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Joint Realizability of Monotone Boolean Functions.

Peter Crawford-Kahrl1, Bree Cummins1, Tomáš Gedeon1

  • 1Department of Mathematical Sciences, Montana State University, Bozeman, MT 59715.

Theoretical Computer Science
|November 28, 2025
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Summary
This summary is machine-generated.

This study connects monotone Boolean functions (MBFs) with ordinary differential equation (ODE) models in gene regulation. It shows that restricting ODE complexity limits the number of realizable MBF collections.

Keywords:
34C2392C4294C11Monotone Boolean functionsgene regulatory networksswitching systems

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Area of Science:

  • Computational Biology
  • Systems Biology
  • Discrete Mathematics

Background:

  • Monotone Boolean functions (MBFs) are fundamental in discrete mathematics and computer science.
  • Ordinary differential equation (ODE) models are widely used to represent gene regulatory network dynamics.
  • The realization problem connects discrete functions (MBFs) with continuous dynamical systems (ODEs).

Purpose of the Study:

  • To explore the connection between monotone Boolean functions (MBFs) and ODE models of gene regulation.
  • To investigate the joint realizability of MBF collections by parameterized ODE dynamics.
  • To determine how restrictions on ODE algebraic complexity affect the class of jointly realizable MBFs.

Main Methods:

  • Formulating the problem of joint realizability for collections of MBFs.
  • Analyzing parameterized dynamics of ODEs and their connection to MBFs.
  • Utilizing a combination of theoretical analysis and explicit examples to demonstrate results.

Main Results:

  • Established a connection between ODE dynamics and collections of MBFs for joint realizability.
  • Demonstrated that as the algebraic complexity of ODEs is restricted, the class of jointly realizable MBF collections strictly decreases.
  • Provided explicit examples illustrating the relationship between ODE complexity and MBF realizability.

Conclusions:

  • The study reveals a trade-off between the complexity of ODE models and the types of MBFs they can represent.
  • Results have implications for understanding regulatory network dynamics and advancing the theory of MBFs.
  • Identified potential extensions and conjectures for future research in this interdisciplinary area.