Measuring the complexity of the law: the United States Code


Einstein’s razor, a corollary of Ockham’s razor, is often paraphrased as follows: make everything as simple as possible, but not simpler. This rule of thumb describes the challenge that designers of a legal system face—to craft simple laws that produce desired ends, but not to pursue simplicity so far as to undermine those ends. Complexity, simplicity’s inverse, taxes cognition and increases the likelihood of suboptimal decisions. In addition, unnecessary legal complexity can drive a misallocation of human capital toward comprehending and complying with legal rules and away from other productive ends. While many scholars have offered descriptive accounts or theoretical models of legal complexity, most empirical research to date has been limited to simple measures of size, such as the number of pages in a bill. No extant research rigorously applies a meaningful model to real data. As a consequence, we have no reliable means to determine whether a new bill, regulation, order, or precedent substantially effects legal complexity. In this paper, we begin to address this need by developing a proposed empirical framework for measuring relative legal complexity. This framework is based on “knowledge acquisition”, an approach at the intersection of psychology and computer science, which can take into account the structure, language, and interdependence of law. We then demonstrate the descriptive value of this framework by applying it to the U.S. Code’s Titles, scoring and ranking them by their relative complexity. We measure various features of a title including its structural size, the net flow of its intra-title citations and its linguistic entropy. Our framework is flexible, intuitive, and transparent, and we offer this approach as a first step in developing a practical methodology for assessing legal complexity.

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  1. 1.

    Recent evidence points to a potential bipartisan political constituency in favor of at least basic overtures toward simplicity. H.R. 946: Plain Writing Act of 2010 (Signed by President Obama on October 13, 2010) is designed “[T]o enhance citizen access to Government information and services by establishing that Government documents issued to the public must be written clearly, and for other purposes”.

  2. 2.

    This data set was provided by the Cornell Legal Information Institute and can be accessed at The United States Code features a total of fifty Titles. However, Title 34—Navy has been repealed. With the recent approval of Title 51—National and Commercial Space Programs the United States Code will once again feature a total of fifty active Titles. All code and additional replication materials are available here

  3. 3.

    End users of the Code are actors who interact directly with its text. End users include not only sophisticated parties, such as lawyers and lawmakers, but also laypersons, public interest groups, and businesses.

  4. 4.

    A “slip law” is the first print of a new law in pamphlet form, usually available 2–3 days after enactment. The Government Printing Office (GPO) offers a useful description of this process see

  5. 5.

    While we do not incorporate these regulations into our analysis, we recognize that their incorporation would paint a more complete picture of the relevant legal landscape. Through a process similar to the compilation of the United States Code, federal regulations are subsequently compiled by topic in the Code of Federal Regulations (C.F.R.). Of course, administrative regulations and the United States Code are not the only sources of federal legal materials. There also exist additional materials such as judicial decisions, executive orders, revenue rulings, etc.

  6. 6.

    As Title 9 is the smallest Title in the United States Code, it allows us to clearly indicate these distinctions that would otherwise be obscured by the size of the tree for other Titles.

  7. 7.

    While Chapter 1 is explicitly labeled, the remaining Chapters are located at the same horizontal level of the hierarchy.

  8. 8.

    Since Tn is a tree as in Fig. 1, As must be |V| − 1.

  9. 9.

    All code and additional replication materials are available here

  10. 10.

    The online appendix can be access here:

  11. 11.

    One obvious weakness with our proposed measure of linguistic complexity is its failure to capture the underlying semantics. As this is an introductory effort, we would invite future work focused upon this particular dimension of the question.

  12. 12.

    We acknowledge the extensive literature on text complexity (e.g., Flesch and Gould 1949; Kincaid et al. 1975; Si and Callan 2001). Much of this work, however, is directed at reading comprehension of standard sentences and paragraphs. The United States Code is a specialized document with its passages separated by the unique presentation formatting used to display statutes.

  13. 13.

    We offer this as a ceteris paribus proposition across the millions of words contained in the United States code.

  14. 14.

    We selected tokens rather than other alternative length measures, such as pages, as we believe these are far less likely to be impacted by formatting conventions.

