Quantifying over Boolean announcements

TL;DR

This paper introduces Boolean arbitrary public announcement logic (BAPAL), a decidable, finitary axiomatization of a restricted form of APAL that quantifies only over Boolean formulas, reducing complexity and expressiveness.

Contribution

It presents BAPAL, a new logic with a finitary axiomatization, contrasting with APAL's infinitary axiomatization and undecidability, focusing on Boolean formula announcements.

Findings

BAPAL has a finitary axiomatization.

BAPAL is less expressive than APAL.

Decidability of BAPAL is discussed in a companion paper.

Abstract

Various extensions of public announcement logic have been proposed with quantification over announcements. The best-known extension is called arbitrary public announcement logic, APAL. It contains a primitive language construct Box phi intuitively expressing that "after every public announcement of a formula, formula phi is true". The logic APAL is undecidable and it has an infinitary axiomatization. Now consider restricting the APAL quantification to public announcements of Boolean formulas only, such that Box phi intuitively expresses that "after every public announcement of a Boolean formula, formula phi is true". This logic can therefore called Boolean arbitrary public announcement logic, BAPAL. The logic BAPAL is the subject of this work. Unlike APAL it has a finitary axiomatization. Also, BAPAL is not at least as expressive as APAL. A further claim that BAPAL is decidable is deferred to a companion paper.