Reasoning about knowledge and probability

Reviewer: Calin Lucaciu

The way toward a model for reasoning about knowledge and probability begins in this paper with a review of classical possible-worlds semantics for knowledge. Next, the authors provide an extended language and a complete axiomatization for knowledge and probability. The general conclusions are preceded by a revision of the language to allow common knowledge and probability. A good bibliography completes the paper. To introduce their language for knowledge and probability, the authors briefly review the standard S5 possible-worlds semantics for knowledge. The discussion is extended to a distributed system environment, where every agent associated with a process can give its probabilistic measure of truth about knowledge in this system at a given time. In the core of the paper, the authors introduce their axiom system about reasoning modularized into four components (propositional, knowledge, linear inequalities, and probabilities). The theorems that follow give the limits of the complete axiomatization and the resources of the language for knowledge and probability. In this construction, the dynamic of the knowledge system is reflected in the fluctuation of its probability measure. This characteristic makes the system sensitive to a particular representation. In the last part of the paper, the authors introduce a new module of axioms about common knowledge. The proof of completeness of the whole axiomatic system is delayed to a later paper. The authors see their axiomatization effort as applicable in economics and game theory: “We expect that further progress can be made by combining the intuitions from both computer science and game theory.”