We explore a fuzzy modal logic that can formalise probabilistic reasoning about actions and knowledge. In particular, we deal with contexts involving statements about events expressed via modal formulas, e.g., ‘after doing a, the probability of A knowing that p holds increases / decreases / is equal to 0.25’, ‘according to A, p is equally likely to happen after doing a or b’, etc. We define the semantics of the logic on Kripke frames equipped with probability measures. We analyse the complexity of deciding the satisfiability of formulas of our logic over finitely branching models, for the full language and its fragments of varying expressivity. In particular, we identify several fragments of our logic where satisfiability is decidable in polynomial time.

Abstract We show that Kozen and Tiuryn’s substructural logic of partial correctness $\mathsf{S}$ embeds into the equational theory of Kleene algebra with domain, $\mathsf{KAD}$ . We provide an implicational formulation of $\mathsf{KAD}$ which sets $\mathsf{S}$ in the context of implicational extensions of Kleene algebra.

Abstract Arbitrary public announcement logic (APAL) is a logic of change of knowledge with modalities representing quantification over announcements. We present two rather different versions of APAL wherein this quantification is restricted to formulas only containing a subset of all propositional variables: SAPAL and SCAPAL. Such restrictions are relevant in principle for the specification of multi-agent system dynamics. We also present another version of APAL, quantifying over all announcements implied by or implying a given formula: IPAL. We then determine the relative expressivity of all these logics and APAL. We also present complete axiomatizations of SAPAL and SCAPAL and show undecidability of satisfiability for all logics involved, by arguments nearly identical to those for APAL. We show that the IPAL quantifier, motivated by the satisfaction clause for substructural implication, yields a new substructural dynamic consequence relation.

This special issue contains extended versions of selected papers presented at the 3rd International Workshop on Dynamic Logic: New Trends and Applications (DaLí 2020). The workshop took place on 9–10 October 2020 online due to the COVID-19 pandemic. The organization of the workshop was based at the Institute of Computer Science of the Czech Academy of Sciences in Prague, Czech Republic. The submissions to DaLí 2020 and to the special issue span across various areas including dynamic logic and dynamic algebra, dynamic logic and public announcement logic and epistemic and modal logic in general. We hope the reader will enjoy the special issue, and we offer a small appetizer in the form of an outline of its contents. In their paper The Expressivity of Quantified Group Announcements, Natasha Alechina, Hans van Ditmarsch, Tim French and Rustam Galimullin examine the expressivity of group announcement logic (GAL) and coalition announcement logic (CAL), involving the closely related logic of arbitrary public announcements (APAL). These logics are important tools for reasoning about whether groups and coalitions of agents can achieve their epistemic goals through truthful public communication. They also explore how memory impacts the dynamics between groups and coalitions.

Abstract We study a version of public announcement logic with common knowledge based on the relevant logic $\textsf {R}$. Public announcements, as represented in our framework, are not necessarily truthful and accepted by all agents, nor is it assumed that beliefs are preserved under announcements. We establish a completeness result with respect to a relational semantics, and we show that an alternative semantics based on information states is dual to the relational one. We add a question-forming inquisitive disjunction operator to the language and prove a completeness result with respect to the information semantics.

Propositional Dynamic Logic, PDL, is a well known modal logic formalizing reasoning about complex actions. We study many-valued generalizations of PDL based on relational models where satisfaction of formulas in states and accessibility between states via action execution are both seen as graded notions, evaluated in a finite Łukasiewicz chain. For each n>1, the logic PDŁn is obtained using the n-element Łukasiewicz chain, PDL being equivalent to PDŁ2. These finitely-valued dynamic logics can be applied in formalizing reasoning about actions specified by graded predicates, reasoning about costs of actions, and as a framework for certain graded description logics with transitive closure of roles. Generalizing techniques used in the case of PDL we obtain completeness and decidability results for all PDŁn. A generalization of Pratt's exponential-time algorithm for checking validity of formulas is given and EXPTIME-hardness of each PDŁn validity problem is established by embedding PDL into PDŁn.

Abstract In this paper, we study the epistemic extensions of distributive substructural inquisitive logics. Substructural inquisitive logics are logics of questions based on substructural logics of declarative sentences. They generalize basic inquisitive logic which is based on the classical logic of declaratives. We show that if the underlying substructural logic is distributive, the generalization can be extended to embrace also the epistemic modalities ‘knowing whether’ and ‘wondering whether’ that are applicable to questions. We construct a semantic framework for a language of propositional substructural logics enriched with a question-forming operator (inquisitive disjunction) and epistemic modalities. We show that within this framework, one can define a canonical model with suitable properties for any (syntactically defined) epistemic inquisitive logic. This leads to a general approach to completeness proofs for such logics. A deductive system for the weakest epistemic inquisitive logic is described and completeness proved for this special case using the general method.