The complexity of the law has been the topic of both scholarly writing and scientific investigation, with the main challenge being the proper definition of “complexity”. Historically, articles in law journals took a conceptual and non-technical approach toward the “complexity of the law”, motivated by practical reasons, such as the ever increasing amount of legislation produced every year and the resulting cost of knowledge acquisition, e.g. [1, 2]. Although this approach is important, it remains technically vague and not accessible to quantitative analysis and measurement. Over the past decade, with the increasing availability of digitized (legal) data and the steady growth of computational power, a new type of literature has emerged within legal theory, the authors of which use various mathematical notions that come from areas as diverse as physics and information theory or graph theory, to analyze the complexity of the law, cf. e.g. [3–5]. The complexity considered results mainly from the exogenous structure of the positive law, i.e. the tree-like hierarchical organization of the legal texts in a forest consisting of codes (root nodes), chapters, sections, etc., but also from the associated reference network.

According to the dichotomy introduced by [6]; one can distinguish between structure-based measures and content-based measures of complexity, with the former pertaining to the field of knowledge representation (knowledge engineering) and the latter relating to the complexity of the norms, which includes, e.g. the (certainty of) legal commands, their efficiency and socio-economic impact.

In this article, we advance the measurement of legal complexity by focusing on the language using a method originating in the physics literature, cf. [7]. So, we map legal documents from several major legal systems into a two-dimensional complexity-entropy plane, spanned by the (normalized) vocabulary entropy and the compression factor, cf. Section 2.1. Using an abstract and rigorous measurement of the complexity of the law, should have significant practical benefits for policy, as discussed previously by, e.g. [1, 2]. For example, it could potentially identify parts of the law that need to be rewritten in order to remain manageable, thereby reducing the costs for citizens and firms who are supposed to comply. Most notably, the French Constitutional Court has ruled that articles of unjustified “excessive-complexity” are unconstitutional ¹ . However, in order to render the notion of “excessive complexity” functional, quantitative methods are needed such as those used by [5, 8]; and which our version of the complexity-entropy plane ideally complements.

A non-trivial question that arises in several disciplines is how the complexity of a hierarchical structure, i.e. of a multi-scale object, can be measured. Different areas of human knowledge are coded as written texts that are organized hierarchically, e.g. each book’s Table of Contents reflects its inherent hierarchical organization as a tree, and all books together form a forest. Furthermore, a tree-like structure appears again at the sentence level in the form of the syntax tree and its semantics as an additional degree of freedom. Although various measures of complexity have been introduced that are specially adapted to a particular class of problems, there is still no unified theory. The first concept we consider is Shannon entropy, [9]; which is a measure of information. It is an observable on the space of probability distributions with values in the non-negative real numbers. For a discrete probability distribution P : = { p 1 , … , p N } , with p i > 0 , for all i, and ∑ ​ i = 1 N p i = 1 , the Shannon entropy H ( P ) , is defined as: H ( P ) : = − ∑ i = 1 N p i ⁡ log 2 ( p i ) , (1) with log 2 , the logarithm with base 2. The normalized Shannon entropy H n ( P ) , is given by H n ( P ) : = H ( P ) log 2 ( N ) , (2)

i.e. by dividing H ( P ) by the entropy H ( P N ) of the discrete uniform distribution P N : = { 1 / N , … , 1 / N } , for N different outcomes. We shall use the normalized entropy in order to measure the information content of the vocabulary of individual legal texts, for details cf. Section 6.3. Word entropies have previously been used by various authors. In the legal domain [5], calculated the word entropy, after removing stop words, for the individual Titles of the U.S. Code. [10] used word entropies to gauge Shakespeare’s and Jin Yong’s writing capacity, based on the 100 most frequent words in each text.

The second concept we consider is related to Kolmogorov complexity (cf. [11, 12] and references therein), which is the prime example of algorithmic (computational) complexity. Heuristically, the complexity of an object is defined as the length of the shortest of all possible descriptions. Further fundamental examples of algorithmic complexity include Lempel-Ziv complexity C 76 , [13]; or Wolfram’s complexity measure of a regular language, [14]. The latter is defined as (logarithm of) the minimal number of nodes of a deterministic finite automaton (DFA) that recognizes the language (Meyhill-Nerode theorem). In order to facilitate the discussion, let us propose a set of axioms for a complexity measure. This measure is basically a general form of an outer measure.

Let X be (at least) a monoid ( X , ∘ , ε ) , with binary composition ∘ : X × X → X , and identity element ε , and additionally, let ≥ be a partial order on X.

