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