Satisfiability of Arbitrary Public Announcement Logic with Common Knowledge is Σ11\Sigma^1_1-hard

Arbitrary Public Announcement Logic with Common Knowledge (APALC) is an extension of Public Announcement Logic with common knowledge modality and quantifiers over announcements. We show that the satisfiability problem of APALC on S5-models, as well as that of two other related logics with quantification and common knowledge, is $\Sigma^1_1$-hard. This implies that neither the validities nor the satisfiable formulas of APALC are recursively enumerable. Which, in turn, implies that APALC is not finitely axiomatisable.Comment: In Proceedings TARK 2023, arXiv:2307.0400

Arbitrary Arrow Update Logic with Common Knowledge is neither RE nor co-RE

Arbitrary Arrow Update Logic with Common Knowledge (AAULC) is a dynamic epistemic logic with (i) an arrow update operator, which represents a particular type of information change and (ii) an arbitrary arrow update operator, which quantifies over arrow updates. By encoding the execution of a Turing machine in AAULC, we show that neither the valid formulas nor the satisfiable formulas of AAULC are recursively enumerable. In particular, it follows that AAULC does not have a recursive axiomatization.Comment: In Proceedings TARK 2017, arXiv:1707.0825

THE EXPRESSIVITY OF FACTUAL CHANGE IN DYNAMIC EPISTEMIC LOGIC

A commonly used dynamic epistemic logic is one obtained by adding commonknowledge and public announcements to a basic epistemic logic. It is known from Kooi (2007) that adding public substitutions to such a logic adds expressivity over the class K of models. Here I show that substitutions also add expressivity over the classes KD45, S4 and S5 of models. Since the combination of common knowledge, public announcements and substitutions, was shown in Kooi (2007) to be equally expressive to relativized common knowledge these results also show that relativized common knowledge is more expressive than common knowledge and public announcements over KD45, S4 and S5. These results therefore extend the result from van Benthem et al. (2006) that shows that relativized common knowledge is more expressive than common knowledge and public announcements over K

Simple Axioms for Local Properties

Correspondence theory allows us to create sound and complete axiomatizations for modal logic on frames with certain properties. For example, if we restrict ourselves to transitive frames we should add the axiom $\square \phi \rightarrow \square\square\phi$ which, among other things, can be interpreted as positive introspection. One limitation of this technique is that the frame property and the axiom are assumed to hold globally, i.e., the relation is transitive throughout the frame, and the agent's knowledge satisfies positive introspection in every world. In a modal logic with local properties, we can reason about properties that are not global. So, for example, transitivity might hold only in certain parts of the model and, as a result, the agent's knowledge might satisfy positive introspection in some worlds but not in others. Van Ditmarsch et al. (2012) introduced sound and complete axiomatizations for modal logics with certain local properties. Unfortunately, those axiomatizations are rather complex. Here, we introduce far simpler axiomatizations for a wide range of local properties.Comment: In Proceedings TARK 2023, arXiv:2307.0400

HyperLTL Satisfiability is Σ11Σ_1^1-complete, HyperCTL* Satisfiability is Σ12Σ_1^2-complete

Temporal logics for the specification of information-flow properties are able to express relations between multiple executions of a system. The two most important such logics are HyperLTL and HyperCTL*, which generalise LTL and CTL* by trace quantification. It is known that this expressiveness comes at a price, i.e. satisfiability is undecidable for both logics. In this paper we settle the exact complexity of these problems, showing that both are in fact highly undecidable: we prove that HyperLTL satisfiability is $\Sigma_1^1$-complete and HyperCTL* satisfiability is $\Sigma_1^2$-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove $\Sigma_1^2$-membership for HyperCTL*, we prove that every satisfiable HyperCTL* sentence has a model that is equinumerous to the continuum, the first upper bound of this kind. We prove this bound to be tight. Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is $\Pi_1^1$-complete