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A Logic of Secrecy on Simplicial Models

Shanxia Wang

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.

A Logic of Secrecy on Simplicial Models

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 . Semantically, holds when agent knows in the ordinary simplicial sense and, moreover, the truth set of belongs to one of the designated secrecy neighborhoods associated with '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 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 , we prove completeness by first constructing an auxiliary-colour canonical model and then representing it inside the original class of pure -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.

Paper Structure

This paper contains 33 sections, 30 theorems, 298 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation: secrecy beyond ordinary epistemic knowledge
  3. Why simplicial semantics
  4. Main idea of the paper
  5. Main results and contributions
  6. Related Work
  7. Logical analyses of secrecy
  8. Simplicial models for epistemic logic
  9. Neighborhood semantics for non-normal modalities
  10. Position of the present work
  11. Syntax and Semantics
  12. Underlying simplicial epistemic structures
  13. Simplicial secrecy models
  14. Language and truth conditions
  15. A running example
  16. ...and 18 more sections

Key Result

Lemma 3.7

For every agent $a\in A$, the relation $\sim_a$ on $\mathsf{Fac}(M)$ is an equivalence relation. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A two-agent running example. Rows are $a$-equivalence classes and columns are $b$-equivalence classes. The shaded event is $U=\{x_1,x_2,x_3,y_1\}.$ The top row is $\mathsf{St}(u_0)$. In each column there is a facet outside $U$, namely $z_1$, $y_2$, and $y_3$, witnessing condition (SN) for the owner $a$ at the local state $u_0$.

Theorems & Definitions (82)

  • Definition 3.1: Simplicial complex
  • Definition 3.2: Facets
  • Definition 3.3: Chromatic simplicial complex
  • Definition 3.4: Chromatic simplicial epistemic model
  • Definition 3.5: Star
  • Definition 3.6: Epistemic indistinguishability
  • Lemma 3.7
  • proof
  • Definition 3.8: Simplicial secrecy model
  • Remark 3.9
  • ...and 72 more