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Knowledge and Simplicial Complexes

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Part of the Philosophical Studies Series book series (PSSP,volume 143)

Abstract

Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in a distributed fashion. Thus, we can define: knowledge, belief, bisimulation, the group notions of mutual, distributed and common knowledge, and also dynamics in the shape of simplicial action models. We give a survey on how to interpret all such notions on simplicial complexes, building upon the foundations laid in Goubault et al. (Inf Comput 278:104597, 2021).

Keywords

  • Simplicial complexes
  • Epistemic models
  • Knowledge and belief
  • Bisimulation
  • Logical tools

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Notes

  1. 1.

    Instead of atoms P, considers atoms P × A, where we write p a for (p, a). For each agent a, let P a be the set of all p a that are local for a. Now, all P a are disjoint.

  2. 2.

    In a subdivision of a (pure chromatic) simplicial complex we replace simplexes by sets of simplexes of the same dimension. For general definitions see Herlihy et al. (2013).

  3. 3.

    There is no agreement on terminology here. The notion that everybody knows φ (which is unambiguous) is in different communities called: shared knowledge, mutual knowledge, general knowledge; where in some communities some of these terms mean common knowledge instead, creating further confusion.

  4. 4.

    The intuition puts one on the wrong foot for distributedly known ignorance: it may be distributed knowledge between a and b that ‘p is true and b is ignorant of p’, but this cannot be made common knowledge between them. If a were to inform b of this, they would then have common knowledge between them of p, but they would not have common knowledge between them of ‘p is true and b does not know p’.

  5. 5.

    Local semantics can also be given for distributed knowledge and for common knowledge. For example, to determine what a and b commonly know on the edge {a0, b1} we need to consider facets bordering on the chain a0—b1—a1—b0 only. For example, \(\mathcal C, \{a0,b1\} \models K_a \neg p_a\) whereas \(\mathcal C,\{a0,b1\} \not \models C_{ab} \neg p_a\).

  6. 6.

    The epistemic model equivalent would be: \({\sim ^{*\cap }_m} := (\bigcup _{B \subseteq A}^{|B|=m+1} \sim ^\cap _B)^*\).

  7. 7.

    Personal communication by Alexandru Baltag.

  8. 8.

    In logics that contains modalities for such action models, we need to require that \(\mathcal V(C)\) is finite and that p o s t is a partial function defined for a finite subset P′⊆ P only.

  9. 9.

    Using local semantics, instead of \(\mathcal C,X \models \bigwedge _{v' \in X'} \mathsf {pre}'(v')\) we can require that \(\mathcal C, v \models \mathsf {pre}'(v')\) for all v ∈ X and v′∈ X′ with χ(v) = χ(v′). This may be more elegant.

  10. 10.

    If we were to incorporate modalities for simplicial action models into the logical language, we can also express propositions such as ‘before the update c did not know the value of b’s variable, but afterwards she knows’: \(\mathcal C, F_4 \models \neg (K_c p_b \vee K_c \neg p_b) \wedge [\mathcal C',F^{\prime }_2] (K_c p_b \vee K_c \neg p_b)\).

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Acknowledgements

We thank the reviewers for their very detailed comments. The second author was partially supported by AID project “Validation of Autonomous Drones and Swarms of Drones” and the academic chair “Complex Systems Engineering” of École Polytechnique-ENSTA-Télécom-Thalès-Dassault-Naval Group-DGA-FX-Fondation ParisTech. The fourth author was partially supported by grants UNAM-PAPIIT IN106520 and CONACYT-LASOL.

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Ditmarsch, H.v., Goubault, É., Ledent, J., Rajsbaum, S. (2022). Knowledge and Simplicial Complexes. In: Lundgren, B., Nuñez Hernández, N.A. (eds) Philosophy of Computing. Philosophical Studies Series, vol 143. Springer, Cham. https://doi.org/10.1007/978-3-030-75267-5_1

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