Practical Reasoning for Uncertain Agents

Abstract

Logical formalisation of agent behaviour is desirable, not only in order to provide a clear semantics of agent-based systems, but also to provide the foundation for sophisticated reasoning techniques to be used on, and by, the agents themselves. The possible worlds semantics offered by modal logic has proved to be a successful framework in which to model mental attitudes of agents such as beliefs, desires and intentions. The most popular choices for modeling the informational attitudes involves annotating the agent with an $\mathit{S5}$-like logic for knowledge, or a $\mathit{KD45}$-like logic for belief. However, using these logics in their standard form, an agent cannot distinguish situations in which the evidence for a certain fact is ‘equally distributed’ over its alternatives, from situations in which there is only one, almost negligible, counterexample to the ‘fact’. Probabilistic modal logics are a way to address this, but they easily end up being both computationally and conceptually complex, for example often lacking the property of compactness. In this paper, we propose a probabilistic modal logic $\mathit{P_F KD45}$, in which the probabilities of the possible worlds range over a finite domain of values, while still allowing the agent to reason about infinitely many options. In this way, the logic remains compact, implying that the agent still has to consider only finitely many possibilities for probability distributions during a reasoning task. We demonstrate a sound, compact and complete axiomatisation for $\mathit{P_F KD45}$ and show that it has several appealing features. Then, we discuss an implemented decision procedure for the logic, and provide a small example. Finally we show that, rather than specifying them beforehand, the finite set of possible probabilities can be obtained directly from the problem specification.