  15. 15.

    There exist additional potential complications. For example, in some instances, longer words are more specific and thus their use can result in less ambiguity.

  16. 16.

    It should be acknowledged that compression ratios are a common alternative to entropy measures. However, due to the large variation in compression algorithms and their implementation-specific behaviors, we felt that simple Shannon entropy was a more reproducible measure than compression.

  17. 17.

    This is the red, green, blue or RGB value. A pure black canvas has an RGB value = #000000.

  18. 18.

    In the random signal case, each pixel is assigned a random color assignment. The pseudocode for this assignment requires a randomly generated string of numbers where the assigned number corresponds to an RGB value and the length of the string is equal to the number of pixels on the canvas.

  19. 19.

    In expectation, given an initial random assignment of pixel colors and a reasonably large canvas, there is likely to be at least some clustering of RGB values. This implies that at least some form of reduced representation is possible. However, this compression will be nominal.

  20. 20.

    In the context of message compression, the fragment “orange in of going the not large kick more end to …” does not easily lend itself to reduced form representation.

  21. 21.

    In the case of the uniform signal, the first fragment “dog” is the only new information content that is imparted to the end user. With only the first fragment and the total length of the message the signal could be quickly compressed.

  22. 22.

    While a number of alternative and more sophisticated forms of entropy exist, the original Shannon entropy measure is the most straight-forward measure and is still commonly used in the information science literature. Thus, for the purpose of comparing the distribution of words within Titles, we apply the Shannon entropy. For additional work on entropy see Tsallis (1988) and Rényi ( 1961 ).

  23. 23.

    Following upon common practice in the field of information retrieval and computational linguistics, we use the stopword list from the Natural Language Toolkit (NLTK) available at

  24. 24.

    There has been a significant amount of recent work on statutory citations including but not limited to Bourcier and Mazzega (2007b); Bommarito and Katz (2010); Boulet et al. (2011); Mazzega et al. (2011).

  25. 25.

    Of course, if an element contains no citations whatsoever, then the protocol above collapses to only the first two rules. However, given many elements of the Code do contain citations, we embed this consideration into our analysis.

  26. 26.

    This “walk” is by no means a random walk. Rather, it could better be described as a special case of graph traversal. These extended citation paths can grow to be quite lengthy. The maximum path length from 46 USC §51510 and 7 USC §87e requires thirty-two separate steps to complete.

  27. 27.

    As an additional complication, note that when a named Act like the IRC of 1986 is cited, one must consult a short name list in order to determine where the Act was codified.

  28. 28.

    Instead, the citation graph disobeys the hierarchical or vertical tree and memorializes various horizontal connections between elements.

  29. 29.

    In the strongly connected component of the graph, there is a directed path from each vertex in the graph to every other vertex. In the weakly connected component, there is an undirected path from each vertex in the graph to every other vertex.

  30. 30.

    A given section can feature internal references to other internal provisions. For purposes of this measurement, we do not distinguish this case from the more general case of interdependence.

  31. 31.

    It is really important to highlight the wide set of potential composite complexity measures that one could contemplate. The purpose of this article is to set forth some of the core components that might be contemplated in a future application.

  32. 32.

    The online appendix can be access here:

  33. 33.

    In this case, this is akin to assigning each measure a weight of \(\frac{1}{3}\).

  34. 34.

    While mere averaging has a certain attraction, it also represents a somewhat arbitrary approach. Given that we do not have any specific theoretical grounds that justify a departure, we have chosen this naïve approach.

  35. 35.

    In this case, “normalization” implies that in all components that comprise the composite measure the size of the Title is controlled for in one respect or another. Therefore, the measured highlighted Table 12 all measures feature a “per section” or some other analogous form of standardization.

  36. 36.

    Again, these are structure, interdependence and language.

  37. 37.

    The online appendix can be access here:

  38. 38.

    The online appendix can be access here:


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Katz, D.M., Bommarito, M.J. Measuring the complexity of the law: the United States Code. Artif Intell Law 22, 337–374 (2014).

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  • Legal complexity
  • Measuring complexity
  • Political economy
  • Artificial intelligence and law