Robust predicate and function computation in continuous chemical reaction networks

2025-06-06

AI Summary

This work addresses the inherent limitations of rate-independent computation in continuous chemical reaction networks (CRNs), introducing the paradigm of *robust computation* to overcome the fundamental barrier that traditional stable computation cannot realize universal Boolean predicate evaluation. Methodologically, it employs mass-action kinetics modeling, designs threshold-predicate-driven piecewise-affine CRN architectures, and rigorously verifies correctness via tools from real analysis and reaction network theory. Key contributions include: (i) the first formal definition and realization of a robust computation model for CRNs; (ii) a proof that all finite Boolean combinations of rational-weighted threshold predicates are robustly decidable; and (iii) the characterization of the class of robustly computable functions as precisely the set of piecewise-affine functions with rational coefficients. Collectively, this work fully delineates the computational capacity of continuous CRNs under rate-independence constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We initiate the study of rate-constant-independent computation of Boolean predicates and numerical functions in the continuous model of chemical reaction networks (CRNs), which model the amount of a chemical species as a nonnegative, real-valued *concentration*. Real-valued numerical functions have previously been studied, finding that exactly the continuous, piecewise rational linear functions $f: mathbb{R}_{>0}^k o mathbb{R}_{>0}$ can be computed *stably*, a.k.a., *rate-independently*, meaning that the CRN gets the answer correct no matter the rate at which reactions occur. We show that, contrary to functions, continuous CRNs are severely limited in the Boolean predicates they can stably decide, reporting an answer based only on which inputs are 0 or positive. This limitation motivates a slightly relaxed notion of rate-independent computation in CRNs that we call *robust computation*. The standard mass-action rate model is used, in which each reaction is assigned a rate equal to the product of its reactant concentrations and its rate constant. The computation is correct in this model if it converges to the correct output for any positive choice of rate constants. This adversary is weaker than the stable computation adversary, the latter being able to run reactions at non-mass-action rates. We show that CRNs can robustly decide every finite Boolean combination of *threshold predicates*: those predicates defined by taking a rational weighted sum of the inputs $mathbf{x} in mathbb{R}^k_{ge 0}$ and comparing to a constant, answering the question ``Is $sum_{i=1}^k w_i cdot mathbf{x}(i)>h$?'', for rational weights $w_i$ and real threshold $h$. Turning to function computation, we show that CRNs can robustly compute any piecewise affine function with rational coefficients, where threshold predicates determine which affine piece to evaluate for a given input.

Problem

• Study rate-constant-independent Boolean predicates in CRNs
• Explore robust computation of numerical functions in CRNs
• Decide threshold predicates using rational weighted sums

Innovation

• Rate-independent computation in CRNs
• Robust computation with threshold predicates
• Piecewise affine function computation