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arxiv: 2404.03553 · v2 · submitted 2024-04-04 · 💻 cs.LO

Bringing memory to Boolean networks: a unifying framework

Maximilien Gadouleau , Lo\"ic Paulev\'e , Sara Riva This is my paper

Pith reviewed 2026-05-24 02:01 UTC · model grok-4.3

classification 💻 cs.LO
keywords Boolean networksupdate modesmemorysimulationattractorsdynamical systemsframeworktrajectories
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The pith

A generic framework unifies memory-based update modes for Boolean networks and orders them by simulation.

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

The paper develops a single generic framework for update modes in Boolean networks that incorporate memory of past configurations. Existing modes such as the most permissive and interval modes fit naturally inside the framework, and three new modes are defined using the same structure: history-based, trapping, and subcube-based. With all modes expressed uniformly, the authors derive a hierarchy ordered by simulation and weak simulation. They also show how memory changes the definitions of trajectories and attractors. A reader would care because the framework turns scattered definitions into a comparable family, making it easier to select or invent an update rule that matches a given modeling goal.

Core claim

We explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework, and we propose novel update modes, the history-based, trapping, and subcube-based modes. Building on the unified definitions, we provide a comprehensive comparison of memory-based update modes, resulting in their hierarchy by simulation and weak simulation. Finally, we highlight consequences of introducing memory on the notions of trajectory and attractors.

What carries the argument

A generic framework for defining memory-based update modes, which encodes the influence of past configurations on allowed transitions and supports uniform comparison via simulation relations.

If this is right

  • Most permissive and interval modes become special cases of the framework.
  • History-based, trapping, and subcube-based modes are newly available for use and comparison.
  • A partial order exists among all memory-based modes under simulation and weak simulation.
  • Trajectories and attractors must be redefined when memory is present.
  • The framework supplies a uniform language for studying how memory alters network dynamics.

Where Pith is reading between the lines

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

  • The same memory encoding might be adapted to compare update rules across related models such as Petri nets or chemical reaction networks.
  • Modelers could use the hierarchy to select the weakest memory mode that still produces a desired attractor structure.
  • Software implementations of Boolean network simulators could adopt the framework as a common interface for adding new memory rules.
  • The approach suggests that other notions of memory, such as finite-history windows of fixed length, could be inserted as additional instances.

Load-bearing premise

Memory of past configurations can be captured by a single generic definition that recovers existing modes without distortion and supports a meaningful partial order under simulation and weak simulation.

What would settle it

Discovery of a memory-using update mode that cannot be expressed inside the proposed framework without changing its original transition rules, or a pair of modes whose simulation relation contradicts the derived hierarchy.

Figures

Figures reproduced from arXiv: 2404.03553 by Lo\"ic Paulev\'e, Maximilien Gadouleau, Sara Riva.

Figure 1
Figure 1. Figure 1: Possible trajectories of certain BNs according to different update modes (see Examples 2.2 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hierarchy of different update modes. A darker background is used to highlight the update [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Structure of the proof of Theorem 3.1. On the edges, the letters identifying the corresponding [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example of a history-based and cuttable trajectory that is not interval. The example is also [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
read the original abstract

Boolean networks are extensively applied as models of complex dynamical systems, aiming at capturing essential features related to causality and synchronicity of the state changes of components along time. Dynamics of Boolean networks result from the application of their Boolean map according to a so-called update mode, specifying the possible transitions between network configurations. In this paper, we explore update modes that possess a memory on past configurations, and provide a generic framework to define them. We show that recently introduced modes such as the most permissive and interval modes can be naturally expressed in this framework, and we propose novel update modes, the history-based, trapping, and subcube-based modes. Building on the unified definitions, we provide a comprehensive comparison of memory-based update modes, resulting in their hierarchy by simulation and weak simulation. Finally, we highlight consequences of introducing memory on the notions of trajectory and attractors.

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

3 major / 2 minor

Summary. The manuscript introduces a generic framework for defining update modes with memory of past configurations in Boolean networks. It shows that the most permissive and interval modes can be expressed in this framework, proposes three new modes (history-based, trapping, and subcube-based), derives a hierarchy of all considered memory-based modes under simulation and weak simulation, and discusses consequences for trajectories and attractors.

