On Quantifying Literals in Boolean Logic and Its Applications to Explainable AI

TL;DR

This paper explores universal literal quantification in Boolean logic, introduces a new semantics, and applies these concepts to enhance explainable AI, offering a more fine-grained approach than variable quantification.

Contribution

It introduces universal literal quantification, a novel semantics, and efficient classes for quantification, advancing the theoretical foundation and applications in explainable AI.

Findings

Universal literal quantification complements existing variable quantification.

A new semantics for quantification is proposed.

Certain classes of Boolean formulas allow efficient quantification.

Abstract

Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/literal and existential/universal quantification. We further identify some classes of Boolean formulas and circuits on which quantification can be done efficiently. Literal quantification is more fine-grained than variable quantification as the latter can be defined in terms of the former. This leads to a refinement of quantified Boolean logic with literal quantification as its primitive.