A complexity measure C on X, is a functional C : X → ℝ + , such that for all a , b ∈ X , we have

pointed: C ( ε ) = 0 , (3)

monotone: if   a ≤ b   then   C ( a ) ≤ C ( b ) , (4)

sub-additive: C ( a ∘ b ) ≤ C ( a ) + C ( b ) . (5)

Examples satisfying the above axioms include tree structures, with the (simple) complexity measure given by the number of levels, i.e. the depth from the baseline. Then the empty tree has zero complexity, the partial order being given by being a sub-tree and composition being given by grafting trees. Further, the Lempel-Ziv complexity C 76 , and Wolfram’s complexity measure for regular languages, if slightly differently defined via recognizable series, satisfy the axioms. However, plain Kolmogorov complexity does not satisfy, e.g. sub-additivity, cf. the discussion by [12].

[24] (pp. 10–11) intuitively describe the broad difference between classical information theory and algorithmic complexity, which we summarize next. Whereas information theory (entropy), as conceived by Shannon, determines the minimal number of bits needed to transmit a set of messages, it does not provide the number of bits necessary to transmit a particular message from the set. Kolmogorov complexity on the other hand, focuses on the information content of an individual finite object, e.g. a play by Shakespeare, accounting for the (empirical) fact that strings which are meaningful to humans, are compressible, cf. [19]. In order to relate entropy, Kolmogorov complexity or Ziv-Lempel compression to one another, various mathematical assumptions such as stationarity, ergodicity or infinity are required, cf. [11, 17, 25]. Also, the convergence of various quantities found in natural languages, e.g. entropy estimates, [26]; are based on some of these assumptions. Despite the fact that the different approximations and assumptions proved valuable for language models, natural language is not necessarily generated by a stationary ergodic process, cf. [11]; as e.g., cf. [25]; the probability of upcoming words can depend on words which are far away. But, as argued by [27]; it is precisely due to the non-ergodic nature of natural language that one can empirically distinguish different topics, e.g. by determining the uneven distribution of keywords in texts, cf. also [28]. [29] considered a model of a random languages and showed how structure emerges as a result of the competition between energy and entropy.

Finally, let us comment on the relation between relative frequencies and probabilities in the context of entropy. Given a standard n-simplex, Δ n , i.e. ( x 0 , … , x n ) ∈ ℝ n + 1 , ∑ ​ i = 0 n x i = 1 , and x i ≥ 0 , for i = 0 , … , n , its points can either be interpreted as discrete probability distributions on ( n + 1 ) elements or as the set of relative frequencies of ( n + 1 ) elements. The distinction between the two concepts is relevant as the Shannon entropy H, provides in both cases a functional (observable) H : Δ n → ℝ + , which, in our context, has two possible interpretations. Namely, as a component of a coordinate system on (law) texts, which is the interpretation in the present study, but also as an estimate of the Shannon entropy of the language used if considered as a sample from the space of all (law) texts of a certain type. In the latter case, it is known that the “naive” estimation of the Shannon entropy Eq. 1 from finite samples is biased. Therefore, several estimators have been developed to solve this problem. We utilize the entropy estimator introduced by [30]; in order to reexamine some of our results in the light of a probabilistic interpretation, and find that it has no qualitative effect on the outcome, cf. Supplementary Material.

Complex systems, e.g. biological, physical or social ones, are high-dimensional multi-scale objects. [31]; and [32] realized that in order to describe them, entropy is not enough, and an independent complexity measure is needed. Guided by the insight that the intuitive notion of complexity for patterns, when ordered by the degree of disorder, is at odds with its algorithmic description, the notion of the physical complexity of a system emerged, cf. [7, 31, 33]. The corresponding physical complexity measure, pioneered by [33]; should not be a monotone function of the disorder or the entropy, but should attain its maximum between complete order (perfect crystal) and total disorder (isolated ideal gas). [7]; introduced the excess Shannon entropy as a statistical complexity measure for physical systems, and later [34] introduced another physical complexity measure, the product of a system’s entropy with its disequilibrium measure. [35]; introduced a novel approach to handle the complexity of patterns on multiple scales using a multi-level renormalization technique to quantify the complexity of a (two- or three-dimensional) pattern by a scalar quantity that should ultimately better fit the intuitive notion of complexity.

[7]; paired both the entropy and the physical complexity measure into what has become a complexity-entropy diagram, in order to describe non-linear dynamical systems; for a review cf. [36]. Remarkably, these low-dimensional coordinates are often sufficient to characterize such systems (in analogy to principal component analysis), since they capture the inherent randomness, but also the degree of organization. Several variants of entropy-complexity diagrams are now widely used, even outside the original context. [37]; by combining the normalized word entropy, cf. Eq. 7, with a version of a statistical complexity measure, quantitatively study Shakespeare and other English Renaissance authors. [23]; used for the complexity-entropy plane the entropy rate and the entropy density and studied the organization of literary texts (Shakespeare, Abbott and Doyle) at different levels of the hierarchy. In order to calculate the entropy rate and density, which are asymptotic quantities, they used the Lempel-Ziv complexity C 76 . Strictly speaking this approach would require the source to be stationary and ergodic, cf. [11].

