Ramon Ferrer‐i‐Cancho

Stuart Semple, Ramon Ferrer-i-Cancho, Morgan L. Gustison

Linguistic laws, the common statistical patterns of human language, have been investigated by quantitative linguists for nearly a century. Recently, biologists from a range of disciplines have started to explore the prevalence of these laws beyond language, finding patterns consistent with linguistic laws across multiple levels of biological organisation, from molecular (genomes, genes, and proteins) to organismal (animal behaviour) to ecological (populations and ecosystems). We propose a new conceptual framework for the study of linguistic laws in biology, comprising and integrating distinct levels of analysis, from description to prediction to theory building. Adopting this framework will provide critical new insights into the fundamental rules of organisation underpinning natural systems, unifying linguistic laws and core theory in biology.

Sonia Petrini, Antoni Casas-i-Muñoz, Jordi Cluet-i-Martinell et al.

Zipf's law of abbreviation, the tendency of more frequent words to be shorter, is one of the most solid candidates for a linguistic universal, in the sense that it has the potential for being exceptionless or with a number of exceptions that is vanishingly small compared to the number of languages on Earth. Since Zipf's pioneering research, this law has been viewed as a manifestation of a universal principle of communication, i.e. the minimization of word lengths, to reduce the effort of communication. Here we revisit the concordance of written language with the law of abbreviation. Crucially, we provide wider evidence that the law holds also in speech (when word length is measured in time), in particular in 46 languages from 14 linguistic families. Agreement with the law of abbreviation provides indirect evidence of compression of languages via the theoretical argument that the law of abbreviation is a prediction of optimal coding. Motivated by the need of direct evidence of compression, we derive a simple formula for a random baseline indicating that word lengths are systematically below chance, across linguistic families and writing systems, and independently of the unit of measurement (length in characters or duration in time). Our work paves the way to measure and compare the degree of optimality of word lengths in languages.

Sonia Petrini, Ramon Ferrer-i-Cancho

The syntactic structure of a sentence can be represented as a graph, where vertices are words and edges indicate syntactic dependencies between them. In this setting, the distance between two linked words is defined as the difference between their positions. Here we wish to contribute to the characterization of the actual distribution of syntactic dependency distances, which has previously been argued to follow a power-law distribution. Here we propose a new model with two exponential regimes in which the probability decay is allowed to change after a break-point. This transition could mirror the transition from the processing of word chunks to higher-level structures. We find that a two-regime model - where the first regime follows either an exponential or a power-law decay - is the most likely one in all 20 languages we considered, independently of sentence length and annotation style. Moreover, the break-point exhibits low variation across languages and averages values of 4-5 words, suggesting that the amount of words that can be simultaneously processed abstracts from the specific language to a high degree. The probability decay slows down after the breakpoint, consistently with a universal chunk-and-pass mechanism. Finally, we give an account of the relation between the best estimated model and the closeness of syntactic dependencies as function of sentence length, according to a recently introduced optimality score.

Sonia Petrini, Antoni Casas-i-Muñoz, Jordi Cluet-i-Martinell et al.

Zipf's law of abbreviation, namely the tendency of more frequent words to be shorter, has been viewed as a manifestation of compression, i.e. the minimization of the length of forms -- a universal principle of natural communication. Although the claim that languages are optimized has become trendy, attempts to measure the degree of optimization of languages have been rather scarce. Here we present two optimality scores that are dualy normalized, namely, they are normalized with respect to both the minimum and the random baseline. We analyze the theoretical and statistical pros and cons of these and other scores. Harnessing the best score, we quantify for the first time the degree of optimality of word lengths in languages. This indicates that languages are optimized to 62 or 67 percent on average (depending on the source) when word lengths are measured in characters, and to 65 percent on average when word lengths are measured in time. In general, spoken word durations are more optimized than written word lengths in characters. Our work paves the way to measure the degree of optimality of the vocalizations or gestures of other species, and to compare them against written, spoken, or signed human languages.

