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.

Highlights

  • Public announcement logic (PAL) [GG97, Pla89] extends epistemic logic with operators for reasoning about the effects of specific public announcements

  • We prove that: for all φ ∈ Lbapal, and for all Ms, Ns : if Ms ↔Ns, Ms |= φ iff Ns |= φ; from which the required follows by restricting the scope of φ to the consequent of the implication

  • We show that Boolean arbitrary public announcement logic (BAPAL) is more expressive than EL and that BAPAL is not as least as expressive as two other logics with quantification over announcements: Arbitrary public announcement logic (APAL), and group announcement logic (GAL) [ ̊ABvDS10]

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Summary

Introduction

Public announcement logic (PAL) [GG97, Pla89] extends epistemic logic with operators for reasoning about the effects of specific public announcements. The formula [ψ]φ means that “φ is true after the truthful announcement of ψ”. Arbitrary public announcement logic (APAL) [BBvD+08] augments this with operators for quantifying over public announcements. The formula 2φ means that “φ is true after the truthful announcement of any formula that does not contain 2”. Synthesis problems can be solved by specifying them as formulas in the logic, and applying modelchecking or satisfiability procedures. In the case of APAL, while there is a PSpacecomplete model-checking procedure [ ̊ABvDS10], the satisfiability problem is undecidable in the presence of multiple agents [FvD08].

LOGICAL METHODS IN COMPUTER SCIENCE
Boolean arbitrary public announcement logic
Semantic results
Arbitrary announcement normal form and necessity form
BAPAL validities involving the quantifier
Boolean closure
Expressivity
Axiomatization
Findings
Conclusions and further research
Full Text
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