Artificial Intelligence and Law

, Volume 22, Issue 4, pp 337–374 | Cite as

Measuring the complexity of the law: the United States Code

  • Daniel Martin KatzEmail author
  • M. J. BommaritoII


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.


Legal complexity Measuring complexity Political economy Artificial intelligence and law 


  1. Achen CH (1978) Measuring representation. Am J Polit Sci 22:475–510Google Scholar
  2. Ansolabehere S, Snyder JM Jr, & Stewart C III (2001) Candidate positioning in US House elections. Am J Polit Sci 45:136–159Google Scholar
  3. Arrow KJ (1963) Social choice and individual values. Yale University Press, New HavenGoogle Scholar
  4. Austen-Smith D, Banks JS (1996) Information aggregation, rationality, and the Condorcet jury theorem. Am Polit Sci Rev 90:34–45Google Scholar
  5. Barton BH (2008) Judges, lawyers, and a predictive theory of legal complexity. University of Tennessee Legal Studies Research Paper No. 31Google Scholar
  6. Bates JE, Shepard HK (1993) Measuring complexity using information fluctuation. Phys Lett A 172(6):416–425CrossRefGoogle Scholar
  7. Becker GS (1983) A theory of competition among pressure groups for political influence. Q J Econ 98(3):371–400CrossRefGoogle Scholar
  8. Bibel LW (2004) AI and the conquest of complexity in law. Artif Intell Law 12(3):159–180CrossRefMathSciNetGoogle Scholar
  9. Bittker BI (1974) Tax reform and tax simplification. U Miami L Rev 29:1Google Scholar
  10. Black D (1948) On the rationale of group decision-making. J Polit Econ 56(1):23CrossRefGoogle Scholar
  11. Bommarito MJ II, Katz DM (2010) A mathematical approach to the study of the United States code. Physics A 389(19):4195–4200CrossRefGoogle Scholar
  12. Bommarito II, Michael J, Katz DM (2009) Properties of the United States code citation network. arXiv preprint arXiv:0911.1751Google Scholar
  13. Bonanno C, Collet P (2007) Complexity for extended dynamical systems. Commun Math Phys 275(3):721–748CrossRefzbMATHMathSciNetGoogle Scholar
  14. Boose JH (1989) A survey of knowledge acquisition techniques and tools. Knowl Acquis 1(1):3–37CrossRefGoogle Scholar
  15. Boose JH, Gaines BR (1990) The foundation of knowledge acquisition. Academic Press Professional, San DiegoGoogle Scholar
  16. Bose R (2002) Information theory, coding and cryptography. Tata McGraw-Hill EducationGoogle Scholar
  17. Boulet R, Mazzega P, Bourcier D (2011) A network approach to the French system of legal codes—part I: analysis of a dense network. Artif Intell Law 19(4):333–355CrossRefGoogle Scholar
  18. Bourcier D, Mazzega P (2007) Toward measures of complexity in legal systems. In: Proceedings of the 11th international conference on artificial intelligence and law. ACM, pp 211–215Google Scholar
  19. Bourcier D, Mazzega P (2007b) Codification law article and graphs. In: Lodder AR, Mommers L (eds) Legal knowledge and information systems. IOS Press, pp 29–38; ISBN 978-1-58603-810-6Google Scholar
  20. Buchanan JM, Tullock G (1965) The calculus of consent: logical foundations of constitutional democracy, vol 100. University of Michigan Press, Ann ArborGoogle Scholar
  21. Buckley JJ (1984) The multiple judge, multiple criteria ranking problem: a fuzzy set approach. Fuzzy Sets Syst 13(1):25–37CrossRefzbMATHGoogle Scholar
  22. Cecil MA (1999) Toward adding further complexity to the internal revenue code: a new paradigm for the deductibility of capital losses. U Ill L Rev 1083–1139Google Scholar
  23. Cimiano P, Hotho A, Staab S (2005) Learning concept hierarchies from text corpora using formal concept analysis. J Artif Intell Res 24:305–339zbMATHGoogle Scholar
  24. Cox GW, McCubbins MD (2007) Legislative leviathan: party government in the House. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. Csiszar I (1991) Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann Stat 19(4):2032–2066CrossRefzbMATHMathSciNetGoogle Scholar
  26. Donaldson SA (2003) Easy case against tax simplification. Va Tax Rev 22:645Google Scholar
  27. Downs A (1957) An economic theory of democracy. Harper & Brothers, New YorkGoogle Scholar