Lluís Alemany-Puig, Ramon Ferrer-i-Cancho

Dependency trees have proven to be a very successful model to represent the syntactic structure of sentences of human languages. In these structures, vertices are words and edges connect syntactically-dependent words. The tendency of these dependencies to be short has been demonstrated using random baselines for the sum of the lengths of the edges or its variants. A ubiquitous baseline is the expected sum in projective orderings (wherein edges do not cross and the root word of the sentence is not covered by any edge), that can be computed in time $O(n)$. Here we focus on a weaker formal constraint, namely planarity. In the theoretical domain, we present a characterization of planarity that, given a sentence, yields either the number of planar permutations or an efficient algorithm to generate uniformly random planar permutations of the words. We also show the relationship between the expected sum in planar arrangements and the expected sum in projective arrangements. In the domain of applications, we derive a $O(n)$-time algorithm to calculate the expected value of the sum of edge lengths. We also apply this research to a parallel corpus and find that the gap between actual dependency distance and the random baseline reduces as the strength of the formal constraint on dependency structures increases, suggesting that formal constraints absorb part of the dependency distance minimization effect. Our research paves the way for replicating past research on dependency distance minimization using random planar linearizations as random baseline.

Lluís Alemany-Puig, Juan Luis Esteban, Ramon Ferrer-i-Cancho

A linear arrangement is a mapping $π$ from the $n$ vertices of a graph $G$ to $n$ distinct consecutive integers. Linear arrangements can be represented by drawing the vertices along a horizontal line and drawing the edges as semicircles above said line. In this setting, the length of an edge is defined as the absolute value of the difference between the positions of its two vertices in the arrangement, and the cost of an arrangement as the sum of all edge lengths. Here we study two variants of the Maximum Linear Arrangement problem (MaxLA), which consists of finding an arrangement that maximizes the cost. In the planar variant for free trees, vertices have to be arranged in such a way that there are no edge crossings. In the projective variant for rooted trees, arrangements have to be planar and the root of the tree cannot be covered by any edge. In this paper we present algorithms that are linear in time and space to solve planar and projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements, and show that caterpillar trees maximize planar MaxLA over all trees of a fixed size thereby generalizing a previous extremal result on trees.

Lluís Alemany-Puig, Juan Luis Esteban, Ramon Ferrer-i-Cancho

The new and growing field of Quantitative Dependency Syntax has emerged at the crossroads between Dependency Syntax and Quantitative Linguistics. One of the main concerns in this field is the statistical patterns of syntactic dependency structures. These structures, grouped in treebanks, are the source for statistical analyses in these and related areas; dozens of scores devised over the years are the tools of a new industry to search for patterns and perform other sorts of analyses. The plethora of such metrics and their increasing complexity require sharing the source code of the programs used to perform such analyses. However, such code is not often shared with the scientific community or is tested following unknown standards. Here we present a new open-source tool, the Linear Arrangement Library (LAL), which caters to the needs of, especially, inexperienced programmers. This tool enables the calculation of these metrics on single syntactic dependency structures, treebanks, and collection of treebanks, grounded on ease of use and yet with great flexibility. LAL has been designed to be efficient, easy to use (while satisfying the needs of all levels of programming expertise), reliable (thanks to thorough testing), and to unite research from different traditions, geographic areas, and research fields.

Ramon Ferrer-i-Cancho

We address the linguistic problem of the sequential arrangement of a head and its dependents from an information theoretic perspective. In particular, we consider the optimal placement of a head that maximizes the predictability of the sequence. We assume that dependents are statistically independent given a head, in line with the open-choice principle and the core assumptions of dependency grammar. We demonstrate the optimality of harmonic order, i.e., placing the head last maximizes the predictability of the head whereas placing the head first maximizes the predictability of dependents. We also show that postponing the head is the optimal strategy to maximize its predictability while bringing it forward is the optimal strategy to maximize the predictability of dependents. We unravel the advantages of the strategy of maximizing the predictability of the head over maximizing the predictability of dependents. Our findings shed light on the placements of the head adopted by real languages or emerging in different kinds of experiments.

Ramon Ferrer-i-Cancho, Savithry Namboodiripad

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.

Víctor Franco-Sánchez, Arnau Martí-Llobet, Ramon Ferrer-i-Cancho

Consider 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 whether 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. We present average swap distance, a novel score for research on swap distance minimization. We find strong evidence of pressure for entropy minimization and swap distance minimization with respect to a die rolling experiment in distinct linguistic structures with $n=3$ or $n=4$. Evidence with respect to a Polya urn process is strong for $n=4$ but weaker for $n=3$. We still find evidence consistent with the action of swap distance minimization when word order frequencies are shuffled, indicating that swap distance minimization effects are beyond pressure to reduce word order entropy.

