Complexity and Entropy in Legal Language
Open accessAbstract
We study the language of legal codes from different countries and legal traditions, using concepts from physics, algorithmic complexity theory and information theory. We show that vocabulary entropy, which measures the diversity of the author’s choice of words, in combination with the compression factor, which is derived from a lossless compression algorithm and measures the redundancy present in a text, is well suited for separating different writing styles in different languages, in particular also legal language. We show that different types of (legal) text, e.g. acts, regulations or literature, are located in distinct regions of the complexity-entropy plane, spanned by the information and complexity measure. This two-dimensional approach already gives new insights into the drafting style and structure of statutory texts and complements other methods.
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Legal language is notoriously complex.
“Legal language is notoriously complex [3–5]”
From GPT-4 passes the bar exam (2024) · cites this paper as [3]
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Cites
A Mathematical Theory of Communication
Shannon, C. E. (1948) · Bell System Technical Journal · doi:10.1002/j.1538-7305.1948.tb01338.x · Cited by 88,486 · see citing works
A universal algorithm for sequential data compression
Ziv, J., & Lempel, A. (1977) · IEEE Transactions on Information Theory · doi:10.1109/tit.1977.1055714 · Cited by 5,456 · see citing works
An Introduction to Kolmogorov Complexity and Its Applications
Li, M., & Vitanyi, P. M. B. (2019) · Texts in computer science · doi:10.1007/978-3-030-11298-1 · Cited by 3,752 · see citing works
Speech and language processing
Politt, R., Pollock, J., & Waller, E. (2010) · Taylor & Francis eBooks · doi:10.4324/9780203461891_chapter_3 · Cited by 3,594 · see citing works
Compression of individual sequences via variable-rate coding
Ziv, J., & Lempel, A. (1978) · IEEE Transactions on Information Theory · doi:10.1109/tit.1978.1055934 · Cited by 3,432 · see citing works
On the Complexity of Finite Sequences
Lempel, A., & Ziv, J. (1976) · IEEE Transactions on Information Theory · doi:10.1109/tit.1976.1055501 · Cited by 2,713 · see citing works
Prediction and Entropy of Printed English
Shannon, C. E. (1951) · Bell System Technical Journal · doi:10.1002/j.1538-7305.1951.tb01366.x · Cited by 2,686 · see citing works
Data Compression: The Complete Reference
(2009) · Kybernetes · doi:10.1108/k.2009.06738eae.001 · Cited by 1,597 · see citing works
Inferring statistical complexity
Crutchfield, J. P., & Young, K. (1989) · Physical Review Letters · doi:10.1103/physrevlett.63.105 · Cited by 987 · see citing works
Algorithmic Information Theory
Chaitin, G. J. (1977) · IBM Journal of Research and Development · doi:10.1147/rd.214.0350 · Cited by 982 · see citing works
1 A Statistical Measure of ComplexityPreprint
López‐Ruiz, R., Mancini, H., & Calbet, X. (1995) · Cited by 939 · see citing works
A Statistical Measure of Complexity
López-Ruiz, R., Mancini, H. L., & Calbet, X. (1995) · Physics Letters A · doi:10.1016/0375-9601(95)00867-5 · Cited by 920 · see citing works
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Cover, T. M., & Thomas, J. A. (2006) · Wiley-Interscience eBooks · Cited by 787 · see citing works
Toward a quantitative theory of self-generated complexity
Grassberger, P. (1986) · International Journal of Theoretical Physics · doi:10.1007/bf00668821 · Cited by 702 · see citing works
Computation theory of cellular automata
Wolfram, S. (1984) · Communications in Mathematical Physics · doi:10.1007/bf01217347 · Cited by 634 · see citing works
Crutchfield, J. P. (2011) · Nature Physics · doi:10.1038/nphys2190 · Cited by 435 · see citing works
Entropy estimation of symbol sequences
Schürmann, T., & Grassberger, P. (1996) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.166191 · Cited by 313 · see citing works
Huberman, B. A., & Hogg, T. (1986) · Physica D Nonlinear Phenomena · doi:10.1016/0167-2789(86)90308-1 · Cited by 175 · see citing works
Feldman, D. P., McTague, C. S., & Crutchfield, J. P. (2008) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.2991106 · Cited by 164 · see citing works
Shannon Information and Kolmogorov ComplexityPreprint
Grünwald, P., & Vitanyi, P. M. B. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.cs/0410002 · Cited by 154 · see citing works
Measuring the complexity of the law: the United States Code
Katz, D. M., & Bommarito, M. J. (2014) · Artificial Intelligence and Law · doi:10.1007/s10506-014-9160-8 · Cited by 117 · see citing works
Legal Complexity: Some Causes, Consequences, and Cures
Schuck, P. H. (1992) · Duke Law Journal · doi:10.2307/1372753 · Cited by 117 · see citing works
