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arxiv: 2604.02673 · v1 · submitted 2026-04-03 · 💻 cs.LO · math.LO

A Logic of Secrecy on Simplicial Models

Shanxia Wang This is my paper

Pith reviewed 2026-05-13 19:04 UTC · model grok-4.3

classification 💻 cs.LO math.LO
keywords epistemic logicsimplicial modelssecrecymulti-agent systemsmodal logicknowledge and belief
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0 comments X

The pith

A secrecy operator can be added directly to simplicial models of knowledge by attaching agent-specific neighborhood functions to local states.

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

The paper equips standard simplicial epistemic models with secrecy neighborhood functions attached to each agent's local state. It defines a primitive operator S_a φ that holds when a knows φ in the usual simplicial sense and the truth set of φ lies inside one of the designated neighborhoods for a's current local state. A sound axiomatization SSL is given for the resulting language. For two or more agents the authors prove completeness by constructing an auxiliary-colour canonical model and then embedding it into the class of pure A-chromatic simplicial secrecy models.

Core claim

Simplicial secrecy models are obtained by enriching ordinary chromatic simplicial complexes with agent-relative secrecy neighborhood functions on the coloured vertices that represent local states. The operator S_a φ is interpreted so that it requires both ordinary simplicial knowledge of φ by a and membership of the truth set of φ in a secrecy neighborhood of a's local state. The frame condition guarantees that every designated secrecy event remains non-trivial for every other agent. The system SSL is sound on the entire class of these models; when at least two agents are present, completeness holds because every consistent set can be realized first in an auxiliary-colour canonical model and

What carries the argument

Simplicial secrecy models, which enrich a chromatic simplicial complex with agent-relative secrecy neighborhood functions attached to each local state (coloured vertex).

If this is right

  • SSL provides a sound and, for |A| ≥ 2, complete calculus for reasoning about secrecy inside geometrically presented multi-agent states.
  • Secrecy remains strictly local to each agent's current vertex rather than being defined by global properties of the complex.
  • The combination of simplicial knowledge with a neighborhood condition yields a primitive modality that cannot be reduced to ordinary knowledge operators alone.
  • Any formula valid on all simplicial secrecy models is provable in SSL once the number of agents is at least two.

Where Pith is reading between the lines

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

  • The geometric representation may allow direct visualization of which local states permit secrecy of which facts.
  • The same neighborhood construction could be applied to dynamic epistemic logics that update the simplicial complex during communication.
  • If the frame condition is relaxed, weaker notions of secrecy that tolerate triviality for some observers become definable inside the same language.

Load-bearing premise

The secrecy neighborhoods attached to each local state must keep every designated secrecy event non-trivial from the viewpoint of every other agent.

What would settle it

A concrete simplicial secrecy model in which some secrecy neighborhood renders an event trivial for another agent would violate the frame condition and falsify the claimed semantics for S_a φ.

Figures

Figures reproduced from arXiv: 2604.02673 by Shanxia Wang.

Figure 1
Figure 1. Figure 1: A two-agent running example. Rows are a-equivalence classes and columns are b￾equivalence classes. The shaded event is U = {x1, x2, x3, y1}. The top row is St(u0). In each column there is a facet outside U, namely z1, y2, and y3, witnessing condition (SN) for the owner a at the local state u0. Using JψKM = JψKMloc and M, X |= Kaψ ⇐⇒ Mloc, X |= Kaψ, we obtain M, X |= Saψ. This completes the induction. 3.4 A… view at source ↗
read the original abstract

We develop a logic of secrecy on simplicial models for multi-agent systems. Standard simplicial models provide a geometric semantics for knowledge by representing global states as facets of a chromatic simplicial complex and agents' local states as coloured vertices. However, secrecy cannot in general be captured as a genuinely new modality by relying on the ordinary simplicial knowledge structure alone. This motivates the introduction of an additional secrecy layer. To this end, we define \emph{simplicial secrecy models}, which enrich standard simplicial epistemic models with agent-relative secrecy neighborhood functions attached to local states. On this basis, we introduce a primitive secrecy operator $S_a\varphi$. Semantically, $S_a\varphi$ holds when agent $a$ knows $\varphi$ in the ordinary simplicial sense and, moreover, the truth set of $\varphi$ belongs to one of the designated secrecy neighborhoods associated with $a$'s current local state. The clause for secrecy thus combines an ordinary knowledge requirement with an additional local-state-based neighborhood requirement, while the frame condition ensures that designated secrecy events remain non-trivial from the perspective of every other agent. We formulate a system $\mathsf{SSL}$ for the resulting language and show that it is sound with respect to the class of simplicial secrecy models. For the genuinely multi-agent case $|A|\ge 2$, we prove completeness by first constructing an auxiliary-colour canonical model and then representing it inside the original class of pure $A$-chromatic simplicial secrecy models. The resulting framework yields a primitive, local-state-based, and geometrically grounded account of secrecy on simplicial models, together with a sound axiomatization and, in the genuinely multi-agent case, a complete one.

