Here we consider the problem of all the possible orders of a linguistic
structure formed by $n$ elements, for instance, subject, direct object and verb
($n=3$) or subject, direct object, indirect object and verb ($n=4$). We
investigate if the frequency of the $n!$ possible orders is constrained by two
principles. First, entropy minimization, a principle that has been suggested to
shape natural communication systems at distinct levels of organization. Second,
swap distance minimization, namely a preference for word orders that require
fewer swaps of adjacent elements to be produced from a source order. Here we
present average swap distance, a novel score for research on swap distance
minimization, and investigate the theoretical distribution of that score for
any $n$: its minimum and maximum values and its expected value in die rolling
experiments or when the word order frequencies are shuffled. We investigate
whether entropy and average swap distance are significantly small in distinct
linguistic structures with $n=3$ or $n=4$ in agreement with the corresponding
minimization principles. We find strong evidence of entropy minimization and
swap distance minimization with respect to a die rolling experiment. The
evidence of these two forces with respect to a Polya urn process is strong for
$n=4$ but weaker for $n=3$. We still find evidence of swap distance
minimization when word order frequencies are shuffled, indicating that swap
distance minimization effects are beyond pressure to minimize word order
entropy.

考虑语言结构的所有可能排列顺序的问题，研究了熵最小化和交换距离最小化原则对于 $n=3$ 或 $n=4$ 的语言结构中的影响，并通过掷骰实验和词序频率随机排列验证了熵最小化和交换距离最小化的强有力证据。

在词序变异中最小化交换距离超越熵最小化

Swap distance minimization beyond entropy minimization in word order  variation

Distance minimization is a general principle of language. A special case of
this principle in the domain of word order is swap distance minimization. This
principle predicts that variations from a canonical order that are reached by
fewer swaps of adjacent constituents are lest costly and thus more likely. Here
we investigate the principle in the context of the triple formed by subject
(S), object (O) and verb (V). We introduce the concept of word order rotation
as a cognitive underpinning of that prediction. When the canonical order of a
language is SOV, the principle predicts SOV < SVO, OSV < VSO, OVS < VOS, in
order of increasing cognitive cost. We test the prediction in three flexible
order SOV languages: Korean (Koreanic), Malayalam (Dravidian), and Sinhalese
(Indo-European). Evidence of swap distance minimization is found in all three
languages, but it is weaker in Sinhalese. Swap distance minimization is
stronger than a preference for the canonical order in Korean and especially
Malayalam.

语言中的距离最小化是一般原则，其特例是词序中的交换距离最小化。本文探讨了该原则在主语（S）、宾语（O）和动词（V）三者组成的三元组中的表现，并引入了词序旋转的概念作为预测的认知基础。结果表明，在三种灵活顺序 SOV 语言（韩语、马拉雅拉姆语和僧伽罗语）中均存在交换距离最小化的证据，但在僧伽罗语中较为弱。在韩语和尤其是马拉雅拉姆语中，交换距离最小化比对规范顺序的偏好更为突出。