skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Polynomial Equivalence of Extended Chemical Reaction Models
The ability to detect whether a species (or dimension) is zero in Chemical Reaction Networks (CRN), Vector Addition Systems, or Petri Nets is known to increase the power of these models - making them capable of universal computation. While this ability may appear in many forms, such as extending the models to allow transitions to be inhibited, prioritized, or synchronized, we present an extension that directly performs this zero checking. We introduce a new void genesis CRN variant with a simple design that merely increments the count of a specific species when any other species' count goes to zero. As with previous extensions, we show that the model is Turing Universal. We then analyze several other studied CRN variants and show that they are all equivalent through a polynomial simulation with the void genesis model, which does not merely follow from Turing-universality. Thus, inhibitor species, reactions that occur at different rates, being allowed to run reactions in parallel, or even being allowed to continually add more volume to the CRN, does not add additional simulation power beyond simply detecting if a species count becomes zero.  more » « less
Award ID(s):
2329918
PAR ID:
10657117
Author(s) / Creator(s):
; ; ; ; ; ; ; ; ;
Editor(s):
Chen, Ho-Lin; Hon, Wing-Kai; Tsai, Meng-Tsung
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
359
ISSN:
1868-8969
Page Range / eLocation ID:
7:1-7:21
Subject(s) / Keyword(s):
Chemical Reaction Networks Simulations Petri-nets Vector Addition Systems Bi-simulation Turing-universality Inhibitors Theory of computation → Models of computation
Format(s):
Medium: X Size: 21 pages; 1086534 bytes Other: application/pdf
Size(s):
21 pages 1086534 bytes
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Chen, Ho-Lin; Evans, Constantine G. (Ed.)
    Discrete chemical reaction networks formalize the interactions of molecular species in a well-mixed solution as stochastic events. Given their basic mathematical and physical role, the computational power of chemical reaction networks has been widely studied in the molecular programming and distributed computing communities. While for Turing-universal systems there is a universal measure of optimal information encoding based on Kolmogorov complexity, chemical reaction networks are not Turing universal unless error and unbounded molecular counts are permitted. Nonetheless, here we show that the optimal number of reactions to generate a specific count x ∈ ℕ with probability 1 is asymptotically equal to a "space-aware" version of the Kolmogorov complexity of x, defined as K̃s(x) = min_p {|p|/log|p| + log(space(𝒰(p))) : 𝒰(p) = x}, where p is a program for universal Turing machine 𝒰. This version of Kolmogorov complexity incorporates not just the length of the shortest program for generating x, but also the space usage of that program. Probability 1 computation is captured by the standard notion of stable computation from distributed computing, but we limit our consideration to chemical reaction networks obeying a stronger constraint: they "know when they are done" in the sense that they produce a special species to indicate completion. As part of our results, we develop a module for encoding and unpacking any b bits of information via O(b/log{b}) reactions, which is information-theoretically optimal for incompressible information. Our work provides one answer to the question of how succinctly chemical self-organization can be encoded - in the sense of generating precise molecular counts of species as the desired state. 
    more » « less
  2. ; ; ; ; ; ; ; (, Springer Nature)
    Step Chemical Reaction Networks (step CRNs) are an augmentation of the Chemical Reaction Network (CRN) model where additional species may be introduced to the system in a sequence of “steps.” We study step CRN systems using a weak subset of reaction rules, void rules, in which molecular species can only be deleted. We demonstrate that step CRNs with only void rules of size (2,0) can simulate threshold formulas (TFs) under linear resources. These limited systems can also simulate threshold circuits (TCs) by modifying the volume of the system to be exponential. We then prove a matching exponential lower bound on the required volume for simulating threshold circuits in a step CRN with (2,0)-size rules under a restricted gate-wise simulation, thus showing our construction is optimal for simulating circuits in this way. 
    more » « less
  3. ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Alistarh, Dan (Ed.)
    Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as A + B → C, representing that if a molecule of A and B collide, they disappear and a molecule of C is produced. CRNs can compute Boolean-valued predicates ϕ:ℕ^d → {0,1} and integer-valued functions f:ℕ^d → ℕ; for instance X₁ + X₂ → Y computes the function min(x₁,x₂), since starting with x_i copies of X_i, eventually min(x₁,x₂) copies of Y are produced. We study the computational power of execution bounded CRNs, in which only a finite number of reactions can occur from the initial configuration (e.g., ruling out reversible reactions such as A ⇌ B). The power and composability of such CRNs depend crucially on some other modeling choices that do not affect the computational power of CRNs with unbounded executions, namely whether an initial leader is present, and whether (for predicates) all species are required to "vote" for the Boolean output. If the CRN starts with an initial leader, and can allow only the leader to vote, then all semilinear predicates and functions can be stably computed in O(n log n) parallel time by execution bounded CRNs. However, if no initial leader is allowed, all species vote, and the CRN is "non-collapsing" (does not shrink from initially large to final O(1) size configurations), then execution bounded CRNs are severely limited, able to compute only eventually constant predicates. A key tool is a characterization of execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction [Czerner et al., 2024]. 
    more » « less
  4. ; ; ; ; ; ; ; ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Schaeffer, Josie; Zhang, Fei (Ed.)
    For general discrete Chemical Reaction Networks (CRNs), the fundamental problem of reachability - the question of whether a target configuration can be produced from a given initial configuration - was recently shown to be Ackermann-complete. However, many open questions remain about which features of the CRN model drive this complexity. We study a restricted class of CRNs with void rules, reactions that only decrease species counts. We further examine this regime in the motivated model of step CRNs, which allow additional species to be introduced in discrete stages. With and without steps, we characterize the complexity of the reachability problem for CRNs with void rules. We show that, without steps, reachability remains polynomial-time solvable for bimolecular systems but becomes NP-complete for larger reactions. Conversely, with just a single step, reachability becomes NP-complete even for bimolecular systems. Our results provide a nearly complete classification of void-rule reachability problems into tractable and intractable cases, with only a single exception. 
    more » « less
  5. ; ; (, DNA Computing and Molecular Programming. DNA 2018.)
    Synthetic biology is a rapidly emerging research area, with expected wide-ranging impact in biology, nanofabrication, and medicine. A key technical challenge lies in embedding computation in molecular contexts where electronic micro-controllers cannot be inserted. This necessitates effective representation of computation using molecular components. While previous work established the Turing-completeness of chemical reactions, defining representations that are faithful, efficient, and practical remains challenging. This paper introduces CRN++, a new language for programming deterministic (mass-action) chemical kinetics to perform computation. We present its syntax and semantics, and build a compiler translating CRN++ programs into chemical reactions, thereby laying the foundation of a comprehensive framework for molecular programming. Our language addresses the key challenge of embedding familiar imperative constructs into a set of chemical reactions happening simultaneously and manipulating real-valued concentrations. Although some deviation from ideal output value cannot be avoided, we develop methods to minimize the error, and implement error analysis tools. We demonstrate the feasibility of using CRN++ on a suite of well-known algorithms for discrete and real-valued computation. CRN++ can be easily extended to support new commands or chemical reaction implementations, and thus provides a foundation for developing more robust and practical molecular programs. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email