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

1 major / 2 minor

Summary. The paper develops simplicial secrecy models by enriching standard chromatic simplicial epistemic models with agent-relative secrecy neighborhood functions attached to local states. It introduces a primitive secrecy modality S_a φ whose semantics requires both ordinary simplicial knowledge of φ and membership of the truth set of φ in a designated secrecy neighborhood of a's local state, subject to a frame condition ensuring non-triviality from other agents' perspectives. The system SSL is shown sound directly from the semantics; for |A|≥2, completeness is proved by constructing an auxiliary-colour canonical model and representing it inside the class of pure A-chromatic simplicial secrecy models.

Significance. If the results hold, the work supplies a geometrically grounded, local-state-based primitive modality for secrecy that is not reducible to the ordinary knowledge structure on simplicial complexes. The explicit auxiliary-colour canonical-model construction followed by representation into the target class constitutes a non-routine but self-contained completeness argument, free of ad-hoc parameters or circular definitions, and yields both soundness for all cases and completeness in the genuinely multi-agent setting.

major comments (1)
  1. [Completeness proof for |A|≥2] Completeness section (representation step): the high-level description states that the auxiliary-colour canonical model is represented inside pure A-chromatic simplicial secrecy models while preserving neighborhood functions and frame conditions, but supplies no explicit verification that the non-triviality clause (secrecy events remain non-trivial from every other agent's perspective) is maintained under the representation map; this step is load-bearing for the completeness claim when |A|≥2.
minor comments (2)
  1. [Abstract and §2] The abstract and introductory paragraphs could clarify the precise set-theoretic status of the secrecy neighborhood functions (e.g., whether they are required to be closed under certain simplicial operations) to make the semantic clause for S_a φ easier to compare with standard neighborhood semantics.
  2. [Definition of simplicial secrecy models] Notation for the secrecy neighborhood function attached to a local state could be introduced with an explicit symbol (e.g., N_a(v)) at first use rather than described only in prose, to improve readability of subsequent semantic definitions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. The single major comment concerns the completeness argument for |A|≥2; we address it directly below.

read point-by-point responses

  1. Referee: [Completeness proof for |A|≥2] Completeness section (representation step): the high-level description states that the auxiliary-colour canonical model is represented inside pure A-chromatic simplicial secrecy models while preserving neighborhood functions and frame conditions, but supplies no explicit verification that the non-triviality clause (secrecy events remain non-trivial from every other agent's perspective) is maintained under the representation map; this step is load-bearing for the completeness claim when |A|≥2.

    Authors: We agree that the representation step requires an explicit verification of the non-triviality clause to make the argument fully self-contained. In the revised manuscript we will insert a dedicated lemma immediately after the definition of the representation map. The lemma will show that if a secrecy neighborhood N in the auxiliary-colour model satisfies the non-triviality condition with respect to every other agent b≠a, then its image under the representation map satisfies the same condition in the target pure A-chromatic model. The argument proceeds by noting that the representation is colour-preserving and that the frame condition is defined pointwise on local states; hence any pair of facets that witness non-triviality in the auxiliary model maps to a pair that witnesses it in the target model. This addition closes the gap without altering the overall proof strategy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from explicit semantic definitions of simplicial secrecy models (enriching standard simplicial epistemic models with agent-relative secrecy neighborhood functions) to the introduction of the primitive operator S_a φ whose clause combines ordinary knowledge with a neighborhood requirement. Soundness of SSL follows directly by induction on the semantic clauses. Completeness for |A|≥2 is obtained via an auxiliary-colour canonical model that is then represented inside the class of pure A-chromatic simplicial secrecy models; the representation step preserves neighborhood functions and frame conditions by explicit construction. No step reduces by definition to its own inputs, no fitted parameters are renamed as predictions, and no load-bearing premise rests solely on self-citation. The argument is self-contained against the stated semantics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the enrichment of simplicial models with neighborhood functions and the semantic clause combining knowledge and neighborhood membership; no free parameters are fitted to data.

axioms (2)
  • standard math Standard axioms of epistemic logic (S5 properties for knowledge operators)
    The secrecy operator builds directly on ordinary knowledge, inheriting standard modal axioms for K_a.
  • domain assumption Frame conditions ensuring secrecy neighborhoods remain non-trivial for other agents
    Invoked in the semantic definition to guarantee the secrecy clause behaves correctly in multi-agent settings.
invented entities (1)
  • Agent-relative secrecy neighborhood functions no independent evidence
    purpose: Attached to local states to define the additional secrecy condition beyond standard knowledge
    Introduced as part of the model enrichment to capture secrecy; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.0 · 5603 in / 1437 out tokens · 72843 ms · 2026-05-13T19:04:46.248509+00:00 · methodology

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