Epistemic Updates on Bilattices

The Logic of Epistemic Actions and Knowledge (EAK) has been introduced by Baltag, Moss and Solecki [1] as a framework for reasoning about knowledge in a dynamic setting. It is thus a language expansion of (classical) modal logic having, besides the usual modal operators that represent knowledge and beliefs of agents, dynamic operators used to represent the epistemic change that can be brought about by epistemic actions such as, e.g., announcements. Formally, epistemic changes are modeled via the so-called product update construction on the Kripke-style models that constitute the relational semantics of EAK. Through the product update, a Kripke model encoding the current epistemic setup of a group of agents is replaced by an updated model, which encodes the setup of the agents after an epistemic action has taken place. In [3, 4] product updates are dually characterized as a construction (called epistemic update) that transforms the complex algebra associated with a given Kripke model into the complex algebra associated with the model updated by means of an action structure; in this way EAK is endowed with an algebraic semantics that is dual to the relational one via a Jónsson-Tarski-type duality. Moreover, the methods of [3, 4] can be used to define a logic of Epistemic Actions and Knowledge on a propositional basis that is weaker than classical logic. This provides us with a more flexible logical formalism, which can be applied to a variety of contexts where classical reasoning is not suitable. This line of research has been further pursued in [5, 6], which extends the mechanism of updates to the bilattice modal logic of [2], obtaining a bilattice public announcement logic. In the present contribution we report on ongoing research that aims at further extending the methods of [5, 6] to introduce a suitable notion of product update on relational and algebraic models of bilattice modal logic, thus providing a semantics and a complete axiomatization for a bilattice-based Logic of Epistemic Action and Knowledge (BEAK). Bilattice modal logic is a logic defined by Kripke models hW,R, vi in which both valuations and the accessibility relation R : W ⇥W ! FOUR take values into the four-element Belnap bilattice FOUR. The language of bilattice modal logic h^,_,!,¬,⌃, t,>, f,?i is essentially the same as that of classical modal logic (augmented with constants representing elements of FOUR), but the propositional connectives as well as the modal operator ⌃ are interpreted using the algebraic operations of FOUR. This logic can be extended to define bilattice-