Entailment Relations on Distributions

TL;DR

This paper explores partial orders on probability distributions to define information content, with applications in natural language entailment and potential extensions to quantum states, advancing theoretical understanding of ordering in probabilistic and quantum spaces.

Contribution

It introduces a generalized framework of partial orders on distributions that unify and extend existing notions of information content and entailment, with implications for language and quantum theory.

Findings

Partial orders are directed complete and form domains.

Framework can be applied to natural language word entailment.

Potential for creating orderings on density operators.

Abstract

In this paper we give an overview of partial orders on the space of probability distributions that carry a notion of information content and serve as a generalisation of the Bayesian order given in (Coecke and Martin, 2011). We investigate what constraints are necessary in order to get a unique notion of information content. These partial orders can be used to give an ordering on words in vector space models of natural language meaning relating to the contexts in which words are used, which is useful for a notion of entailment and word disambiguation. The construction used also points towards a way to create orderings on the space of density operators which allow a more fine-grained study of entailment. The partial orders in this paper are directed complete and form domains in the sense of domain theory.