The undecidability of arbitrary arrow update logic

Arbitrary Arrow Update Logic is a dynamic modal logic with a modality to quantify over arrow updates. Some properties of this logic have already been established, but until now it remained an open question whether the logic's satisfiability problem is decidable. Here, we show by a reduction of the tiling problem that the satisfiability problem of Arbitrary Arrow Update Logic is co-RE hard, and therefore undecidable

Arbitrary Arrow Update Logic

In this paper we introduce arbitrary arrow update logic (AAUL). The logic AAUL takes arrow update logic, a dynamic epistemic logic where the accessibility relations of agents are updated rather than the set of possible worlds, and adds a quantifier over such arrow updates. We investigate the relative expressivity of AAUL compared to other logics, most notably arbitrary public announcement logic (APAL). Additionally, we show that the model checking problem for AAUL is PSPACE-complete. Finally, we introduce a proof system for AAUL, and prove it to be sound and complete

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

To Be Announced

In this survey we review dynamic epistemic logics with modalities for quantification over information change. Of such logics we present complete axiomatizations, focussing on axioms involving the interaction between knowledge and such quantifiers, we report on their relative expressivity, on decidability and on the complexity of model checking and satisfiability, and on applications. We focus on open problems and new directions for research

Quantifying over information change with common knowledge

Public announcement logic (PAL) extends multi-agent epistemic logic with dynamic operators modelling the effects of public communication. Allowing quantification over public announcements lets us reason about the existence of an announcement that reaches a certain epistemic goal. Two notable examples of logics of quantified announcements are arbitrary public announcement logic (APAL) and group announcement logic (GAL). While the notion of common knowledge plays an important role in PAL, and in particular in characterisations of epistemic states that an agent or a group of agents might make come about by performing public announcements, extensions of APAL and GAL with common knowledge still haven’t been studied in detail. That is what we do in this paper. In particular, we consider both conservative extensions, where the semantics of the quantifiers is not changed, as well as extensions where the scope of quantification also includes common knowledge formulas. We compare the expressivity of these extensions relative to each other and other connected logics, and provide sound and complete axiomatisations. Finally, we show how the completeness results can be used for other logics with quantification over information change.publishedVersio

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

Arrow update synthesis

In this contribution we present arbitrary arrow update model logic (AAUML). This is a dynamic epistemic logic or update logic. In update logics, static/basic modalities are interpreted on a given relational model whereas dynamic/update modalities induce transformations (updates) of relational models. In AAUML the update modalities formalize the execution of arrow update models, and there is also a modality for quantification over arrow update models. Arrow update models are an alternative to the well-known action models. We provide an axiomatization of AAUML. The axiomatization is a rewrite system allowing to eliminate arrow update modalities from any given formula, while preserving truth. Thus, AAUML is decidable and equally expressive as the base multi-agent modal logic. Our main result is to establish arrow update synthesis: if there is an arrow update model after which φ, we can construct (synthesize) that model from φ. We also point out some pregnant differences in update expressivity between arrow update logics, action model logics, and refinement modal logic

Coalition and Relativised Group Announcement Logic

There are several ways to quantify over public announcements. The most notable are reflected in arbitrary, group, and coalition announcement logics (APAL, GAL, and CAL correspondingly), with the latter being the least studied so far. In the present work, we consider coalition announcements through the lens of group announcements, and provide a complete axiomatisation of a logic with coalition announcements. To achieve this, we employ a generalisation of group announcements. Moreover, we study some logical properties of both coalition and group announcements that have not been studied before.acceptedVersio

Quantifying over Boolean announcements

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

Second-order propositional modal logic: expressiveness and completeness results

In this paper we advance the state-of-the-art on the application of second-order propositional modal logic (SOPML) in the representation of individual and group knowledge, as well as temporal and spatial reasoning. The main theoretical contributions of the paper can be summarised as follows. Firstly, we introduce the language of (multi-modal) SOPML and interpret it on a variety of different classes of Kripke frames according to the features of the accessibility relations and of the algebraic structure of the quantification domain of propositions. We provide axiomatisations for some of these classes, and show that SOPML is unaxiomatisable on the remaining classes. Secondly, we introduce novel notions of (bi)simulations and prove that they indeed preserve the interpretation of formulas in (the universal fragment of) SOPML. Then, we apply this formal machinery to study the expressiveness of Second-order Propositional Epistemic Logic (SOPEL) in representing higher-order knowledge, i.e., the knowledge agents have about other agents’ knowledge, as well as graph-theoretic notions (e.g., 3-colorability, Hamiltonian paths, etc.). The final outcome is a rich formalism to represent and reason about relevant concepts in artificial intelligence, while still having a model checking problem that is no more computationally expensive than that of the less expressive quantified boolean logic