---
title: "Asymptotic Conditional Probability in Modal Logic: A Probabilistic Reconstruction of Nonmonotonic Logic"
authors:
  - "Riccardo Rosati"
  - "Georg Gottlob"
doi: ""
url: "https://ijcai.org/proceedings/2005/185"
pdf: "https://ijcai.org/proceedings/2005/0185.pdf"
venue: "Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence"
year: 2005
pages: "1378-1382"
track: ""
keywords:
---

# Asymptotic Conditional Probability in Modal Logic: A Probabilistic Reconstruction of Nonmonotonic Logic

Riccardo Rosati, Georg Gottlob

*Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence*, pages 1378-1382.

## Abstract

We analyze the asymptotic conditional validity of modal formulas, i.e., the probability that a formula ψ is valid in the finite Kripke structures in which a given modal formula φ is valid, when the size of these Kripke structures grows to infinity. We characterize the formulas ψ that are almost surely valid (i.e., with probability 1) in case φ is a flat, S5consistent formula, and show that these formulas ψ are exactly those which follow from φ according to the nonmonotonic modal logic S5G. Our results provide – for the first time – a probabilistic semantics to a well-known nonmonotonic modal logic, establishing a new bridge between nonmonotonic and probabilistic reasoning, and give a computational account of the asymptotic conditional validity problem in Kripke structures.