Ability and knowledge: from epistemic transition systems to labelled stit models

Abstract

It is possible to know that one can guarantee a certain result and yet not know how to guarantee it. In such cases one has the ability to guarantee something in a causal sense, but not in an epistemic sense. In this paper we focus on two formalisms used to model both conceptions of ability: one formalism based on epistemic transition systems and the other on labelled stit models. We show a strong correspondence between the two formalisms by providing mappings from the former to the latter for both the languages and the structures. Moreover, we demonstrate that our extension of labelled stit logic is more expressive than the logic of epistemic transition systems.

Keywords: Ability; Knowledge; Modal logic; Multiagent system; Stit theory; Transition system.

Fig. 1

ETS depicting causal ability to q at $w_1$

Fig. 2

ETS depicting epistemic ability to q at $w_1$ and non-determinism of $\mathsf{R}$ at $w_2$

Fig. 3

Labelled stit model depicting causal ability to q at $m_1$

Fig. 4

Labelled stit model depicting epistemic ability to q at $m_1$ and non-determinism of $K_4$ of type $\mathsf{R}$ at $m_2$

Fig. 5

ETS to be transformed into a GX.kstit model

Fig. 6

Transform GX.kstit model based on the ETS in Fig. 5

Fig. 7

Epistemic transition system $\mathcal{N}$

Fig. 8

Transform discrete labelled stit model ${\mathcal{M}}_{\mathcal{N}}$ based on the epistemic transition system $\mathcal{N}$ in Fig. 7