Ramon Ferrer-i-Cancho

The word order of a sentence is shaped by multiple principles. The principle of syntactic dependency distance minimization is in conflict with the principle of surprisal minimization (or predictability maximization) in single head syntactic dependency structures: while the former predicts that the head should be placed at the center of the linear arrangement, the latter predicts that the head should be placed at one of the ends (either first or last). A critical question is when surprisal minimization (or predictability maximization) should surpass syntactic dependency distance minimization. In the context of single head structures, it has been predicted that this is more likely to happen when two conditions are met, i.e. (a) fewer words are involved and (b) words are shorter. Here we test the prediction on the noun phrase when it is composed of a demonstrative, a numeral, an adjective and a noun. We find that, across preferred orders in languages, the noun tends to be placed at one of the ends, confirming the theoretical prediction. We also show evidence of anti locality effects: syntactic dependency distances in preferred orders are longer than expected by chance.

Ramon Ferrer-i-Cancho

The frequency of the preferred order for a noun phrase formed by demonstrative, numeral, adjective and noun has received significant attention over the last two decades. We investigate the actual distribution of the 24 possible orders. There is no consensus on whether it is well-fitted by an exponential or a power law distribution. We find that an exponential distribution is a much better model. This finding and other circumstances where an exponential-like distribution is found challenge the view that power-law distributions, e.g., Zipf's law for word frequencies, are inevitable. We also investigate which of two exponential distributions gives a better fit: an exponential model where the 24 orders have non-zero probability (a geometric distribution truncated at rank 24) or an exponential model where the number of orders that can have non-zero probability is variable (a right-truncated geometric distribution). When consistency and generalizability are prioritized, we find higher support for the exponential model where all 24 orders have non-zero probability. These findings strongly suggest that there is no hard constraint on word order variation and then unattested orders merely result from undersampling, consistently with Cysouw's view.

Ramon Ferrer-i-Cancho, Marta Arias

The syntactic structure of a sentence can be described as a tree that indicates the syntactic relationships between words. In spite of significant progress in unsupervised methods that retrieve the syntactic structure of sentences, guessing the right direction of edges is still a challenge. As in a syntactic dependency structure edges are oriented away from the root, the challenge of guessing the right direction can be reduced to finding an undirected tree and the root. The limited performance of current unsupervised methods demonstrates the lack of a proper understanding of what a root vertex is from first principles. We consider an ensemble of centrality scores, some that only take into account the free tree (non-spatial scores) and others that take into account the position of vertices (spatial scores). We test the hypothesis that the root vertex is an important or central vertex of the syntactic dependency structure. We confirm the hypothesis in the sense that root vertices tend to have high centrality and that vertices of high centrality tend to be roots. The best performance in guessing the root is achieved by novel scores that only take into account the position of a vertex and that of its neighbours. We provide theoretical and empirical foundations towards a universal notion of rootness from a network science perspective.

Ramon Ferrer-i-Cancho, Carlos Gómez-Rodríguez

Dependency distance minimization (DDm) is a well-established principle of word order. It has been predicted theoretically that DDm implies compression, namely the minimization of word lengths. This is a second order prediction because it links a principle with another principle, rather than a principle and a manifestation as in a first order prediction. Here we test that second order prediction with a parallel collection of treebanks controlling for annotation style with Universal Dependencies and Surface-Syntactic Universal Dependencies. To test it, we use a recently introduced score that has many mathematical and statistical advantages with respect to the widely used sum of dependency distances. We find that the prediction is confirmed by the new score when word lengths are measured in phonemes, independently of the annotation style, but not when word lengths are measured in syllables. In contrast, one of the most widely used scores, i.e. the sum of dependency distances, fails to confirm that prediction, showing the weakness of raw dependency distances for research on word order. Finally, our findings expand the theory of natural communication by linking two distinct levels of organization, namely syntax (word order) and word internal structure.

