We develop a formal account of complex group-level epistemic modalities in the situation calculus, with a particular focus on reasoning about common knowledge. Expressions involving common knowledge cannot be handled by standard regression techniques and are thus difficult to reason about effectively. Taking our cue from recent promising work in dynamic epistemic logic, we overcome this limitation by increasing the expressive power of the epistemic language. The syntax of first-order dynamic logic is used to form complex epistemic modalities from the individual-level knowledge operators. Common knowledge is a special case of this syntax, using the unbounded iteration operator. We develop a regression rule for these complex modalities and demonstrate its use to reason about common knowledge in an example domain. The result is a rich multi-agent theory of knowledge and action in which complex group-level epistemic modalities are amenable to effective automated reasoning.

/pdf/complex-epistemic-modalities-in-the-situation-calculus-1zogu0hv5b.pdf

Complex epistemic modalities in the situation calculus

Sequential von Neumann–Morgernstern (VM) games are a very general formalism for representing multi-agent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and complete axiomatization in the situation calculus. This axiomatization allows us to represent game-theoretic reasoning and solution concepts such as Nash equilibrium. We discuss the application of various concepts from VM game theory to the theory of planning and multi-agent interactions, such as representing concurrent actions and using the Baire topology to define continuous payoff functions.

/pdf/representing-von-neumann-morgenstern-games-in-the-situation-25fbzs9whf.pdf

Representing von Neumann–Morgenstern Games in the Situation Calculus

https://people.eng.unimelb.edu.au/adrianrp/papers/kelly14.pdf

Asynchronous knowledge with hidden actions in the situation calculus

Recent articles explore formalizations of the popular board game Clue. For several years, this game was used as a testbed for automated theorem proving exercises in the context of introductory and advanced AI classes at the University of Saskatchewan, by way of motivating the usefulness of Prolog Technology Theorem Provers (PTTPS) as game playing engines. Although the game has a simple axiomatization over a small universe, adapting a PTTP to play a full game of Clue is not trivial. This paper illustrates solutions to problems encountered during this exercise, including some interesting computational rules of inference.

Clue as a Testbed for Automated Theorem Proving

In this paper we outline a formal system for reasoning about agents’ knowledge in knowledge games - a special type of multi-agent system. Knowledge games are card games where the agents’ actions involve an exchange of information with other agents in the game. Our system is modeled using Coq - a formal proof management system. To the best of our knowledge, there are no papers in which knowledge games are considered using a Coq proof assistant. We use the dynamic logic of common knowledge, where we particularly focus on the epistemic consequences of epistemic actions carried out by agents. We observe the changes in the system that result from such actions. Those changes that can occur in such a system that are of interest to us take the form of agents’ knowledge about the state of the system, knowledge about other agents’ knowledge, higher-order agents’ knowledge and so on, up to common knowledge. Besides an axiomatic of epistemic logic, we use a known axiomatization of card games that is extended with some new axioms that are required for our approach. Due to a deficit in implementations grounded in theory that enable players to compute their knowledge in any state of the game, we show how our approach can be used for these purposes.

Reasoning about Epistemic Actions and Knowledge in Multi-agent Systems using Coq

The situation calculus is a logical formalism that has been extensively developed for planning. We apply the formalism in a complex multi-agent domain, modelled on the game of Clue. We find that the situation calculus, with suitable extensions, supplies a unified representation of (1) the interaction protocol, or structure of the game, (2) the dynamics of the knowledge and common knowledge of the agents, and (3) principles of strategic planning.

/pdf/knowledge-and-planning-in-an-action-based-multi-agent-2altyjedgp.pdf

Knowledge and Planning in an Action-Based Multi-agent Framework: A Case Study

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Knowledge and Planning in an Action-Based Multi-agent Framework: A Case Study