  28. Dworkin R (1986) Law’s empire. Harvard University Press, CambridgeGoogle Scholar
  29. Eckenrode RT (1965) Weighting multiple criteria. Manag Sci 12(3):180–192CrossRefGoogle Scholar
  30. Einstein A (1934) On the method of theoretical physics. Philos Sci 1(2):163–169CrossRefGoogle Scholar
  31. Epstein RA (1995) Simple rules for a complex world. Harvard University Press, CambridgeGoogle Scholar
  32. Epstein RA (2004) The optimal complexity of legal rules. Law School, University of Chicago. Olin Working Paper No. 210Google Scholar
  33. Eustice JS (1989) Tax complexity and the tax practitioner. Tax L Rev 45:7Google Scholar
  34. Fainmesser I, Fershtman C, Gandal N, Panunzi F (2005) A consistent weighted ranking scheme with an application to NCAA college football rankings. Centre for Economic Policy ResearchGoogle Scholar
  35. Feldman DP, Crutchfield JP (1998) Measures of statistical complexity: why? Phys Lett A 238(4):244–252CrossRefzbMATHMathSciNetGoogle Scholar
  36. Feltovich PJ, Spiro RJ, Coulson RL, Myers-Kelson A (1995) Reductive bias and the crisis of text (in the law). J Contemp Legal Issues 6:187Google Scholar
  37. Ferstl EC, von Cramon DY (2007) Time, space and emotion: fMRI reveals content-specific activation during text comprehension. Neurosci Lett 427(3):159–164CrossRefGoogle Scholar
  38. Flesch R, Gould AJ (1949) The art of readable writing. Harper, New York, p 196Google Scholar
  39. Flournoy A (1994) Coping with complexity. Loyola of Los Angeles Law Rev 27(3):809Google Scholar
  40. Francesconi E (2011) A learning approach for knowledge acquisition in the legal domain. In: Sartor G, Casanovas P, Biasiotti M, Fernández-Barrera M (eds) Approaches to legal ontologies. Springer, Netherlands, pp 219–233Google Scholar
  41. Frisch D (2011) Commercial law’s complexity. Geo Mason L Rev 18:245Google Scholar
  42. Ganapathi V, Vickrey D, Duchi J, Koller D (2012) Constrained approximate maximum entropy learning of markov random fields. arXiv preprint arXiv:1206.3257Google Scholar
  43. Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41(4):587–601Google Scholar
  44. Golan A, Judge G, Perloff J (1997) Estimation and inference with censored and ordered multinomial response data. J Econom 79(1):23–51CrossRefzbMATHMathSciNetGoogle Scholar
  45. Halford GS, Busby J (2007) Acquisition of structured knowledge without instruction: the relational schema induction paradigm. J Exp Psychol Learn Mem Cogn 33(3):586CrossRefGoogle Scholar
  46. Hamming RW (1986) Coding and information theory. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  47. Harsanyi JC (1955) Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J Polit Econ 63:309–321CrossRefGoogle Scholar
  48. Holsapple CW, Raj V, Wagner WP (2008) An experimental investigation of the impact of domain complexity on knowledge acquisition (KA) methods. Expert Syst Appl 35(3):1084–1094CrossRefGoogle Scholar
  49. Iria J (2009) A core ontology of knowledge acquisition. In: Aroyo L, Traverso P, Ciravegna F, Cimiano P, Heath T, Hyvönen E, Mizoguchi R, Oren E, Sabou M, Simperl E (eds) The semantic web: research and applications. Springer, Berlin, pp 233–247Google Scholar
  50. Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106(4):620CrossRefzbMATHMathSciNetGoogle Scholar
  51. Kades E (1997) Laws of complexity and the complexity of laws: the implications of computational complexity theory for the law. Rutgers L Rev 49:403Google Scholar
  52. Kaplow L (1995) A model of the optimal complexity of legal rules. J Law Econ Organ 11:150Google Scholar
  53. Kincaid JP, Fishburne RP Jr, Rogers RL, Chissom BS (1975) Derivation of new readability formulas (automated readability index, fog count and flesch reading ease formula) for navy enlisted personnel (No. RBR-8-75). Naval Technical Training Command Millington TN Research BranchGoogle Scholar
  54. Kintsch W, Van Dijk TA (1978) Toward a model of text comprehension and production. Psychol Rev 85(5):363CrossRefGoogle Scholar
  55. Kirman AP, Zimmermann JB (2001) Economics with heterogeneous interacting agents, vol 503. Springer, HeidelbergzbMATHGoogle Scholar
