Epistemic Logic in Ensemble Specification

Abstract

Different logics are used to specify the structure, life cycle and interaction of dynamically forming ensembles. Specified aspects include the construction and finalization of an ensemble, joining and leaving of collaborators, acceptable and forbidden behaviors of the ensemble, local and global goals and boundaries, and the permission to access resources. As ensembles are dynamically formed from heterogeneous agents, it is reasonable to assume an evolving information asymmetry between its collaborators. Epistemic logic explicitly considers the concepts of knowledge, as held and developed by the different agents in a scope. In this paper, we explore the idea of applying epistemic logic in the specification of different aspects of an ensemble.

6 Appendix

1.1 6.1 Kripke Structures

An epistemic Kripke structure $\mathbf {W}= (W, Sat ,({\equiv _\alpha })_{\alpha \in \mathcal {A}})$ over atomic propositions $\mathcal {P}$ and agents $A$ is a vertex- and edge-labeled, directed graph, where

• each vertex $w\in W$ is called a world,

• $Sat : \mathcal {P}\rightarrow 2^W$ is the valuation function, mapping atomic proposition $p\in \mathcal {P}$ to the set $Sat (p)$ of worlds in which p is true, and

• each ${\equiv _\alpha }\subseteq W\times W$ is a reflexive, symmetric relations specifying that if in world w, $\alpha$ also considers $w'$ as the current world iff $w\equiv _\alpha w'$.

For an agent $\alpha$ and a world $w\in W$, we define the set $Ind _{\mathbf {W}}(w,\alpha ) := \{w'\mid w'\in W, w\equiv _\alpha w'\}$ of worlds indistinguishable from w by $\alpha$ in $\mathbf {W}$, observing $w\in Ind _{\mathbf {W}}(w,\alpha )$.

1.2 6.2 Semantics of $\mathbb {L}$

The semantics of a formula are defined by the set of worlds in which the formula holds, formalized as the function $Worlds _{}(\mathbf {W},\cdot ) : \mathbb {L}\rightarrow 2^W$ where for each $\phi ,\psi \in \mathbb {L}$:

• 1. $Worlds _{}(\mathbf {W},p) = Sat (p)$ for all $p\in \mathcal {P}$.

• 2. $Worlds _{}(\mathbf {W},\lnot \phi ) = W\setminus Worlds _{}(\mathbf {W},\phi )$.

• 3. $Worlds _{}(\mathbf {W},\phi \vee \psi ) = Worlds _{}(\mathbf {W},\phi )\cup Worlds _{}(\mathbf {W},\psi )$.

• 4. $Worlds _{}(\mathbf {W},\mathop {\mathsf {K}_{\alpha }} \phi ) = \{w\in W \mid Ind _{\mathbf {W}}(w,\alpha )\subseteq Worlds _{}(\mathbf {W},\phi ) \}$.

• 5. $Worlds _{}(\mathbf {W},\mathop {\mathsf {K}_{A}} \phi ) = \bigcap _{\alpha \in A} Worlds _{}(\mathbf {W},\mathop {\mathsf {K}_{\alpha }} \phi )$.

• 6. $Worlds _{}(\mathbf {W},\mathop {\mathsf {CK}_{A}}\phi ) = Worlds _{}(\mathbf {W},\phi )\cap Worlds _{}(\mathbf {W},\mathop {\mathsf {K}_{A}} \phi )\cap Worlds _{}(\mathbf {W},\mathop {\mathsf {K}_{A}} \mathop {\mathsf {K}_{A}} \phi )\dots$.

From $w\in Ind _{\mathbf {W}}(w,\alpha )$, we get $Worlds _{}(\mathbf {W},\mathop {\mathsf {K}_{\alpha }} \phi )\subseteq Worlds _{}(\mathbf {W},\phi )$. Hence, $\mathop {\mathsf {K}_{\alpha }} \phi \Rightarrow \phi$ is a tautology: $\alpha$ knows that the formula $\phi$ is a fact, that is, $\phi$ is true, and $\alpha$ knows that $\phi$ is true.

1.3 6.3 Semantics of $\mathbb {L}_e$

The semantics of a formula $\phi \in \mathbb {L}_e$ are defined by the set of worlds in which the formula holds, formalized as the function $Worlds _{e}(\mathbf {W},\cdot ) : \mathbb {L}_e\rightarrow 2^W$ where for each $\phi ,\psi \in \mathbb {L}$:

• 1. $Worlds _{e}(\mathbf {W},p) = Sat (p)$ for all $p\in \mathcal {P}$.

• 2. $Worlds _{e}(\mathbf {W},\lnot \phi ) = W\setminus Worlds _{e}(\mathbf {W},\phi )$.

• 3. $Worlds _{e}(\mathbf {W},\phi \vee \psi ) = Worlds _{e}(\mathbf {W},\phi )\cup Worlds _{e}(\mathbf {W},\psi )$.

• 4. $Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{\alpha }} \phi ) = \{w\in W \mid Ind _{\mathbf {W}}(w,\alpha )\subseteq Worlds _{e}(\mathbf {W},\phi )\}$.

• 5. $Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{A}} \phi ) = \bigcap _{\alpha \in A} Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{\alpha }} \phi )$.

• 6. $Worlds _{e}(\mathbf {W},\mathop {\mathsf {CK}_{A}}\phi ) = Worlds _{e}(\mathbf {W},\phi )\cap Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{A}} \phi )\cap Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{A}} \mathop {\mathsf {K}_{A}} \phi )\dots$.

• 7. $Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{e}} \phi ) = \bigcap _{\alpha \in \mathcal {A}} Worlds _{e}(\mathbf {W},\phi _\alpha )$ where $\phi _\alpha = member _{e}(\alpha )\Rightarrow \mathop {\mathsf {K}_{\alpha }} \phi$.

• 8. $Worlds _{e}(\mathbf {W},\mathop {\mathsf {CK}_{A}}\phi ) = Worlds _{e}(\mathbf {W},\phi )\cap Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{e}} \phi )\cap Worlds _{e}(\mathbf {W},\mathop {\mathsf {K}_{e}} \mathop {\mathsf {K}_{e}} \phi )\dots$.