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arxiv: 2604.17510 · v1 · submitted 2026-04-19 · 💻 cs.CC

Reachability with Restricted Reactions in Inhibitory Chemical Reaction Networks

Divya Bajaj , Bin Fu , Ryan Knobel , Austin Luchsinger , Aiden Massie , Pablo Santos , Ramiro Santos , Robert Schweller
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Evan Tomai Tim Wylie
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Pith reviewed 2026-05-10 05:12 UTC · model grok-4.3

classification 💻 cs.CC
keywords reachabilitychemical reaction networksinhibitory CRNsNP-completenessPSPACE-completenessdeletion-only reactionsfixed-parameter tractability
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The pith

Inhibitory chemical reaction networks make reachability NP-complete for deletion-only reactions and PSPACE-complete for (1,1)-size reactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the reachability problem in two inhibitory extensions of chemical reaction networks, where certain species can block reactions from firing. It shows that restrictions that keep standard CRN reachability in polynomial time, such as deletion-only reactions or volume-preserving rules of size at most two, become NP-complete or PSPACE-complete once inhibition is introduced. For Priority-Inhibitory CRNs most deletion-only cases stay in P, but one variant is NP-complete. For plain Inhibitory CRNs deletion-only reachability is NP-complete in most settings, and the problem is PSPACE-complete even when every reaction changes exactly one molecule into one molecule. Fixed-parameter tractable algorithms are given for many of the Inhibitory CRN cases, and similar hardness is shown for CRNs augmented with explicit states.

Core claim

Reachability with deletion-only reactions is mostly NP-complete in the Inhibitory CRN model and remains NP-complete for one case in the Priority-Inhibitory model, while even (1,1)-size general reactions are PSPACE-complete in Inhibitory CRNs; most of these problems admit fixed-parameter tractable algorithms.

What carries the argument

The inhibition rule that prevents a reaction from applying whenever a designated inhibitor species is present in the current configuration.

If this is right

  • Deletion-only reachability in Inhibitory CRNs requires super-polynomial time in the worst case unless P equals NP.
  • Deciding reachability for (1,1)-size reactions in Inhibitory CRNs requires polynomial space and is complete for that class.
  • Fixed-parameter algorithms solve most Inhibitory CRN reachability instances efficiently when a small parameter such as the number of species is fixed.
  • Reachability in CRNs with explicit states is already NP-hard for the simplest deletion-only systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Inhibition supplies a lightweight form of conditional control that raises the computational power of even the most restricted reaction rules.
  • The same hardness pattern may appear in other models that combine positive and negative interactions, such as Petri nets with inhibitor arcs.
  • The FPT results indicate that practical instances remain solvable when the number of inhibitor species or the size of the target configuration is small.

Load-bearing premise

The precise definitions of inhibition and of the reaction restrictions such as deletion-only are used exactly as stated without additional unstated constraints that would change the complexity.

What would settle it

A polynomial-time algorithm that correctly decides reachability for deletion-only reactions in the Inhibitory CRN model on arbitrary instances would falsify the NP-completeness claims.

read the original abstract

Chemical Reaction Networks (CRNs) are a well-established model of distributed computing characterized by quantities of molecular species that can transform or change through applications of reactions. A fundamental problem in CRNs is the reachability problem, which asks if an initial configuration of species can transition to a target configuration through an applicable sequence of reactions. It is well-known that the reachability problem in general CRNs was recently proven to be Ackermann-complete. However, if the CRN's reactions are restricted in both power, such as only deleting species (deletion-only rules) or consuming and producing an equal number of species (volume-preserving rules), and size (unimolecular or bimolecular rules), then reachability falls below Ackermann-completeness, and is even solvable in polynomial time for deletion-only systems. In this paper, we investigate reachability under this set of restricted unimolecular and bimolecular reactions, but in the Priority-Inhibitory CRN and Inhibitory CRN models. These models extend a traditional CRN by allowing some reactions to be inhibited from firing in a configuration if certain species are present; the exact inhibition behavior varies between the models. We first show that reachability with Priority iCRNs mostly remains in P for deletion-only systems, but becomes NP-complete for one case. We then show that reachability with deletion-only reactions for iCRNs is mostly NP-complete, and PSPACE-complete even for (1,1)-size (general) reactions. We also provide FPT algorithms for solving most of the reachability problems for the iCRN model. Finally, we show reachability for CRNs with states is already NP-hard for the simplest deletion-only systems, and is PSPACE-complete even for (general) (1,1)-size reactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the reachability problem for reachability in two inhibitory extensions of chemical reaction networks (Priority iCRNs and standard iCRNs) under deletion-only reactions and size restrictions (unimolecular or bimolecular). It shows that deletion-only reachability in Priority iCRNs remains in P except for one NP-complete case. For standard iCRNs, deletion-only reachability is mostly NP-complete and PSPACE-complete even when restricted to (1,1)-size reactions. The authors supply FPT algorithms for most iCRN cases and extend the analysis to CRNs with states, proving NP-hardness for deletion-only systems and PSPACE-completeness for (1,1)-size reactions.

Significance. If the reductions and algorithms are correct, the work provides a fine-grained complexity classification for reachability in inhibitory CRN models, which naturally capture regulatory inhibition in biochemical systems. The results refine the known polynomial-time solvability for restricted standard CRNs and Ackermann-completeness for unrestricted CRNs by showing how inhibition interacts with deletion-only and size bounds. The explicit FPT algorithms constitute a constructive positive contribution that complements the hardness results and supports practical analysis of parameterized instances.

minor comments (3)
  1. [Abstract] Abstract: the qualifier 'mostly' appears in two places ('mostly remains in P' and 'mostly NP-complete') without an immediate enumeration of the exceptional cases; a parenthetical list of the precise exceptions would improve readability.
  2. [Introduction] The definitions of inhibition semantics for Priority iCRNs versus standard iCRNs are referenced but not restated in the abstract; a one-sentence contrast early in the introduction would help readers who are not already familiar with the two models.
  3. [Section on FPT algorithms] The FPT algorithms are claimed for 'most' iCRN reachability problems; a table summarizing which parameterizations admit FPT algorithms and which do not would make the positive results easier to compare with the hardness results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The referee's overview accurately reflects the complexity results for reachability in Priority iCRNs and iCRNs under deletion-only and size-restricted reactions, as well as the FPT algorithms and extensions to CRNs with states.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core results consist of complexity classifications (P, NP-complete, PSPACE-complete) for reachability under deletion-only and size-restricted reactions in Priority-iCRN and iCRN models. These are established via explicit polynomial-time reductions from known hard problems (standard in complexity theory) and direct algorithmic constructions for the upper bounds, including FPT algorithms. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear; the inhibition semantics are defined as straightforward extensions of ordinary CRNs, and the claims remain independent of any internal tautology or prior author result that would collapse the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard complexity theory and prior CRN results without introducing new free parameters or invented entities.

axioms (2)
  • standard math Reachability in general CRNs is Ackermann-complete
    Cited as background to contrast with the restricted cases analyzed.
  • standard math Standard definitions of P, NP-completeness, PSPACE-completeness, and FPT via polynomial reductions and parameterized algorithms
    Basis for all stated classifications and algorithm claims.

pith-pipeline@v0.9.0 · 5654 in / 1319 out tokens · 51570 ms · 2026-05-10T05:12:40.172477+00:00 · methodology

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Reference graph

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