This document discusses measuring the complexity of legal systems using the United States Code as a case study. It proposes measuring complexity based on the structure, language dependence, and size of the Code. Structure is measured by the number of hierarchical elements. Language dependence is measured by the net flow of citations between titles. Size is approximated by the number of words. A weighted composite measure is calculated by ranking titles based on these factors to quantify the complexity of acquiring and understanding the information in different parts of the Code.

1. Complex Systems Models
in the Social Sciences
(Lecture 5)
!
daniel martin katz
illinois institute of technology
chicago kent college of law
@computationaldanielmartinkatz.com computationallegalstudies.com

2. How Complex is a
Particular System?

3. This is an Important
Question Considered
in the Field

4. For an Overview
See Chapter 7

5. Lots of Potential
Candidate Measures

6. http://web.mit.edu/esd.83/www/notebook/Complexity.PDF

7. “1. How hard is it to describe?
2. How hard is it to create?
3. What is its degree of organization”
Questions Typically Considered:

8. Entropy
Algorithmic Complexity
Minimum Description
Length
Fisher Information
Renyi Entropy
Chernoff Information
Dimension
Fractal Dimension
Lempel--Ziv Complexity
Code Length
(prefix-free, Huffman,
Shannon-Fano,
error-correcting,
Hamming)
1. Difficulty of Description
(Typically measured in bits)

9. Computational Complexity
Time Computational Complexity
Space Computational Complexity
Information--Based Complexity
Logical Depth
Thermodynamic Depth
Crypticity
2. Difficulty of Creation
(Typically measured in time, energy, dollars, etc.)

10. Metric Entropy
Fractal Dimension
Excess Entropy
Stochastic Complexity
Sophistication
Effective Measure Complexity
True Measure Complexity
Topological epsilon-machine size
Conditional Information
Conditional Algorithmic Information Content
Schema length
Ideal Complexity
Hierarchical Complexity
Tree-subgraph diversity
Homogeneous Complexity
Grammatical Complexity
3. Degree of organization
(a) Effective Complexity

11. Algorithmic Mutual Information
Channel Capacity
Correlation
Stored Information
Organization
3. Degree of organization
(B) Mutual Information

12. http://vserver1.cscs.lsa.umich.edu/~crshalizi/
notabene/complexity-measures.html

14. Measuring Complexity:
An Applied Case

15. Measuring the
Complexity
of Legal Systems

16. Optimal Precision in Legal Rules
Kaplow (1995) Tullock (1995)
The Complexity of the Law is
Canonical Question
Applied Scholarship --
Tax, Environmental Law, Admin Law, etc.
Legal and Political Theory
Justifications for Law and the State

17. Aligns Incentives / Channels Behavior
Offers Focal Points / Coordination Mechanisms
Encourages Actors to Internalize Costs
Maintains Monopoly on Legitimated Violence
Protects Individual Rights and Liberties
Law as a Means of
Solving Social Dilemmas

18. Complexity of Society &
Complexity of the Law
Economic Exchange
What conditions must be met for law
to Achieve these Ends?
Legal Rules Should Reflect The Nature of:
Social Interaction
Political Behavior

19. Applying the Tools of the ‘Big Data’ Era
A Perspective on the Scope
of Law in a Modern Society
Using an Important Corpus of Written Law
This Paper is an Effort (Albeit Imperfect)
to Measure the Complexity of the Law
That is Large and Cross Cutting

20. This is the United States Code

21. Compiled Version of Federal Statutory Law
This is the United States Code
Drawn from the U.S. Statutes at Large
Does not Include Fed Admin Regulations

22. The 50 titles in US Code

23. How Complex is the Code and its Components?
How has its Composition Changed Over Time?
Some Potential Questions
Can we understand if those changes Scale
to changes in the complexity of broader
Society?
How Large is the United States Code?

24. How Large is the United States Code?

25. How Large is the United States Code?

26. How Large is the United States Code?

27. How Large is the United States Code?

28. ~Title 29 - Labor
How Large is the United States Code?

