From 0f31e1058228612b908fbd912b453f45862e5dec Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Hillerstr=C3=B6m?= Date: Wed, 26 May 2021 01:07:20 +0100 Subject: [PATCH] WIP --- thesis.tex | 31 +++++++++++++++---------------- 1 file changed, 15 insertions(+), 16 deletions(-) diff --git a/thesis.tex b/thesis.tex index fc446f6..9bbc41d 100644 --- a/thesis.tex +++ b/thesis.tex @@ -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