  56. Kolmogorov AN (1965) Three approaches to the quantitative definition ofinformation. Probl Inf Transm 1(1):1–7MathSciNetGoogle Scholar
  57. Koppelman SA (1989) At-risk and passive activity limitations: can complexity be reduced. Tax L Rev 45:97Google Scholar
  58. Lall A, Sekar V, Ogihara M, Xu J, Zhang H (2006) Data streaming algorithms for estimating entropy of network traffic. ACM SIGMETRICS Perform Eval Rev 34(1):145–156. ACMGoogle Scholar
  59. Landauer R (1988) A simple measure of complexity. Nature 336:306–307CrossRefGoogle Scholar
  60. Landauer R (1996) The physical nature of information. Phys Lett A 217(4):188–193CrossRefzbMATHMathSciNetGoogle Scholar
  61. Lazer D, Pentland AS, Adamic L, Aral S, Barabasi AL, Brewer D, Van Alstyne M (2009) Life in the network: the coming age of computational social science. Science (New York, NY) 323(5915):721Google Scholar
  62. Lloyd S, Pagels H (1988) Complexity as thermodynamic depth. Ann Phys 188(1):186–213CrossRefMathSciNetGoogle Scholar
  63. Long SB, Swingen JA (1987) An approach to the measurement of tax law complexity. J Am Tax Assoc 8(2):22–36Google Scholar
  64. Manning CD, Raghavan P, Schütze H (2008) Introduction to information retrieval. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  65. Mazzega P, Bourcier D, Bourgine P, Nadah N, Boulet R (2011) A complex-system approach: legal knowledge, ontology, information and networks. In: Sartor G, Casanovas P, Biasiotti M, Fernández-Barrera M (eds) Approaches to legal ontologies. Springer, Netherlands, pp 117–132Google Scholar
  66. McCaffery EJ (1990) Holy grail of tax simplification. Wis L Rev 1267–1322Google Scholar
  67. McKelvey RD (1976) Intransitivities in multidimensional voting models and some implications for agenda control. J Econ Theory 12(3):472–482CrossRefzbMATHMathSciNetGoogle Scholar
  68. McKelvey RD (1986) Covering, dominance, and institution-free properties of social choice. Am J Polit Sci 30:283–314Google Scholar
  69. Mitchell M (2009) Complexity: a guided tour. Oxford University Press, OxfordGoogle Scholar
  70. Nigam K, Lafferty J, McCallum A (1999) Using maximum entropy for text classification. In: IJCAI-99 workshop on machine learning for information filtering, vol 1, pp 61–67Google Scholar
  71. Ohm P (2009) Computer programming and the law: a new research agenda. Vill L Rev 54:117MathSciNetGoogle Scholar
  72. Ostrom E (1990) Governing the commons: the evolution of institutions for collective action. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  73. Pagallo U (2010) As law goes by: topology, ontology, evolution. In: Casanovas P, Pagallo U, Sartor G, Ajani G (eds) AI approaches to the complexity of legal systems. complex systems, the semantic web, ontologies, argumentation, and dialogue. Springer, Berlin, pp 12–26Google Scholar
  74. Page SE (2008) Uncertainty, difficulty, and complexity. J Theor Polit 20(2):115–149CrossRefGoogle Scholar
  75. Paul DL (1997) Sources of tax complexity: how much simplicity can fundamental tax reform achieve. NCL Rev 76:151Google Scholar
  76. Phelan DR (2009) The effect of complexity of law on litigation strategy. In: Masson A, Shariff MJ (eds) Legal strategies. Springer, Berlin, pp 335–351Google Scholar
  77. Pitt MM, Slemrod J (1989) The compliance cost of itemizing deductions: evidence from individual tax returns. Am Econ Rev 79:1224–1232Google Scholar
  78. Pollock E, Chandler P, Sweller J (2002) Assimilating complex information. Learn Instr 12(1):61–86CrossRefGoogle Scholar
  79. Poole KT, Rosenthal H (1991) Patterns of congressional voting. Am J Polit Sci 35:228–278Google Scholar
  80. Quade D (1979) Using weighted rankings in the analysis of complete blocks with additive block effects. J Am Stat Assoc 74:680Google Scholar
  81. Rényi A (1961) On measures of entropy and information. In: Fourth Berkeley symposium on mathematical statistics and probability, pp 547–561Google Scholar
  82. Riker WH (1962) The theory of political coalitions, vol 578. Yale University Press, New HavenGoogle Scholar
  83. Rook LW (1993) Laying down the law: canons for drafting complex legislation. Or L Rev 72:663Google Scholar
