PDT Logic: A Probabilistic Doxastic Temporal Logic about Reasoning about Beliefs in Mulit-agent Systems

This thesis presents Probabilistic Doxastic Temporal (PDT) Logic, a formalism to represent and reason about probabilistic beliefs and their temporal evolution in multiagent systems. This formalism enables the quantification of agents’ beliefs through probability intervals and incorporates an explicit notion of time. Quantifying probabilistic knowledge through probability intervals instead of single probability values significantly eases the task of formally representing existing knowledge of a human domain expert. In most cases, a domain expert can give reasonable probability estimates of her knowledge, but will inevitably fail at giving correct precise numerical values on these probabilities. Thus, the use of probability intervals provides means to express probabilistic knowledge as precise as possible without enforcing unrealistic precision. The width of a probability interval can then give additional information about the certainty of a probability quantification. Naturally, a narrow interval is associated with a high certainty of the respective probability and vice versa, a wide interval is associated with low certainty. In contrast to related work, PDT Logic employs an explicit notion of time and thereby facilitates the expression of richer temporal relations. Existing approaches of dynamic epistemic logics usually employ an implicit notion of time. This makes it impossible to reason about temporal relations that span over multiple time points. Through the introduction of an appropriate syntax and semantics we discuss how beliefs in multi-agent systems can be formally represented and how their temporal evolutions can be analyzed. We analyze the complexity of decision problems in PDT Logic and develop sound and complete satisfiability checking algorithms. Possible applications of PDT Logic are indicated through the discussion of suitable examples.