Likelihood, Probability, and Knowledge

Joseph Y. Halpern, David A. McAllester

Abstract:

The modal logic LL was introduced by Halpern and Rabin as a means of doing qualitative reasoning about likelihood. Here the relationship between LL and probability theory is examined. It is shown that there is a way of translating probability assertions into LL in a sound manner, so that LL in some sense can capture the probabilistic interpretation of likelihood. However, the translation is subtle; several more obvious attempts are shown to lead to inconsistencies. We also extend LL by adding modal operators for knowledge. This allows us to reason about the interaction between knowledge and likelihood. The propositional version of the resulting logic LLK is shown to have a complete axiomatization and to be decidable in exponential time, provably the best possible.

La logique modale LL a ete proposee par Halpern et Rabin comme moyen de proceder a un raisonnement qualitatif a propos de la vraisemblance. Dans cet article, la relation entre la logique modale LL et la theorie des probabilites est examinee. Les auteurs demontrent qu'il existe une facon de bien traduire des assertions probabilistiques en logique modale LL de facon a ce que cette derniere puisse saisir l'interpretation probabilistique de la vraisemblance. Cependant, cette traduction est subtile; plusieurs tentatives plus evidentes ont entraine des incoherences. Des operateurs modaux ont ete ajoutes a la logique modale LL afin de permettre un raisonnement sur l'interaction de la connaissance et de la vraisemblance. On a constate que la version propositionnelle de la logique resultante possedait une axiomatisation complete et s'averait un facteur decisif en temps exponentiel.