Lluís Alemany-Puig, Ramon Ferrer-i-Cancho

The syntactic structure of a sentence is often represented using syntactic dependency trees. The sum of the distances between syntactically related words has been in the limelight for the past decades. Research on dependency distances led to the formulation of the principle of dependency distance minimization whereby words in sentences are ordered so as to minimize that sum. Numerous random baselines have been defined to carry out related quantitative studies on languages. The simplest random baseline is the expected value of the sum in unconstrained random permutations of the words in the sentence, namely when all the shufflings of the words of a sentence are allowed and equally likely. Here we focus on a popular baseline: random projective permutations of the words of the sentence, that is, permutations where the syntactic dependency structure is projective, a formal constraint that sentences satisfy often in languages. Thus far, the expectation of the sum of dependency distances in random projective shufflings of a sentence has been estimated approximately with a Monte Carlo procedure whose cost is of the order of $Rn$, where $n$ is the number of words of the sentence and $R$ is the number of samples; it is well known that the larger $R$, the lower the error of the estimation but the larger the time cost. Here we present formulae to compute that expectation without error in time of the order of $n$. Furthermore, we show that star trees maximize it, and give an algorithm to retrieve the trees that minimize it.

Neus Català, Jaume Baixeries, Ramon Ferrer-Cancho et al.

In his pioneering research, G. K. Zipf formulated a couple of statistical laws on the relationship between the frequency of a word with its number of meanings: the law of meaning distribution, relating the frequency of a word and its frequency rank, and the meaning-frequency law, relating the frequency of a word with its number of meanings. Although these laws were formulated more than half a century ago, they have been only investigated in a few languages. Here we present the first study of these laws in Catalan. We verify these laws in Catalan via the relationship among their exponents and that of the rank-frequency law. We present a new protocol for the analysis of these Zipfian laws that can be extended to other languages. We report the first evidence of two marked regimes for these laws in written language and speech, paralleling the two regimes in Zipf's rank-frequency law in large multi-author corpora discovered in early 2000s. Finally, the implications of these two regimes will be discussed.

David Carrera-Casado, Ramon Ferrer-i-Cancho

Biosemiosis is a process of choice-making between simultaneously alternative options. It is well-known that, when sufficiently young children encounter a new word, they tend to interpret it as pointing to a meaning that does not have a word yet in their lexicon rather than to a meaning that already has a word attached. In previous research, the strategy was shown to be optimal from an information theoretic standpoint. In that framework, interpretation is hypothesized to be driven by the minimization of a cost function: the option of least communication cost is chosen. However, the information theoretic model employed in that research neither explains the weakening of that vocabulary learning bias in older children or polylinguals nor reproduces Zipf's meaning-frequency law, namely the non-linear relationship between the number of meanings of a word and its frequency. Here we consider a generalization of the model that is channeled to reproduce that law. The analysis of the new model reveals regions of the phase space where the bias disappears consistently with the weakening or loss of the bias in older children or polylinguals. The model is abstract enough to support future research on other levels of life that are relevant to biosemiotics. In the deep learning era, the model is a transparent low-dimensional tool for future experimental research and illustrates the predictive power of a theoretical framework originally designed to shed light on the origins of Zipf's rank-frequency law.

Lluís Alemany-Puig, Juan Luis Esteban, Ramon Ferrer-i-Cancho

The Minimum Linear Arrangement problem (MLA) consists of finding a mapping $π$ from vertices of a graph to distinct integers that minimizes $\sum_{\{u,v\}\in E}|π(u) - π(v)|$. In that setting, vertices are often assumed to lie on a horizontal line and edges are drawn as semicircles above said line. For trees, various algorithms are available to solve the problem in polynomial time in $n=|V|$. There exist variants of the MLA in which the arrangements are constrained. Iordanskii, and later Hochberg and Stallmann (HS), put forward $O(n)$-time algorithms that solve the problem when arrangements are constrained to be planar (also known as one-page book embeddings). We also consider linear arrangements of rooted trees that are constrained to be projective (planar embeddings where the root is not covered by any edge). Gildea and Temperley (GT) sketched an algorithm for projective arrangements which they claimed runs in $O(n)$ but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in $O(n \log d_{max})$ where $d_{max}$ is the maximum degree but did not provide sufficient detail. Here we correct an error in HS's algorithm for the planar case, show its relationship with the projective case, and derive simple algorithms for the projective and planar cases that run without a doubt in $O(n)$ time.

Ramon Ferrer-i-Cancho, Carlos Gómez-Rodríguez, Juan Luis Esteban et al.

