Related Experiment Video
Updated: Jan 10, 2026
14:05
One Dimensional Turing-Like Handshake Test for Motor Intelligence
Published on: December 15, 2010
28.4K
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
Summary
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.
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.
Related Concept Videos
01:28Piecewise-Defined Functions
192
Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function: uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value...
192
01:30Fundamental Theorem of Algebra
198
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
198
01:26Constraints and Statical Determinacy
925
In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...
925
01:28Rational Expressions
286
Rational expressions are algebraic fractions in which both the numerator and the denominator are polynomials. These expressions follow the arithmetic rules of numerical fractions but require extra care due to the presence of variables. A fundamental part of working with rational expressions is identifying values that make the expression undefined, typically those that result in division by zero or undefined radicals.Determining the DomainThe domain of a rational expression includes all real...
286
01:26Singularity Functions for Shear
406
In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the...
406
01:28Synthetic Disvision of Polynomials
120
Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
120