From Structural Description to Interactions in Law: the Case for Simplicial Complexes

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

2. Introductionproblem statement • Several communities focuses on networks • Networks studied via graph theory •  higher dimensional networks • Applications leave out the legal systems Atelier Complexité et Politiques Publiques

3. Introductionobjectives • Generalising to relations among N things • Developments concerning Law Atelier Complexité et Politiques Publiques

4. Introductionmethod • 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 Atelier Complexité et Politiques Publiques

5. Introductionoutline • 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 Atelier Complexité et Politiques Publiques

6. Simplicial complex Vertices (bi) Simplices (aj) Dimension (a4, 2) Faces (a1,<b2,b4>) Nearness (a1 a2 ,b4) Connectivity (a1,a2,a5,a6,a4) Loops (a4,a5,a6) Holes <b6,b7,b8> Atelier Complexité et Politiques Publiques

7. 1st level of interpretation • Simplicial complex construction • vertices  simplices • Choice of invariants • algebraic • topological • combinatorial • Quantitative measures • Qualitative observations Atelier Complexité et Politiques Publiques

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 ? Atelier Complexité et Politiques Publiques

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 Atelier Complexité et Politiques Publiques

10. Multidimensional networks in legal systemStatute 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) Atelier Complexité et Politiques Publiques

11. Multidimensional networks in legal systemStatute 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) Atelier Complexité et Politiques Publiques

12. Multidimensional networks in legal systemStatute 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" Atelier Complexité et Politiques Publiques

13. Multidimensional networks in legal systemStatute 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 Atelier Complexité et Politiques Publiques

14. Multidimensional networks in legal systemStatute 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 Atelier Complexité et Politiques Publiques

15. Multidimensional networks in legal systemCommon 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 Atelier Complexité et Politiques Publiques

16. Multidimensional networks in legal systemCommon 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 Atelier Complexité et Politiques Publiques

17. Multidimensional networks in legal systemCommon 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) Atelier Complexité et Politiques Publiques

18. Multidimensional networks in legal systemCommon 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 Atelier Complexité et Politiques Publiques

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 Atelier Complexité et Politiques Publiques

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 Atelier Complexité et Politiques Publiques