Labelled Natural Deduction for Public Announcement Logic with Common Knowledge

Round 1

Reviewer 1 Report

Summary:
The paper presents a labelled natural deducion system which is shown to be sound and complete for public announcement logic with common knowledge.

General comment:
The only complete systems for this logic known so far are Hilbert calculi. The paper therefore is a significant contribution to the the study of the proof theory of this logic. The results presented are clearly stated and results are proved carefully and convincingly, with few exceptions (see minor comments below). The final discussion suggests that establishing the normalizability of the system is non-trivial and this is left for further work.

I would recommend to publish the paper with minor revision needed to cope with the the following issues.

Minor points:
* In section 3, it would be useful to point out that the semantic definition of |=^\overrightarro{F} A is to be understood as a double induction, a main induction on the length of \overrigharrow{F} and a second induction on the complexity of A.

* The author should recall, before proposition 2, that conjunction is defined and what is the ``standard'' semantic clause for &.\

* In section 5, after line 158 the author should briefly explain the difference between the rules for R_{\overrigharrow{a}} and those for R_{\overrigharrow{a}}*, namely that the former has a finite number of introduction rules and correspondgly the elimination has a finite number of minor premises (namely as many as the number of agents in \overrigharrow{a}). On the other hand, R_{\overrigharrow{a}}* has infinitely many introduction rules and correspondgly the elimination rule has an infinite number of minor premises, thereby making the derivations in the system to be trees with possibly inifinitely many branches (where each branch is however always finite in length).

* In section 7, is it really necessary to atomize the absurdity rule? Could one devise extra reduction to get rid of configurations constituted by Reduction followed by an elimination (in the style of Stalmark for full classical logic)? A reference could be Ana Teresa Martins, Lília Ramalho Martins Natural Deduction for Full S5 Modal Logic with Weak Normalization Electronic Notes in Theoretical Computer Science, Volume 143, 6 January 2006, Pages 129-140.

* In the proof of proposition 6.1 and 6.3. it is not clear how the author is using the rule $\bot$. In table 1, the rule has only one premise, but in the derivations on pages 13-14 it is applied as if it had two.

* The formulation of the second diffculty is not very clear. The author could better clarafy the point.

* English should be checked throughout the paper.

Author Response

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Reviewer 2 Report

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Author Response

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