# Bottlenose Dolphins' Clicks Comply with Three Laws of Efficient Communication
## 4 Discussion  
Our results indicate that the observed repertoire of broadband click sounds emitted by _Tursiops truncatus_ conforms to three essential linguistic laws. The number of occurrences of a particular type of click in the dataset, as identified by the clustering algorithm, is inversely correlated with its frequency rank. This is consistent with the idea that, similarly to human languages, this repertoire is formed due to two opposing selectional pressures, one aiming for greater unification and a reduced number of repetitions, and the other aiming for greater token diversity. The observed slope value of \(-\)0.94 is on par with that previously reported in the domain of dolphins' whistles [15, 35, 36, 37, 38, 39, 40, 41]. This might indicate a similar strategy of use of both signal types in communication, which is conceivable given their common anatomical source, namely, nasal or phonic lips [26, 27, 28, 46, 47]. The distribution of sequence lengths followed a log-normal distribution typical for natural language by the law of brevity.  
Let us suppose that clicks are used for communicative purposes in ways that [PERSON]'s law series predicts. In that case, one can expect structural sophistication in how click sequences are formed to express meanings in the dolphins' putative language. We did not control the contextual variables in the reported study, so it remains impossible to correlate specific clicks or click sequences with conceivable meanings they could convey (cf. [48]). Nevertheless, the structural complexity of utterances can be fruitfully studied independently of associated meaning, as has been abundantly demonstrated for natural language by modern formal syntactic theory [49].  
One pertinent question is what levels of linguistic building blocks are represented by individual clicks and click sequences. Human language is characterised by the duality of patterning, whereby meaningless units such as phonemes are combined to produce minimal meaning-bearing conglomerates, e.g., morphemes, and those further up the structural hierarchy, reaching ultimately the level of a complete message or utterance [50]. In animal communication, the duality of patterning has yet to be attested, but some aspects of it were affirmed in certain species [51]. The sheer variety of click types raises the valid question of whether individual clicks or groups of clicks may possibly serve as units conventionalised for use with a fixed lexical or functional meaning. Under this hypothesis, of particular interest are potential syntactic rules used to concatenate clicks into sequences [51, 52]. Investigation of these syntactic rules requires a fine-grained analysis of the conditional probabilities of occurrence of various click combinations, possibly along the entropy computation suggested for dolphins' whistles [37]. We deem that further studies on the structure of dolphin language are required before we can make conclusions on the multifunctionality of click series. The reader may refer to [53], where a fascinating hypothesis was formulated, and a thorough review of the dolphin repertoire is available in [54, 55].  
Furthermore, the size of a click sequence was shown to correlate inversely with the duration of clicks, in accord with [PERSON]'s law. Previous studies in corpus linguistics demonstrate that [PERSON]'s law holds for word length in terms of the number of letters as well as sentence lengths in terms of the number of words and can be modelled by a version of the generalised gamma distribution, in particular, gamma [6, 7], Weibull [3], or log-normal [56] distributions. Similar shape and scale parameters characterise all three observed distributions, and choosing among them is challenging [57]. However, they all capture the exponential-like character of dependency between frequency and length.