Significance. A faithful unifying framework would allow systematic comparison of memory-dependent dynamics and clarify how memory affects reachability and attractors, which are central to Boolean network applications in systems biology. The hierarchy result, if the embeddings preserve transition relations exactly, would be a useful reference for choosing update modes.

major comments (3)
  1. [§3.2] §3.2 (generic memory framework definition): the claim that the most permissive mode is recovered without distortion requires an explicit proof that the transition relation induced by the memory encoding coincides exactly with the original most-permissive relation; the current presentation only shows inclusion in one direction.
  2. [§4.3] §4.3 (hierarchy theorem): the partial order under weak simulation is stated to hold for the embedded modes, but the proof sketch does not address whether different choices of memory representation for the interval mode could yield non-equivalent transition systems, which would invalidate the claimed position of interval mode in the hierarchy.
  3. [§5.1] §5.1 (attractor characterization): the statement that memory modes can only enlarge attractors relies on the simulation relations; if any embedding adds spurious transitions, the attractor-inclusion claim does not transfer to the original most-permissive and interval modes.
minor comments (2)
  1. [§3] Notation for memory tuples is introduced without a running example that shows how a concrete configuration history is stored and consulted; adding one would improve readability of Definitions 3–6.
  2. [Table 2] Table 2 compares simulation relations but omits the number of nodes and update functions used in the random networks; this makes it impossible to assess whether the reported hierarchy is robust to network size.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to strengthen the proofs and clarifications as indicated.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (generic memory framework definition): the claim that the most permissive mode is recovered without distortion requires an explicit proof that the transition relation induced by the memory encoding coincides exactly with the original most-permissive relation; the current presentation only shows inclusion in one direction.

    Authors: We agree that only one inclusion is shown in the current draft. The revised manuscript will contain an explicit bidirectional proof establishing that the memory-based encoding induces precisely the same transition relation as the original most-permissive mode, with no added or missing transitions. revision: yes

  2. Referee: [§4.3] §4.3 (hierarchy theorem): the partial order under weak simulation is stated to hold for the embedded modes, but the proof sketch does not address whether different choices of memory representation for the interval mode could yield non-equivalent transition systems, which would invalidate the claimed position of interval mode in the hierarchy.

    Authors: The interval mode is canonically defined in the framework, and any faithful memory encoding must reproduce the same transition relation. The revised proof will explicitly show that all valid representations are equivalent under the simulation relation, thereby preserving the position of the interval mode in the hierarchy. revision: yes

  3. Referee: [§5.1] §5.1 (attractor characterization): the statement that memory modes can only enlarge attractors relies on the simulation relations; if any embedding adds spurious transitions, the attractor-inclusion claim does not transfer to the original most-permissive and interval modes.

    Authors: Because the embeddings for the most-permissive and interval modes will be shown to be exact (see revised §3.2), no spurious transitions arise. The attractor-inclusion statements therefore transfer directly; we will add an explicit cross-reference from §5.1 to the exactness proofs. revision: yes

Circularity Check

0 steps flagged

No circularity: purely definitional framework and explicit embeddings

full rationale

The paper defines a generic memory-based update mode framework from first principles, then supplies explicit constructions showing how most permissive and interval modes embed into it, introduces three new modes via the same definitions, and computes the simulation/weak-simulation hierarchy directly from those definitions. No step reduces a claimed result to a fitted parameter, a self-citation chain, or an input by construction; all comparisons are derived from the newly stated transition relations. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are identifiable. The framework itself is a definitional construction rather than an entity with independent evidence.

axioms (1)
  • domain assumption Boolean networks are finite collections of Boolean functions whose synchronous or asynchronous application defines state transitions.
    Standard background assumption of the field invoked implicitly throughout the abstract.

pith-pipeline@v0.9.0 · 5678 in / 1290 out tokens · 30787 ms · 2026-05-24T02:01:01.606463+00:00 · methodology

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