Universal Entropy of Word Ordering Across Linguistic Families
Montemurro, M. A., & Zanette, D. H. (2011) · PLoS ONE · doi:10.1371/journal.pone.0019875 · Cited by 95 · see citing works
Measuring, Monitoring, and Managing Legal ComplexityPreprint
Ruhl, J. B., & Katz, D. M. (2015) · SSRN Electronic Journal · Cited by 74 · see citing works
Entropy Estimates from Insufficient Samplings
Grassberger, P. (2003) · arXiv · doi:10.48550/arxiv.physics/0307138 · Cited by 74 · see citing works
A mathematical approach to the study of the United States Code
Bommarito, M. J., & Katz, D. M. (2010) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2010.05.057 · Cited by 68 · see citing works
Multiscale structural complexity of natural patterns
Bagrov, A. A., Iakovlev, I. A., Iliasov, A. A., Katsnelson, M. I., & Mazurenko, V. V. (2020) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2004976117 · Cited by 67 · see citing works
Rosso, O. A., Craig, H., & Moscato, P. (2008) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2008.11.018 · Cited by 66 · see citing works
Schuck, P. H. (2018) · Routledge eBooks · doi:10.4324/9780429499180-1 · Cited by 54 · see citing works
The network of French legal codes
Mazzéga, P., Bourcier, D., & Boulet, R. (2009) · International Conference on Artificial Intelligence and Law · doi:10.1145/1568234.1568271 · Cited by 47 · see citing works
Toward measures of complexity in legal systems
Bourcier, D., & Mazzega, P. (2007) · Proceedings of the 11th international conference on Artificial intelligence and law · doi:10.1145/1276318.1276359 · Cited by 32 · see citing works
DeGiuli, E. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.128301 · Cited by 31 · see citing works
Hansel, G., Dominique Perrin, A., & Simon, I. (1992) · Lecture notes in computer science · doi:10.1007/3-540-55210-3_209 · Cited by 24 · see citing works
Measuring, Monitoring and Managing Legal Complexity
Katz, D. M., & Ruhl, J. B. (2015) · bepress Legal Repository · Cited by 22 · see citing works
Is Natural Language a Perigraphic Process? The Theorem about Facts and Words Revisited
Dębowski, Ł. (2018) · Entropy · doi:10.3390/e20020085 · Cited by 18 · see citing works
Complexity-entropy analysis at different levels of organisation in written language
Rams, E. E., Mesa-Rodríguez, A., & Estévez-Moya, D. (2019) · PLoS ONE · doi:10.1371/journal.pone.0214863 · Cited by 15 · see citing works
Correlation Lengths in the Language of Computable Information
Martiniani, S., Lemberg, Y., Chaikin, P. M., & Levine, D. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.170601 · Cited by 13 · see citing works
Measuring information-based energy and temperature of literary texts
Chang, M.-C., Yang, A. C., Stanley, H. E., et al. (2016) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2016.11.106 · Cited by 12 · see citing works
Friedrich, R., Luzzatto, M., & Ash, E. (2020) · Repository for Publications and Research Data (ETH Zurich) · doi:10.3929/ethz-b-000427800 · Cited by 1 · see citing works
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Research works printed but not linked: 2
- Kelsen H. Reine Rechtslehre: Mit einem Anhang: Das Problem der Gerechtigkeit. Tübingen: Mohr Siebeck (2017). doi:10.33196/9783704683991 · DOI
- Koehn P. Conference Proceedings: The Tenth Machine Translation Summit. AAMT. Phuket, Thailand: AAMT (2005). p. 79-86. Europarl: A Parallel Corpus for Statistical Machine Translation.
Cited by
Sorted by citations
Katz, D. M., Bommarito, M. J., Gao, S., & Arredondo, P. (2024) · Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences · doi:10.1098/rsta.2023.0254 · Cited by 420 · see citing works
- Authors
- Friedrich, R.
- Affiliations
- ETH Zurich (CH)
- Corresponding author
- Friedrich, Roland
- Venue
- Frontiers in Physics
- Publisher
- Frontiers Media SA
- Published
- 2021-06-04
- Volume
- 9
- ISSN
- 2296-424X (ISSN-L)
- OpenAlex
- W3168273140
- Cited by
- 19
- References
- 39
- Citations by year
- 2026: 4 · 2025: 4 · 2024: 6 · 2023: 3 · 2022: 1 · 2021: 1
- FWCI
- 0.97
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Frontiers in Physics (publishedVersion, cc-by)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Repository for Publications and Research Data (ETH Zurich) (submittedVersion, cc-by)
- ETH Zürich Research Collection (cc-by)
- Funders
- Harvard University
- Eidgenössische Technische Hochschule Zürich
- Max-Planck-Institut für Physik Komplexer Systeme
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Authorship Attribution and Profiling
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Fractal and DNA sequence analysis
- Social Sciences › Law › Legal Language and Interpretation
- Keywords
- legal language, legal codes, algorithmic complexity, lossless compression, statutory text, legal tradition, information theory
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