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arxiv: 2606.30270 · v1 · pith:F4IMBU2Nnew · submitted 2026-06-29 · 💻 cs.CC · cs.DM· math.DS

Cyclic Attractor Detection in Boolean Network Dynamics under Local Logical Constraints

Alexander Drobyshev , Grigoriy Bokov This is my paper

Pith reviewed 2026-06-30 03:16 UTC · model grok-4.3

classification 💻 cs.CC cs.DMmath.DS
keywords Boolean networksattractor detectionPost's latticecomputational complexitycyclic attractorsdichotomy theoremNP-completeness
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The pith

The problem of detecting a cyclic attractor of exact period k in a Boolean network is NP-complete or polynomial-time solvable depending on which closed class of local rules from Post's lattice is used.

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

This paper maps the computational complexity of finding whether a Boolean network has a cycle of exact length k under parallel updates. The local rules are circuits from a fixed closed class of Boolean functions, classified by Post's lattice. For each fixed k at least 2, the problem is NP-complete if the class includes majority-like self-dual rules or mixed conjunctive-disjunctive monotone families. It is solvable in polynomial time for the other classes, including affine rules and pure conjunctive or disjunctive rules with constants. The boundary arises because certain rule types preserve structure that allows efficient search while others can express hard global constraints.

Core claim

For every fixed k ≥ 2, the exact-k-cyclic-attractor problem over Boolean networks is NP-complete whenever the local rule class contains majority-like self-dual rules or one of the two mixed conjunctive-disjunctive monotone families, and it is polynomial-time solvable in all remaining Post classes, with affine rules and pure conjunctive or pure disjunctive rules with constants providing the boundary tractable cases.

What carries the argument

The classification of closed Boolean function classes given by Post's lattice, which determines whether the local rules preserve enough algebraic or ordering structure to make attractor search tractable.

If this is right

  • Networks whose update rules are affine admit polynomial-time detection of period-k cycles because the preserved linear structure allows efficient solving.
  • Rule classes with self-dual majority functions allow reduction of hard consistency problems to the attractor detection task.
  • Pure conjunctive rules with constants permit attractor detection by checking consistency in an order structure in polynomial time.
  • The dichotomy holds for every fixed period k at least 2.

Where Pith is reading between the lines

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

  • The findings indicate that models of gene networks using majority voting rules may require different analysis techniques than those using linear or one-sided rules.
  • Similar complexity boundaries might appear in related problems such as finding fixed points rather than cycles.
  • Relaxing the assumption of synchronous updates could produce a different set of tractable rule classes.
  • Practical implementations could first check the rule class to decide which algorithm to apply.

Load-bearing premise

The coordinate functions are realized by circuits over a fixed finite basis of a closed Boolean-function class and the network evolves under parallel synchronous update.

What would settle it

An explicit Post class containing mixed conjunctive-disjunctive monotone rules together with a polynomial-time algorithm for the period-k problem in that class, or conversely an affine class in which the problem is NP-complete.

Figures

Figures reproduced from arXiv: 2606.30270 by Alexander Drobyshev, Grigoriy Bokov.

Figure 1
Figure 1. Figure 1: From local Boolean rules to the attractor landscape for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Complexity regions in Theorem 7 for exact-period cyclic-attractor existence with fixed [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Layered construction of the D2-network Nψ used in Lemma 10. The literal vertices are fixed by self-loops. The checking vertices encode clause satisfaction and consistency by maj￾gates. The cycle layer is a circular shift register; δ1 receives δk, n copies of each Gj , and m copies of each Bi . These multiplicities make transmission through δ1 depend on the balance between clause-test and consistency-test o… view at source ↗
read the original abstract

Boolean networks are finite discrete nonlinear systems whose long-term behaviour is organised by fixed-point and cyclic attractors. Detecting such recurrent states is important in applications ranging from gene regulation and neural computation to complex-network models, but the computational boundary between tractable and intractable attractor analysis is still not fully understood. We study that boundary from the perspective of local logical rules. We consider Boolean networks under parallel update whose coordinate functions are given by circuits over a fixed finite basis of a closed Boolean-function class, and ask whether the network has a cyclic attractor of prescribed exact period $k$. For every fixed $k\ge 2$, we obtain a complete complexity dichotomy over Post's lattice. The problem is $\mathrm{NP}$-complete whenever the local rule class contains majority-like self-dual rules or one of the two mixed conjunctive-disjunctive monotone families. In all remaining Post classes it is polynomial-time solvable, with affine rules and pure conjunctive or pure disjunctive rules with constants providing the boundary tractable cases. The results show that exact attractor detection is governed not only by the network architecture but also by the logical mechanism of local update: affine and one-sided rules preserve algebraic or order structure, whereas majority-like and mixed monotone rules can encode global Boolean consistency constraints.

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 / 1 minor

Summary. The paper claims a complete complexity dichotomy, for every fixed k≥2, for the problem of deciding whether a Boolean network (with parallel synchronous update) whose coordinate functions are circuits over a fixed finite basis of a closed class in Post's lattice has a cyclic attractor of exact period k. The problem is NP-complete precisely when the class contains majority-like self-dual rules or one of the two mixed conjunctive-disjunctive monotone families; it is polynomial-time solvable in all remaining classes, with affine rules and pure conjunctive or pure disjunctive rules with constants as the boundary tractable cases.

Significance. If the claimed dichotomy holds, the result is significant: it supplies a full classification of exact-period cyclic attractor detection over the standard algebraic structure of Post's lattice, showing that tractability is governed by whether the local rules preserve affine structure or one-sided order properties versus the ability to encode global Boolean consistency constraints via majority-like or mixed monotone rules. This refines the computational boundary for attractor analysis in applications such as gene regulation.

minor comments (1)
  1. Ensure that the main text explicitly enumerates the Post classes falling on each side of the dichotomy (rather than relying solely on the abstract's high-level description) so that readers can immediately verify the boundary cases without reconstructing the lattice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive recommendation to accept and for the accurate summary of the complexity dichotomy over Post's lattice.

Circularity Check

0 steps flagged

No significant circularity; classification is external to the paper

full rationale

The paper establishes a complexity dichotomy for cyclic attractor detection by partitioning over the standard, externally defined Post's lattice of Boolean function classes. The tractable and NP-complete cases are identified by algebraic properties of those classes (affine, pure conjunctive/disjunctive with constants vs. majority-like self-dual or mixed monotone families), with no equations, reductions, or predictions that collapse back to fitted parameters or self-defined quantities. No self-citations are invoked as load-bearing uniqueness theorems, and the problem formulation (circuits over a fixed basis, parallel update, exact period k) introduces no internal self-reference that would make the claimed boundaries tautological. The result is therefore a standard complexity classification against an independent mathematical structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard classification of Boolean functions by Post's lattice and on the modeling choice of parallel update; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Post's lattice enumerates all closed classes of Boolean functions under composition
    Invoked throughout the abstract as the organizing structure for the dichotomy
  • domain assumption Network evolution uses parallel (synchronous) update
    Explicitly stated as the update scheme under which the attractor question is posed

pith-pipeline@v0.9.1-grok · 5756 in / 1414 out tokens · 59376 ms · 2026-06-30T03:16:26.562748+00:00 · methodology

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