It is often stated that human languages, as other biological systems, are shaped by cost-cutting pressures but, to what extent? Attempts to quantify the degree of optimality of languages by means of an optimality score have been scarce and focused mostly on English. Here we recast the problem of the optimality of the word order of a sentence as an optimization problem on a spatial network where the vertices are words, arcs indicate syntactic dependencies and the space is defined by the linear order of the words in the sentence. We introduce a new score to quantify the cognitive pressure to reduce the distance between linked words in a sentence. The analysis of sentences from 93 languages representing 19 linguistic families reveals that half of languages are optimized to a 70% or more. The score indicates that distances are not significantly reduced in a few languages and confirms two theoretical predictions, i.e. that longer sentences are more optimized and that distances are more likely to be longer than expected by chance in short sentences. We present a new hierarchical ranking of languages by their degree of optimization. The new score has implications for various fields of language research (dependency linguistics, typology, historical linguistics, clinical linguistics and cognitive science). Finally, the principles behind the design of the score have implications for network science.

Ramon Ferrer-i-Cancho, Carlos Gómez-Rodríguez, Juan Luis Esteban

A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems on the sum of edge lengths in trees of a fixed size: the minimum and the maximum value of the sum for specific trees, the minimum and the maximum in classes of trees (bistar trees and caterpillar trees) and finally the minimum and the maximum for any tree. We establish some foundations for research on optimality scores for spatial networks in one dimension.

Carlos Gómez-Rodríguez, Morten H. Christiansen, Ramon Ferrer-i-Cancho

The ability to produce and understand an unlimited number of different sentences is a hallmark of human language. Linguists have sought to define the essence of this generative capacity using formal grammars that describe the syntactic dependencies between constituents, independent of the computational limitations of the human brain. Here, we evaluate this independence assumption by sampling sentences uniformly from the space of possible syntactic structures. We find that the average dependency distance between syntactically related words, a proxy for memory limitations, is less than expected by chance in a collection of state-of-the-art classes of dependency grammars. Our findings indicate that memory limitations have permeated grammatical descriptions, suggesting that it may be impossible to build a parsimonious theory of human linguistic productivity independent of non-linguistic cognitive constraints.

Xinying Chen, Carlos Gómez-Rodríguez, Ramon Ferrer-i-Cancho

A comment on "Neurophysiological dynamics of phrase-structure building during sentence processing" by Nelson et al (2017), Proceedings of the National Academy of Sciences USA 114(18), E3669-E3678.

Antoni Lozano, Bernardino Casas, Chris Bentz et al.

Entropy is a fundamental property of a repertoire. Here, we present an efficient algorithm to estimate the entropy of types with the help of Zhang's estimator. The algorithm takes advantage of the fact that the number of different frequencies in a text is in general much smaller than the number of types. We justify the convenience of the algorithm by means of an analysis of the statistical properties of texts from more than 1000 languages. Our work opens up various possibilities for future research.

Ramon Ferrer-i-Cancho

Comment on "Dependency distance: a new perspective on syntactic patterns in natural language" by Haitao Liu et al

Ramon Ferrer-i-Cancho

The minimization of the length of syntactic dependencies is a well-established principle of word order and the basis of a mathematical theory of word order. Here we complete that theory from the perspective of information theory, adding a competing word order principle: the maximization of predictability of a target element. These two principles are in conflict: to maximize the predictability of the head, the head should appear last, which maximizes the costs with respect to dependency length minimization. The implications of such a broad theoretical framework to understand the optimality, diversity and evolution of the six possible orderings of subject, object and verb are reviewed.

Ramon Ferrer-i-Cancho, Carlos Gomez-Rodriguez, J. L. Esteban

The syntactic structure of a sentence can be modelled as a tree, where vertices correspond to words and edges indicate syntactic dependencies. It has been claimed recurrently that the number of edge crossings in real sentences is small. However, a baseline or null hypothesis has been lacking. Here we quantify the amount of crossings of real sentences and compare it to the predictions of a series of baselines. We conclude that crossings are really scarce in real sentences. Their scarcity is unexpected by the hubiness of the trees. Indeed, real sentences are close to linear trees, where the potential number of crossings is maximized.

R. Ferrer-i-Cancho, D. Lusseau, B. McCowan

A short review of similarities between dolphins and humans with the help of quantitative linguistics and information theory.

