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Epistemic Ensembles in Semantic and Symbolic Environments (Extended Version with Proofs)

Rolf Hennicker, Alexander Knapp, Martin Wirsing

TL;DR

This paper develops a formal framework for epistemic ensembles, i.e., families of knowledge-based agents that dynamically share and update beliefs about themselves and others. It introduces a unified syntactic operational semantics for ensembles and develops two complementary mathematical interpretations: semantic environments built from classes of pointed Kripke states and symbolic environments grounded in a finite symbolic knowledge base. A central construct, Φ-equivalence, relates semantic and symbolic states by ensuring that a selected focus of formulas holds exactly as specified in the symbolic state, enabling updates to preserve this equivalence. The main result proves that Φ-equivalent ensemble configurations simulate each other and satisfy the same dynamic ensemble formulæ, providing a robust bridge between semantic and symbolic reasoning for dynamic epistemic ensembles. The work lays groundwork for scalable behavioural ensembles, epistemic planning, and integration with distributed local-state models and LLM-assisted development.

Abstract

An epistemic ensemble is composed of knowledge-based agents capable of retrieving and sharing knowledge and beliefs about themselves and their peers. These agents access a global knowledge state and use actions to communicate and cooperate, altering the collective knowledge state. We study two types of mathematical semantics for epistemic ensembles based on a common syntactic operational ensemble semantics: a semantic environment defined by a class of global epistemic states, and a symbolic environment consisting of a set of epistemic formulæ. For relating these environments, we use the concept of Φ-equivalence, where a class of epistemic states and a knowledge base are Φ-equivalent, if any formula of Φ holds in the class of epistemic states if, and only if, it is an element of the knowledge base. Our main theorem shows that Φ-equivalent configurations simulate each other and satisfy the same dynamic epistemic ensemble formulae.

Epistemic Ensembles in Semantic and Symbolic Environments (Extended Version with Proofs)

TL;DR

This paper develops a formal framework for epistemic ensembles, i.e., families of knowledge-based agents that dynamically share and update beliefs about themselves and others. It introduces a unified syntactic operational semantics for ensembles and develops two complementary mathematical interpretations: semantic environments built from classes of pointed Kripke states and symbolic environments grounded in a finite symbolic knowledge base. A central construct, Φ-equivalence, relates semantic and symbolic states by ensuring that a selected focus of formulas holds exactly as specified in the symbolic state, enabling updates to preserve this equivalence. The main result proves that Φ-equivalent ensemble configurations simulate each other and satisfy the same dynamic ensemble formulæ, providing a robust bridge between semantic and symbolic reasoning for dynamic epistemic ensembles. The work lays groundwork for scalable behavioural ensembles, epistemic planning, and integration with distributed local-state models and LLM-assisted development.

Abstract

An epistemic ensemble is composed of knowledge-based agents capable of retrieving and sharing knowledge and beliefs about themselves and their peers. These agents access a global knowledge state and use actions to communicate and cooperate, altering the collective knowledge state. We study two types of mathematical semantics for epistemic ensembles based on a common syntactic operational ensemble semantics: a semantic environment defined by a class of global epistemic states, and a symbolic environment consisting of a set of epistemic formulæ. For relating these environments, we use the concept of Φ-equivalence, where a class of epistemic states and a knowledge base are Φ-equivalent, if any formula of Φ holds in the class of epistemic states if, and only if, it is an element of the knowledge base. Our main theorem shows that Φ-equivalent configurations simulate each other and satisfy the same dynamic epistemic ensemble formulae.
Paper Structure (28 sections, 24 theorems, 28 equations, 1 figure, 3 tables)

This paper contains 28 sections, 24 theorems, 28 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Related work.
  3. Structure of the paper.
  4. Epistemic Ensembles
  5. Epistemic formulæ.
  6. Epistemic ensemble signatures.
  7. Epistemic processes.
  8. Epistemic ensembles.
  9. Dynamic Epistemic Ensemble Logic
  10. Action Models for Agent Actions
  11. Epistemic choice actions.
  12. Epistemic Ensembles in a Semantic Environment
  13. Epistemic states.
  14. Epistemic updates.
  15. Semantic environments.
  16. ...and 13 more sections

Key Result

lemma 1

If $P \mathrel{\xhookrightarrow{\widehat{\varphi} \cln \eta}{}\mkern-5mu_{\check{\Sigma}}} P'$, then $\mathrm{ags}_{\mkern-1.5mu\check{\Sigma}}(P) \subseteq \mathrm{ags}_{\mkern-1.5mu\Sigma}(\widehat{\varphi}) \cap \mathrm{ags}_{\mkern-1.5mu\check{\Sigma}}(\eta) \cap \mathrm{ags}_{\mkern-1.5mu\check

Figures (1)

  • Figure 1: Transition system for the syntactic bit transmission ensemble.

Theorems & Definitions (36)

  • lemma 1
  • lemma 2
  • lemma 3
  • proposition 1
  • lemma 4
  • corollary 1
  • lemma 5
  • lemma 6
  • lemma 7
  • proposition 2
  • ...and 26 more