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arxiv: 2605.19488 · v1 · pith:NTHP55BWnew · submitted 2026-05-19 · 💻 cs.DC · cs.DM· math.CO

A parallel wakeup problem and multi-room light switch strategies

John Haslegrave , Paul A. Russell , Mark Walters This is my paper

Pith reviewed 2026-05-20 02:25 UTC · model grok-4.3

classification 💻 cs.DC cs.DMmath.CO
keywords wakeup problemdistributed computingprisoners and light switchesshared registerssymmetric protocolsmulti-room coordinationadversarial scheduling
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The pith

Multi-room switch problems are solved with a precise count of states, and symmetric wakeup works exactly when processor and register numbers meet derived conditions.

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

This paper tackles the wakeup problem in distributed systems where processors must confirm others have run using only shared registers without clocks or symmetry. It answers open questions on the fewest switch states needed when prisoners visit multiple indistinguishable rooms in an unknown order to signal their presence. For the symmetric version of the wakeup problem, it pinpoints the exact pairs of processor counts and register counts that allow a solution. Understanding these boundaries helps clarify how limited shared memory can achieve coordination in fully adversarial settings.

Core claim

The authors establish the minimum number of switch states sufficient for the prisoners to solve the multi-room problem and prove that symmetric wakeup protocols exist if and only if the number of processors and registers satisfy specific conditions derived from their analysis.

What carries the argument

The shared multi-state switches in parallel indistinguishable rooms, serving as the sole communication channel under adversarial visit sequences.

Load-bearing premise

The rooms are indistinguishable to the prisoners and the sequence of room visits is completely unknown and adversarial.

What would settle it

Finding a specific pair of processor and register counts where a symmetric wakeup solution exists or fails contrary to the claimed exact characterization.

read the original abstract

The wakeup problem in distributed computing asks for a symmetric protocol that enables one of several processors to eventually guarantee that all (or, in a more general setting, enough) other processors have acted, using a shared register but no global clock. Dropping the symmetry requirement gives a well-known exercise often phrased in terms of prisoners entering, in an unknown sequence, a room equipped with a single binary switch, and using it to communicate. Kane and Kominers recently analysed a more general version of the latter with multiple parallel and indistinguishable rooms. We answer some open questions of Kane and Kominers regarding the minimum number of switch states needed for the prisoners to solve the problem. We also consider the symmetric ``wakeup'' version of this scenario, and establish exactly for which numbers of processors and registers a solution is possible.

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 resolves open questions from Kane and Kominers on the minimum number of switch states required to solve the multi-room prisoner problem under adversarial scheduling and indistinguishable rooms. It also analyzes the symmetric wakeup problem and gives an exact characterization of the processor-register pairs for which solutions exist.

Significance. If the results hold, the work is significant for distributed computing theory: it supplies explicit constructions for the minimum switch states in the parallel prisoner setting and a complete characterization of solvable (processor, register) pairs for the symmetric wakeup variant. These parameter-free characterizations and constructions strengthen the combinatorial foundations of coordination protocols without clocks or distinguishable resources and may serve as building blocks for further results on anonymous networks.

minor comments (3)
  1. The abstract states that exact conditions are established but does not preview the form of the characterization (e.g., whether it is a simple arithmetic condition on n and m). Adding one sentence would improve readability.
  2. Ensure that all references to Kane and Kominers include the full bibliographic details (title, venue, year) at first citation.
  3. In the section presenting the wakeup constructions, verify that the notation for registers and processors is introduced before its first use and remains consistent throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report highlights the significance of our results on minimum switch states for the multi-room prisoner problem and the characterization of solvable processor-register pairs for the symmetric wakeup problem. No specific major comments were enumerated in the report, so we provide no point-by-point responses below. We will prepare a revised version incorporating any minor editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity; results rest on explicit constructions and characterizations

full rationale

The paper develops explicit constructions for the minimum switch states required in the multi-room prisoner problem and provides a complete characterization of solvable (processor, register) pairs for the symmetric wakeup variant. These rest on the standard adversarial scheduling and indistinguishable-room assumptions stated in the setup, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The reference to Kane and Kominers serves only as motivation for resolving their open questions; the derivations are self-contained against external benchmarks and do not import uniqueness theorems or ansatzes from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper extends the standard asynchronous distributed computing model with shared registers and indistinguishable rooms; no new free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • domain assumption Standard model of distributed computing with shared registers, no global clock, and adversarial unknown visit order
    Invoked throughout the problem definition in the abstract.

pith-pipeline@v0.9.0 · 5669 in / 1147 out tokens · 51293 ms · 2026-05-20T02:25:57.018421+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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