Ramon Ferrer-i-Cancho

Here we sketch a new derivation of Zipf's law for word frequencies based on optimal coding. The structure of the derivation is reminiscent of Mandelbrot's random typing model but it has multiple advantages over random typing: (1) it starts from realistic cognitive pressures (2) it does not require fine tuning of parameters and (3) it sheds light on the origins of other statistical laws of language and thus can lead to a compact theory of linguistic laws. Our findings suggest that the recurrence of Zipf's law in human languages could originate from pressure for easy and fast communication.

Carlos Gómez-Rodríguez, Ramon Ferrer-i-Cancho

The structure of a sentence can be represented as a network where vertices are words and edges indicate syntactic dependencies. Interestingly, crossing syntactic dependencies have been observed to be infrequent in human languages. This leads to the question of whether the scarcity of crossings in languages arises from an independent and specific constraint on crossings. We provide statistical evidence suggesting that this is not the case, as the proportion of dependency crossings of sentences from a wide range of languages can be accurately estimated by a simple predictor based on a null hypothesis on the local probability that two dependencies cross given their lengths. The relative error of this predictor never exceeds 5% on average, whereas the error of a baseline predictor assuming a random ordering of the words of a sentence is at least 6 times greater. Our results suggest that the low frequency of crossings in natural languages is neither originated by hidden knowledge of language nor by the undesirability of crossings per se, but as a mere side effect of the principle of dependency length minimization.

Ramon Ferrer-i-Cancho

Word order evolution has been hypothesized to be constrained by a word order permutation ring: transitions involving orders that are closer in the permutation ring are more likely. The hypothesis can be seen as a particular case of Kauffman's adjacent possible in word order evolution. Here we consider the problem of the association of the six possible orders of S, V and O to yield a couple of primary alternating orders as a window to word order evolution. We evaluate the suitability of various competing hypotheses to predict one member of the couple from the other with the help of information theoretic model selection. Our ensemble of models includes a six-way model that is based on the word order permutation ring (Kauffman's adjacent possible) and another model based on the dual two-way of standard typology, that reduces word order to basic orders preferences (e.g., a preference for SV over VS and another for SO over OS). Our analysis indicates that the permutation ring yields the best model when favoring parsimony strongly, providing support for Kauffman's general view and a six-way typology.

Ramon Ferrer-i-Cancho, Carlos Gómez-Rodríguez

A commentary on the article "Large-scale evidence of dependency length minimization in 37 languages" by Futrell, Mahowald & Gibson (PNAS 2015 112 (33) 10336-10341).

Ramon Ferrer-i-Cancho

In a recent article, Christiansen and Chater (2016) present a fundamental constraint on language, i.e. a now-or-never bottleneck that arises from our fleeting memory, and explore its implications, e.g., chunk-and-pass processing, outlining a framework that promises to unify different areas of research. Here we explore additional support for this constraint and suggest further connections from quantitative linguistics and information theory.

R. Ferrer-i-Cancho, C. Bentz, C. Seguin

Languages across the world exhibit Zipf's law of abbreviation, namely more frequent words tend to be shorter. The generalized version of the law - an inverse relationship between the frequency of a unit and its magnitude - holds also for the behaviours of other species and the genetic code. The apparent universality of this pattern in human language and its ubiquity in other domains calls for a theoretical understanding of its origins. To this end, we generalize the information theoretic concept of mean code length as a mean energetic cost function over the probability and the magnitude of the types of the repertoire. We show that the minimization of that cost function and a negative correlation between probability and the magnitude of types are intimately related.

Ramon Ferrer-i-Cancho

A family of information theoretic models of communication was introduced more than a decade ago to explain the origins of Zipf's law for word frequencies. The family is a based on a combination of two information theoretic principles: maximization of mutual information between forms and meanings and minimization of form entropy. The family also sheds light on the origins of three other patterns: the principle of contrast, a related vocabulary learning bias and the meaning-frequency law. Here two important components of the family, namely the information theoretic principles and the energy function that combines them linearly, are reviewed from the perspective of psycholinguistics, language learning, information theory and synergetic linguistics. The minimization of this linear function is linked to the problem of compression of standard information theory and might be tuned by self-organization.