We introduce a new variant Γ of the complexity-entropy plane, spanned by the normalized word entropy and the compression factor, in order to study text data. So, every text t, can be represented by a point in Γ , via the map t ↦ ( H n ( t ) , r ( t ) ) , with coordinates H n , the normalized Shannon entropy of the underlying vocabulary, and r, the compression factor. Let us note, that Γ is naturally a metric space, e.g. with the Euclidean metric, but other metrics may be more appropriate, depending on the particular question at hand.

Let us now motivate some of our research questions from the perspective of Legal Theory.

[38] and his school introduced and formalized the notion of the “Stufenbau der Rechtsordnung”, ² which led to the concept of the hierarchy of norms. The hierarchy starts with the Constitution (often originating in a revolutionary charter written by the “pouvoir constituant”), which governs the creation of statutes or acts, which themselves govern the creation (by delegation) of regulations, administrative actions, and also the judiciary. At the national level these (abstract) concepts are taken into account, e.g. Guide de légistique [39]; when drafting positive law. This is valid for, e.g. Austria, France, Germany, Italy, Switzerland and the European Union, although strictly speaking, it does not have a formal Constitution. Every new piece of legislation has to fit the preexisting order, so at each level, the content outlined at an upper level, has to be made more precise, which leads to the supposed linguistic gradient of abstraction. A new phenomenon can be observed for regulations, namely that the legislature, or more precisely its drafting agencies, is being forced to abandon the realm of natural language and take an approach that is common to all scientific writing, namely the inclusion of images, figures and formulae. The purpose of figures, tables and formulae is not only the ability to succinctly visualize or summarize large amounts of abstract information, but most often it is the only mean to convey complex scientific information at all. As regulations increasingly leave the domain of jurisprudence, novel methods should be adopted. For example [2], advocated the inclusion of mathematical formulae in a statute if this statue contains a computation that is based on this formula. Ultimately, a natural scientific approach (including the writing style) to law would be beneficial, however, this might be at odds with the idea of law being intelligible to a wide audience.

Our hypothesis is that these functional differences between the levels of the hierarchy of legal norms should manifest themselves as differences in vocabulary entropy or in the compression factor.

Our first analysis, cf. Table 1, is a summary of the sizes of the different corpora, the languages used, the number of individual items, the mean text sizes and standard deviations. The analysis shows different approaches to the organization of national law, namely either by thousands of small texts of around 50 KB (Canada, Germany, United Kingdom) or less than a hundred large codes, several MB in size (France, United States), with the regulations significantly exceeding the number of acts. Note that the French codes contain both the law and the corresponding regulation in the same text. The size of a corpus within the same category, i.e. act or regulation, differs from country to country by an order of magnitude or even two, which is noteworthy as broadly similar or even identical areas are regulated within the law, e.g. banking, criminal, finance or tax law. This begs the question of what an efficient codification should ideally look like. The Swiss Federal codification is remarkably compact, despite the fact that the English version does not contain all acts or regulations available in German, French or Italian (which are the official languages); nevertheless all important and recent ones are included, cf. Section 6.1.

We introduced a tool that is new to the legal field but has already served other areas of scientific research well. Its main strength is the ability to simultaneously capture and visualize independent and fundamental information, namely entropy and complexity, of large collections of data, and to track changes over time. By devising a novel variant of the complexity-entropy plane, we were not only able to show that legal texts of different types and languages are located in distinguishable regions, but also to identify different drafting approaches with regard to laws and regulations. In addition, we have taken the first steps to follow the spatial evolution of the legislation over time. Although we observe that constitutions tend to have lower compression factors than acts and regulations, and regulations on average have higher compression factors than acts, which corresponds to the hierarchy of norms, we could not fully capture the assumed abstraction gradient. This suggests that other language-specific methods should also be used to investigate (possible) differences. On the other hand, the high(er) redundancy of the regulations reflects the increasing need to leave the realm of natural language and to borrow tools from the natural sciences. The analysis we perform can be modified in a number of ways to provide even more specific information. So, one might include n-grams, or perform additional pre-processing steps, or choose different compression algorithms. Also, one might add a third coordinate for even more visual information. In combination with other quantitative methods such as citation networks or the consideration of additional (internal) degrees of freedom such as local entropy, new types of quantitative research questions could be formulated, which may lead to more efficient and manageable legislation. In summary, we expect a broad range of further applications of complexity-entropy diagrams within the legal domain.