Epistemic logics model how agents reason about their beliefs and the beliefs of other agents. Existing logics typically assume the ability of agents to reason perfectly about propositions of unbounded modal depth. We present DBEL, an extension of S5 that models agents that can reason about epistemic formulas only up to a specific modal depth. To support explicit reasoning about agent depths, DBEL includes depth atoms Ead (agent a has depth exactly d) and Pad (agent a has depth at least d). We provide a sound and complete axiomatization of DBEL. We extend DBEL to support public announcements for bounded depth agents and show how the resulting DPAL logic generalizes standard axioms from public announcement logic. We present two alternate extensions and identify two undesirable properties, amnesia and knowledge leakage, that these extensions have but DPAL does not. We provide axiomatizations of these logics as well as complexity results for satisfiability and model checking. Finally, we use these logics to illustrate how agents with bounded modal depth reason in the classical muddy children problem, including upper and lower bounds on the depth knowledge necessary for agents to successfully solve the problem.

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reasoning in the target logic (public announcement logic), required e.g. to obtain a convincing encoding and automation of the wise men puzzle, can be realized. Key to the presented semantical embedding is that evaluation domains are modeled explicitly and treated as an additional parameter in the encodings of the constituents of the embedded target logic; in previous related works, e.g. on the embedding of normal modal logics, evaluation domains were implicitly shared between meta-logic and target logic. The work presented in this article constitutes an important addition to the pluralist LogiKEy knowledge engineering methodology, which enables experimentation with logics and their combinations, with general and domain knowledge, and with concrete use cases -- all at the same time.

A shallow semantical embedding for public announcement logic with relativized common knowledge is presented. This embedding enables the first-time automation of this logic with off-the-shelf theorem provers for classical higher-order logic. It is demonstrated (i) how meta-theoretical studies can be automated this way, and (ii) how non-trivial reasoning in the target logic (public announcement logic), required e.g. to obtain a convincing encoding and automation of the wise men puzzle, can be realized. Key to the presented semantical embedding -- in contrast, e.g., to related work on the semantical embedding of normal modal logics -- is that evaluation domains are modeled explicitly and treated as additional parameter in the encodings of the constituents of the embedded target logic, while they were previously implicitly shared between meta logic and target logic.

Current applied intelligent systems have crucial shortcomings either in reasoning the gathered knowledge, or representation of comprehensive integrated information. To address these limitations, we develop a formal transition system which is applied to the common artificial intelligence (AI) systems, to reason about the findings. The developed model was created by combining the Public Announcement Logic (PAL) and the Linear Temporal Logic (LTL), which will be done to analyze both single-framed data and the following time-series data. To do this, first, the achieved knowledge by an AI-based system (i.e., classifiers) for an individual time-framed data, will be taken, and then, it would be modeled by a PAL. This leads to developing a unified representation of knowledge, and the smoothness in the integration of the gathered and external experiences. Therefore, the model could receive the classifier's predefined -- or any external -- knowledge, to assemble them in a unified manner. Alongside the PAL, all the timed knowledge changes will be modeled, using a temporal logic transition system. Later, following by the translation of natural language questions into the temporal formulas, the satisfaction leads the model to answer that question. This interpretation integrates the information of the recognized input data, rules, and knowledge. Finally, we suggest a mechanism to reduce the investigated paths for the performance improvements, which results in a partial correction for an object-detection system.

We propose a logic of asynchronous announcements, where truthful announcements are publicly sent but individually received by agents. Additional to epistemic modalities, the logic therefore contains two types of dynamic modalities, for sending messages and for receiving messages. The semantics defines truth relative to the current state of reception of messages for all agents. This means that knowledge need not be truthful, because some messages may not have been received by the knowing agent. Messages that are announcements may also result in partial synchronization, namely when an agent learns from receiving an announcement that other announcements must already have been received by other agents. We give detailed examples of the semantics, and prove several semantic results, including that: after an announcement an agent knows that a proposition is true, if and only if on condition of the truth of that announcement, the agent knows that after that announcement and after any number of other agents also receiving it, the proposition is true. We show that on multi-agent epistemic models, each formula in asynchronous announcement logic is equivalent to a formula in epistemic logic.

A true lie is a lie that becomes true when announced. In a logic of announcements, where the announcing agent is not modelled, a true lie is a formula (that is false and) that becomes true when announced. We investigate true lies and other types of interaction between announced formulas, their preconditions and their postconditions, in the setting of Gerbrandy's logic of believed announcements, wherein agents may have or obtain incorrect beliefs. Our results are on the satisfiability and validity of instantiations of these semantically defined categories, on iterated announcements, including arbitrarily often iterated announcements, and on syntactic characterization. We close with results for iterated announcements in the logic of knowledge (instead of belief), and for lying as private announcements (instead of public announcements) to different agents. Detailed examples illustrate our lying concepts.

In his seminal work [Plaza, 1989], Plaza proposed the public announcement logic (PAL), which is considered as the pilot logic in the field of dynamic epistemic logic. In the same paper, Plaza also introduced an interesting “know-value” operator Kv and listed a few valid formulas of PAL+Kv. However, it is unknown that whether these formulas, on top of the axioms for PAL, completely axiomatize PAL+Kv. In this paper, we first give a negative answer to this open problem. Moreover, we generalize the Kv operator and show that in the setting of PAL, replacing the Kv operator with its generalized version does not increase the expressive power of the resulting logic. This suggests that we can simply use the more flexible generalization instead of the original PAL+Kv. As the main result, we give a complete proof system for PAL plus the generalized operator based on a complete axiomatization of epistemic logic with the same operator in the single-agent setting.

We propose a dynamic logic of lying, wherein a 'lie that phi' (where phi is a formula in the logic) is an action in the sense of dynamic modal logic, that is interpreted as a state transformer relative to the formula phi. The states that are being transformed are pointed Kripke models encoding the uncertainty of agents about their beliefs. Lies can be about factual propositions but also about modal formulas, such as the beliefs of other agents or the belief consequences of the lies of other agents. We distinguish (i) an outside observer who is lying to an agent that is modelled in the system, from (ii) one agent who is lying to another agent, and where both are modelled in the system. For either case, we further distinguish (iii) the agent who believes everything that it is told (even at the price of inconsistency), from (iv) the agent who only believes what it is told if that is consistent with its current beliefs, and from (v) the agent who believes everything that it is told by consistently revising its current beliefs. The logics have complete axiomatizations, which can most elegantly be shown by way of their embedding in what is known as action model logic or the extension of that logic to belief revision.