Ramon Ferrer-i-Cancho

The use of null hypotheses (in a statistical sense) is common in hard sciences but not in theoretical linguistics. Here the null hypothesis that the low frequency of syntactic dependency crossings is expected by an arbitrary ordering of words is rejected. It is shown that this would require star dependency structures, which are both unrealistic and too restrictive. The hypothesis of the limited resources of the human brain is revisited. Stronger null hypotheses taking into account actual dependency lengths for the likelihood of crossings are presented. Those hypotheses suggests that crossings are likely to reduce when dependencies are shortened. A hypothesis based on pressure to reduce dependency lengths is more parsimonious than a principle of minimization of crossings or a grammatical ban that is totally dissociated from the general and non-linguistic principle of economy.

Ramon Ferrer-i-Cancho

The syntactic structure of a sentence can be modeled as a tree where vertices are words and edges indicate syntactic dependencies between words. It is well-known that those edges normally do not cross when drawn over the sentence. Here a new null hypothesis for the number of edge crossings of a sentence is presented. That null hypothesis takes into account the length of the pair of edges that may cross and predicts the relative number of crossings in random trees with a small error, suggesting that a ban of crossings or a principle of minimization of crossings are not needed in general to explain the origins of non-crossing dependencies. Our work paves the way for more powerful null hypotheses to investigate the origins of non-crossing dependencies in nature.

Ramon Ferrer-i-Cancho

According to Zipf's meaning-frequency law, words that are more frequent tend to have more meanings. Here it is shown that a linear dependency between the frequency of a form and its number of meanings is found in a family of models of Zipf's law for word frequencies. This is evidence for a weak version of the meaning-frequency law. Interestingly, that weak law (a) is not an inevitable of property of the assumptions of the family and (b) is found at least in the narrow regime where those models exhibit Zipf's law for word frequencies.

Ramon Ferrer-i-Cancho

Comment on "Approaching human language with complex networks" by Cong & Liu

Alvaro Corral, Gemma Boleda, Ramon Ferrer-i-Cancho

Zipf's law is a fundamental paradigm in the statistics of written and spoken natural language as well as in other communication systems. We raise the question of the elementary units for which Zipf's law should hold in the most natural way, studying its validity for plain word forms and for the corresponding lemma forms. In order to have as homogeneous sources as possible, we analyze some of the longest literary texts ever written, comprising four different languages, with different levels of morphological complexity. In all cases Zipf's law is fulfilled, in the sense that a power-law distribution of word or lemma frequencies is valid for several orders of magnitude. We investigate the extent to which the word-lemma transformation preserves two parameters of Zipf's law: the exponent and the low-frequency cut-off. We are not able to demonstrate a strict invariance of the tail, as for a few texts both exponents deviate significantly, but we conclude that the exponents are very similar, despite the remarkable transformation that going from words to lemmas represents, considerably affecting all ranges of frequencies. In contrast, the low-frequency cut-offs are less stable.

Ramon Ferrer-i-Cancho

Vocabulary learning by children can be characterized by many biases. When encountering a new word, children as well as adults, are biased towards assuming that it means something totally different from the words that they already know. To the best of our knowledge, the 1st mathematical proof of the optimality of this bias is presented here. First, it is shown that this bias is a particular case of the maximization of mutual information between words and meanings. Second, the optimality is proven within a more general information theoretic framework where mutual information maximization competes with other information theoretic principles. The bias is a prediction from modern information theory. The relationship between information theoretic principles and the principles of contrast and mutual exclusivity is also shown.

Ramon Ferrer-i-Cancho

It is well known that the length of a syntactic dependency determines its online memory cost. Thus, the problem of the placement of a head and its dependents (complements or modifiers) that minimizes online memory is equivalent to the problem of the minimum linear arrangement of a star tree. However, how that length is translated into cognitive cost is not known. This study shows that the online memory cost is minimized when the head is placed at the center, regardless of the function that transforms length into cost, provided only that this function is strictly monotonically increasing. Online memory defines a quasi-convex adaptive landscape with a single central minimum if the number of elements is odd and two central minima if that number is even. We discuss various aspects of the dynamics of word order of subject (S), verb (V) and object (O) from a complex systems perspective and suggest that word orders tend to evolve by swapping adjacent constituents from an initial or early SOV configuration that is attracted towards a central word order by online memory minimization. We also suggest that the stability of SVO is due to at least two factors, the quasi-convex shape of the adaptive landscape in the online memory dimension and online memory adaptations that avoid regression to SOV. Although OVS is also optimal for placing the verb at the center, its low frequency is explained by its long distance to the seminal SOV in the permutation space.

