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From Structural Description to Interactions in Law : the Case for Simplicial Complexes Jacky Legrand Université Paris 2, CERSA

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september Atelier Complexité et Politiques PubliquesJ. Legrand 2 Introduction problem statement Several communities focuses on networks Networks studied via graph theory higher dimensional networks Applications leave out the legal systems

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september Atelier Complexité et Politiques PubliquesJ. Legrand 3 Introduction objectives Generalising to relations among N things Developments concerning Law

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september Atelier Complexité et Politiques PubliquesJ. Legrand 4 Introduction method Syntactic & semantic perspectives 1st level of interpretation –topology Q-analysis* –maths methodology to explicitate 2nd level of interpretation –Q-analysis* applications typology * Ronald Atkin 70's 40 years of applications

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september Atelier Complexité et Politiques PubliquesJ. Legrand 5 Introduction outline Graph –the vertices stand for things –the edges stand for binary relations Hypergraph –the edges no longer connect pairs –the edges enclose subsets of vertices Simplicial complex –multi-dimensional polygonal shapes

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september Atelier Complexité et Politiques PubliquesJ. Legrand 6 Simplicial complex Vertices (b i ) Simplices (a j ) Dimension (a 4, 2) Faces (a 1, ) Nearness (a 1 a 2,b 4 ) Connectivity (a 1,a 2,a 5,a 6,a 4 ) Loops (a 4,a 5,a 6 ) Holes

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september Atelier Complexité et Politiques PubliquesJ. Legrand 7 1st level of interpretation Simplicial complex construction –vertices simplices Choice of invariants –algebraic –topological –combinatorial Quantitative measures Qualitative observations

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september Atelier Complexité et Politiques PubliquesJ. Legrand 8 2nd level of interpretation Applications of Q-analysis Domain-oriented typology –down to earth –a relational world –artefacts, abstractions and concepts Item-oriented typology –significance of simplices in complex ? –polyhedra as reified objects ?

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september Atelier Complexité et Politiques PubliquesJ. Legrand 9 Multidimensional networks in legal system Statute Law –to measure and model substantive complexity –Boulet Bourcier Mazzega (11th & 12th ICAIL) –legal analysis via graph theory Common Law –argumentation frameworks –computational model of legal inference –case-based reasoning

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september Atelier Complexité et Politiques PubliquesJ. Legrand 10 Multidimensional networks in legal system Statute Law Networking of the French legal Codes –legal analysis via graph theory –vertices of the graph are articles of law –the edges stand for quotations –X cites Y (X, Y tied vertices) Extension to a multidimensional network 1 – method for complex construction –simplices = neighbour of X (or Y) –simplices = cliques* of graph * subset of X, Y such every pair connected 2 – to use Q-analysis (for one result)

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september Atelier Complexité et Politiques PubliquesJ. Legrand 11 Multidimensional networks in legal system Statute Law Connectivity extends "degree" –to which extent neighbours of X are connected together –q for each dimension of faces sharing –Q-structure vector number of connected components at each q level large values = globally fragmented structure New or generalised measures (degree, closeness, betweenness, eccentricity)

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september Atelier Complexité et Politiques PubliquesJ. Legrand 12 Multidimensional networks in legal system Statute Law Apart from quantitative techniques To generalise the notion of nearness –stars and hubs A new meaning for connectivity –holes = shomotopic objects (discrete homotopy) The backcloth-traffic paradigm –structural description and dynamics –the backcloth "influences" activity –the backcloth "allows" and "forbids"

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september Atelier Complexité et Politiques PubliquesJ. Legrand 13 Multidimensional networks in legal system Statute Law Two-mode data network –a bipartite graph is two layered –an edge connects only vertices in different sets –Q-analysis is mainly devoted to 2 mode data Legal complexity modelling offer 2-mode networks –a Code cites a Code = one-mode data network –partitioning the corpus introduces 2-mode data –links between legal texts and jurisprudential texts

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september Atelier Complexité et Politiques PubliquesJ. Legrand 14 Multidimensional networks in legal system Statute Law From simplicial complexes to ontologies Hypernetworks (J. Johnson 2000's) Simplicial family –"assembly relation" and "cones" –to map vertices to a higher structured set Iterating the process –to create simplices at a higher level Multilevel lattice hierarchy –sharing faces in the cones allows Q analysis

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september Atelier Complexité et Politiques PubliquesJ. Legrand 15 Multidimensional networks in legal system Common Law Judicial process based on the analysis of the previous cases versus deductive : facts+context "legal text" Case and precedents = arguments Theories to explain cases Argumentation frameworks Recurrent worry for computational models Intense research efforts in AI & Law

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september Atelier Complexité et Politiques PubliquesJ. Legrand 16 Multidimensional networks in legal system Common Law Inventory of well known issues (25 years) –VAFS, CATO, CABARET, GREBE, HYPO To extract the different concepts Unique framework –to produce arguments –to answer, and defeat, the opposite side –interactive production of competing arguments –strategy from one point of view in order to win

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september Atelier Complexité et Politiques PubliquesJ. Legrand 17 Multidimensional networks in legal system Common Law Outcomes (decision) Arguments (used to win) Moves (to proceed from argument to argument ) Factors or Dimensions (organisation and abstraction of facts) Issues ({factors} "warrants" outcome) Values (social value the factors promote) (ordered with a preference relation )

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september Atelier Complexité et Politiques PubliquesJ. Legrand 18 Multidimensional networks in legal system Common Law Two ideas emerge from these investigations "sharing" – connectivity (the sharing of factors among the cases) "trajectory" – adaptative paths (the moves resulting from strategy) The "factors" as "keystone of backcloth" –the structure of cases "explains" dynamics

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september Atelier Complexité et Politiques PubliquesJ. Legrand 19 Conclusion and discussion Several caveats –lack of computer software applications –lack of independent validating yardstick –a naive feeling for Q analysis may be prejudicial –the narrow gap between metaphor and woolliness Many benefits –connectivity is not similarity, covariance or correlation –graph theory and Q analysis can gain from the other –the "backcloth" as a suggestive metaphor –the use of graphics as a language –a powerful thinking tool

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september Atelier Complexité et Politiques PubliquesJ. Legrand 20 Future work Toward multidimensional networks in legal system –applications should be time, funds & corpus consuming –backcloth metaphor for questioning the "legal fabric" Hypernetworks (J. Johnson) –answer to systemic limitations –answer to theory of sets limitations –further research deserves our attention

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