |
|
|
|
|
