A Categorical Approach to DIBI Models

Authors Tao Gu , Jialu Bao , Justin Hsu , Alexandra Silva, Fabio Zanasi



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Author Details

Tao Gu
  • University College London, UK
Jialu Bao
  • Cornell University, Ithaca, NY, USA
Justin Hsu
  • Cornell University, Ithaca, NY, USA
Alexandra Silva
  • Cornell University, Ithaca, NY, USA
Fabio Zanasi
  • University College London, UK
  • University of Bologna, OLAS team (INRIA), Italy

Acknowledgements

We thank the anonymous reviewers for their close reading and detailed feedback.

Cite AsGet BibTex

Tao Gu, Jialu Bao, Justin Hsu, Alexandra Silva, and Fabio Zanasi. A Categorical Approach to DIBI Models. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FSCD.2024.17

Abstract

The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in discrete probability distributions and in relational databases, using a probabilistic DIBI model and a similarly-constructed relational model. Despite the similarity of the two models, there lacks a uniform account. As a result, the laborious case-by-case verification of the frame conditions required for constructing new models hinders them from generalising the results to CI in other useful models such that continuous distribution. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. We show that DIBI models arise from arbitrary symmetric monoidal categories with copy-discard structure. In particular, we use string diagrams - a graphical presentation of monoidal categories - to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a comparison between string diagrammatic approaches to CI in the literature and a logical notion of CI, defined in terms of the satisfaction of specific DIBI formulas. We show that the logical notion is an extension of string diagrammatic CI under reasonable conditions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Theory of computation → Semantics and reasoning
  • Theory of computation → Models of computation
Keywords
  • Conditional Independence
  • Dependence Independence Bunched Implications
  • String Diagrams
  • Markov Categories

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