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Abstract Dialectical Frameworks are Boolean Networks (full version)

Jesse Heyninck, Matthias Knorr, João Leite

TL;DR

The paper investigates the relationship between Abstract Dialectical Frameworks and Boolean Networks, showing that AD F semantics align with BN dynamics through formal mappings between acceptance conditions and regulatory functions. By translating BN concepts such as state transitions, attractors, and trap spaces into ADF terms, it demonstrates that key BN results on counting complexity and fixpoint existence carry over to ADFs, including precise bounds under various structural conditions. The work identifies bipolar/bipolarizable subclasses as a common tractable regime and highlights the potential for cross-domain tools, benchmarks, and implementations. Overall, the correspondence reveals that ADFs essentially behave as a generalization of BNs, enabling fruitful exchange of theory and practice across formal argumentation and systems biology.

Abstract

Dialectical frameworks are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. Their generality allow them to cover a number of different approaches with varying forms of representing the argumentation structure. Boolean regulatory networks are used to model the dynamics of complex biological processes, taking into account the interactions of biological compounds, such as proteins or genes. These models have proven highly useful for comprehending such biological processes, allowing to reproduce known behaviour and testing new hypotheses and predictions in silico, for example in the context of new medical treatments. While both these approaches stem from entirely different communities, it turns out that there are striking similarities in their appearence. In this paper, we study the relation between these two formalisms revealing their communalities as well as their differences, and introducing a correspondence that allows to establish novel results for the individual formalisms.

Abstract Dialectical Frameworks are Boolean Networks (full version)

TL;DR

The paper investigates the relationship between Abstract Dialectical Frameworks and Boolean Networks, showing that AD F semantics align with BN dynamics through formal mappings between acceptance conditions and regulatory functions. By translating BN concepts such as state transitions, attractors, and trap spaces into ADF terms, it demonstrates that key BN results on counting complexity and fixpoint existence carry over to ADFs, including precise bounds under various structural conditions. The work identifies bipolar/bipolarizable subclasses as a common tractable regime and highlights the potential for cross-domain tools, benchmarks, and implementations. Overall, the correspondence reveals that ADFs essentially behave as a generalization of BNs, enabling fruitful exchange of theory and practice across formal argumentation and systems biology.

Abstract

Dialectical frameworks are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. Their generality allow them to cover a number of different approaches with varying forms of representing the argumentation structure. Boolean regulatory networks are used to model the dynamics of complex biological processes, taking into account the interactions of biological compounds, such as proteins or genes. These models have proven highly useful for comprehending such biological processes, allowing to reproduce known behaviour and testing new hypotheses and predictions in silico, for example in the context of new medical treatments. While both these approaches stem from entirely different communities, it turns out that there are striking similarities in their appearence. In this paper, we study the relation between these two formalisms revealing their communalities as well as their differences, and introducing a correspondence that allows to establish novel results for the individual formalisms.
Paper Structure (10 sections, 9 theorems, 3 equations, 3 figures)

This paper contains 10 sections, 9 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Abstract Dialectical Frameworks
  3. Boolean Networks
  4. Syntax
  5. Dynamics
  6. Subclasses of Boolean Networks
  7. Insights from Boolean Networks
  8. Complexity of Counting
  9. Existence of Fixpoints
  10. Conclusion

Key Result

Proposition 1

A space $m$ is a trap set if and only if $S[m]\subseteq S[F[m]]$.

Figures (3)

  • Figure 1: Examples of a) an ADF for Example \ref{['example:travelling']}, and b) a BN for Example \ref{['example:mussels']}.
  • Figure 2: Example of a Boolean logical model.
  • Figure 3: STG of the model in Ex. \ref{['ex:model']} for a) synchronous updates and b) asynchronous updates, both with $v_4$ inactive on the left, and active on the right (adapted from FilipeGouveia_9_2020).

Theorems & Definitions (29)

  • Example 1
  • Example 2
  • Example 3
  • Definition 1
  • Example 4: Ex. \ref{['example:travelling']} ctd.
  • Definition 2
  • Example 5
  • Definition 3
  • Example 6
  • Definition 4
  • ...and 19 more