Ramon Ferrer-i-Cancho

It has been hypothesized that the rather small number of crossings in real syntactic dependency trees is a side-effect of pressure for dependency length minimization. Here we answer a related important research question: what would be the expected number of crossings if the natural order of a sentence was lost and replaced by a random ordering? We show that this number depends only on the number of vertices of the dependency tree (the sentence length) and the second moment about zero of vertex degrees. The expected number of crossings is minimum for a star tree (crossings are impossible) and maximum for a linear tree (the number of crossings is of the order of the square of the sequence length).

Ramon Ferrer-i-Cancho, Łukasz Dębowski, Fermín Moscoso del Prado Martín

Constant entropy rate (conditional entropies must remain constant as the sequence length increases) and uniform information density (conditional probabilities must remain constant as the sequence length increases) are two information theoretic principles that are argued to underlie a wide range of linguistic phenomena. Here we revise the predictions of these principles to the light of Hilberg's law on the scaling of conditional entropy in language and related laws. We show that constant entropy rate (CER) and two interpretations for uniform information density (UID), full UID and strong UID, are inconsistent with these laws. Strong UID implies CER but the reverse is not true. Full UID, a particular case of UID, leads to costly uncorrelated sequences that are totally unrealistic. We conclude that CER and its particular cases are incomplete hypotheses about the scaling of conditional entropies.

Ramon Ferrer-i-Cancho

Here tree dependency structures are studied from three different perspectives: their degree variance (hubiness), the mean dependency length and the number of dependency crossings. Bounds that reveal pairwise dependencies among these three metrics are derived. Hubiness (the variance of degrees) plays a central role: the mean dependency length is bounded below by hubiness while the number of crossings is bounded above by hubiness. Our findings suggest that the online memory cost of a sentence might be determined not just by the ordering of words but also by the hubiness of the underlying structure. The 2nd moment of degree plays a crucial role that is reminiscent of its role in large complex networks.

Ramon Ferrer-i-Cancho, Haitao Liu

Mixing dependency lengths from sequences of different length is a common practice in language research. However, the empirical distribution of dependency lengths of sentences of the same length differs from that of sentences of varying length and the distribution of dependency lengths depends on sentence length for real sentences and also under the null hypothesis that dependencies connect vertices located in random positions of the sequence. This suggests that certain results, such as the distribution of syntactic dependency lengths mixing dependencies from sentences of varying length, could be a mere consequence of that mixing. Furthermore, differences in the global averages of dependency length (mixing lengths from sentences of varying length) for two different languages do not simply imply a priori that one language optimizes dependency lengths better than the other because those differences could be due to differences in the distribution of sentence lengths and other factors.

Stuart Semple, Minna J. Hsu, Govindasamy Agoramoorthy et al.

Words follow the law of brevity, i.e. more frequent words tend to be shorter. From a statistical point of view, this qualitative definition of the law states that word length and word frequency are negatively correlated. Here the recent finding of patterning consistent with the law of brevity in Formosan macaque vocal communication (Semple et al., 2010) is revisited. It is shown that the negative correlation between mean duration and frequency of use in the vocalizations of Formosan macaques is not an artifact of the use of a mean duration for each call type instead of the customary 'word' length of studies of the law in human language. The key point demonstrated is that the total duration of calls of a particular type increases with the number of calls of that type. The finding of the law of brevity in the vocalizations of these macaques therefore defies a trivial explanation.

Ramon Ferrer-i-Cancho, Antoni Hernández-Fernández

Parallels of Zipf's law of brevity, the tendency of more frequent words to be shorter, have been found in bottlenose dolphins and Formosan macaques. Although these findings suggest that behavioral repertoires are shaped by a general principle of compression, common marmosets and golden-backed uakaris do not exhibit the law. However, we argue that the law may be impossible or difficult to detect statistically in a given species if the repertoire is too small, a problem that could be affecting golden backed uakaris, and show that the law is present in a subset of the repertoire of common marmosets. We suggest that the visibility of the law will depend on the subset of the repertoire under consideration or the repertoire size.