  84. Rothkopf MH, Pekeč A, Harstad RM (1998) Computationally manageable combinational auctions. Manag Sci 44(8):1131–1147CrossRefzbMATHGoogle Scholar
  85. Ruhl JB (2008) Law’s complexity: a primer. Ga St UL Rev 24:885Google Scholar
  86. Sanderson M, Croft B (1999) Deriving concept hierarchies from text. In: Proceedings of the 22nd annual international ACM SIGIR conference on research and development in information retrieval, ACM, pp 206–213Google Scholar
  87. Schenk DH (1989) Simplification for individual taxpayers: problems and proposals. Tax L Rev 45:121Google Scholar
  88. Schennach SM (2005) Bayesian exponentially tilted empirical likelihood. Biometrika 92(1):31–46CrossRefzbMATHMathSciNetGoogle Scholar
  89. Schnotz W, Kürschner C (2008) External and internal representations in the acquisition and use of knowledge: visualization effects on mental model construction. Instr Sci 36(3):175–190CrossRefGoogle Scholar
  90. Schuck PH (1992) Legal complexity: some causes, consequences, and cures. Duke Law J 42:1–52Google Scholar
  91. Schuck PE (2000) The limits of law. Westview Press, BoulderGoogle Scholar
  92. Sen A (1970) Collective choice and social welfare. Holden Day, San FranciscoGoogle Scholar
  93. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423CrossRefzbMATHMathSciNetGoogle Scholar
  94. Shannon CE (1951) Prediction and entropy of printed English. Bell Syst Tech J 30(1):50–64CrossRefzbMATHGoogle Scholar
  95. Shoham Y, Leyton-Brown K (2009) Multiagent systems: algorithmic, game-theoretic, and logical foundations. Cambridge University Press, CambridgeGoogle Scholar
  96. Si L, Callan J (2001) A statistical model for scientific readability. In: Proceedings of the tenth international conference on Information and knowledge management. ACM, pp 574–576Google Scholar
  97. Slemrod J (2005) The etiology of tax complexity: evidence from US state income tax systems. Public Financ Rev 33(3):279–299CrossRefGoogle Scholar
  98. Slemrod JB, Blumenthal M (1996) The income tax compliance cost of big business. Public Financ Rev 24(4):411–438CrossRefGoogle Scholar
  99. Soofi ES (2000) Principal information theoretic approaches. J Am Stat Assoc 95(452):1349–1353CrossRefzbMATHMathSciNetGoogle Scholar
  100. Spiro RJ, Jehng JC (1990) Cognitive flexibility and hypertext: theory and technology for the nonlinear and multidimensional traversal of complex subject matter. Cogn Educ Multimed Explor Ideas High Technol 163–205Google Scholar
  101. Stoop R, Stoop N, Bunimovich L (2004) Complexity of dynamics as variability of predictability. J Stat Phys 114(3–4):1127–1137CrossRefzbMATHMathSciNetGoogle Scholar
  102. Surrey SS (1969) Complexity and the internal revenue code: the problem of the management of tax detail. Law Contemp Probl 34:673–710Google Scholar
  103. Sweller J, Chandler P (1994) Why some material is difficult to learn. Cogn Instr 12(3):185–233CrossRefGoogle Scholar
  104. Tang A, Jackson D, Hobbs J, Chen W, Smith JL, Patel H, Beggs JM (2008) A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro. J Neurosci 28(2):505–518CrossRefGoogle Scholar
  105. Tress W (2009) Lost laws: what we can’t find in the United States code. Golden Gate UL Rev 40:129Google Scholar
  106. Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52(1–2):479–487CrossRefzbMATHMathSciNetGoogle Scholar
  107. Tukey JW (1957) Sums of random partitions of ranks. Ann Math Stat 23:987–992Google Scholar
  108. Tullock G (1995) On the desirable degree of detail in the law. Eur J Law Econ 2(3):199–209CrossRefGoogle Scholar
  109. Weingast BR, Marshall WJ (1988) The industrial organization of Congress; or, why legislatures, like firms, are not organized as markets. J Polit Econ 96:132–163Google Scholar
  110. White MJ (1992) Legal complexity and lawyers’ benefit from litigation. Int Rev Law Econ 12(3):381–395CrossRefGoogle Scholar
  111. Wright RG (2000) Illusion of simplicity: an explanation of why the law can’t just be less complex. Fla St UL Rev 27:715Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA
  2. 2.Lex Predict, LLCWayneUSA

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