29. How Large is the United States Code?

30. This is the United
States Code

31. The Case for a
Computational Approach
The United States Code is Large ...
Computational Methods are Arguably Required
Need Tools and Methods that
Scale to the Size and Scope
of this Body of Information

32. Computational Approach
to the Measurement
of Complexity
(2) Generate a Measurement
Strategy for that Object
(1) Provide a Mathematical
Representation for an Object

33. A Mathematical Approach to the
Study of the United States Code
389 Physica A 4195 (2010 Forthcoming)

34. Additional Treatment
Available Here

35. The US Code as a
Mathematical Object
Hierarchical
Structure
Title 26
Subtitle A
Chapter 1
Subchapter F
Part I
Section 501
Subsection (c)
Paragraph (3)

36. Citation
Network
Example: Tax Evasion
Title 26 -
Tax
Might Cite
Title 18 -
Crimes &
Criminal Procedure
The US Code as a
Mathematical Object
http://computationallegalstudies.com/
2009/09/14/the-structure-of-the-united-
states-code/

37. +
Hierarchical
Structure
Citation
Network

38. +
Hierarchical
Structure
Citation
Network

39. Linguistic Content
United States Code
Features
13 Million tokens
+

40. Measuring the Complexity
of the United States Code
http://link.springer.com/article/10.1007%2Fs10506-014-9160-8

41. Knowledge Acquisition
Framework
Knowledge Acquisition
Subfield at Intersection of
Computer Science and Psychology
Interested in protocols used by
subjects as they acquire, store and
analyze information

42. Interested in Mirroring the “Protocols”
Used by a Hypothetical End User
Costs of Executing the Protocols Are
Driven by the Complexity of the Object
We Consider an Individual Engaging
in a Knowledge Acquisition Process
Measuring the Complexity
of the United States Code

43. Protocol for United States
Code Knowledge Acquisition
(1) Select an initial element of the Code
corresponding to a concept of interest
(2) Beginning from this initial element,
recursively assimilate the content of all
sub-elements
(3) When a citation is encountered, apply
this protocol recursively to the cited
element

44. The Complexity of
Knowledge Acquisition
3 Factors Influence Complexity of
Executing the Protocol
Structure Language Dependence

45. Proxy for an Information
Acquisition Process
(1) Work Through a Hierarchy
(2) Look up any Citations you
encounter
(3) Confront Language of
varying length and diversity

46. The Complexity of Law
The sheer difficulty of searching for and
assimilating the information content of a
relevant body of legal rules
Distinct from other important features:
Ambiguity
Uncertainty

47. Structure
The Unnormalized Measure:
Structure
Vertices
(Nodes)

48. Structure
The Unnormalized Measure:
Structure
!
Vertices
(Nodes)

49. Net Flow
Dependence
The Unnormalized Measure:
Net Flow

50. Composite Measure
through Weighted Ranks
(A) For each factor - Measure each of the
49 Active titles
(C) Apply a Weight to each of these factors
(Simplest Case is to Average)
(B) Then, rank each title from
(1) Most to (49) Least

51. Toward a Composite Measure
Structure LanguageDependence
Composite
Measure

52. Two Forms of Weighted Ranks
Unnormalized Measure
Considers the Complexity of
reviewing a full title
Does not Control for Title Size
Normalized Measure
Considers the Average or Emblematic
Provision within the Title
Thus, we control for title size

53. The Unnormalized Measure
Structure LanguageDependence
Unnormalized
Measure
EntropyNet Flow
Vertices
(Provisions)

54. The Unnormalized Measure:
Structure
!
Structure
Vertices
(Provisions)

55. The Unnormalized Measure:
Structure
Structure
Vertices
(Provisions)

56. Net Flow
Dependence
The Unnormalized Measure:
Net Flow

57. The Unnormalized Measure:
Net Flow
Net Flow
Dependence

58. The Unnormalized Measure:
Language
Language
Entropy
!

59. The Unnormalized Measure:
Language
Language
Entropy Uniform Random

60. The Unnormalized Measure:
Language
Language
Entropy
Uniform Random
Dog,
Dog,
Dog,
Dog,
Dog,
Dog,
Dog,
Dog,
Dog,
...
snug
pizza
pasture
it
observation
hand
how
of
dare
...

61. The Unnormalized Measure:
Language
Language
Entropy Title 42 - Public Health & Welfare
Leprosy
Social Security
National Flood Insurance
US Synthetic Fuels Corp
Intl Child Abduction Remedies
Entropy ≈ Diversity

62. The Unnormalized Measure:
Language
Language
Entropy

63. The Unnormalized Measure
Structure LanguageDependence
Unnormalized
Measure
EntropyNet Flow
Vertices
(Provisions)

64. The Unnormalized Measure

65. The Normalized Measure
Structure LanguageDependence
Normalized
Measure
Mean
Depth Entropy
Net Flow
Per
Section
Tokens
Per
Section
Language

66. The Unnormalized Measure

67. Offered a framework designed to
Measure the complexity of the legal rules
Designed to open a dialogue regarding Legal
complexity and its measurement
Demonstrated measurement principles that
are potentially applicable to other classes
of legal documents
Summary
Highlighted what computation might be able
to offer to empirical legal studies