Tags:Counting Quantifiers, Logic and Numbers and Probabilistic Logic

Abstract:

We present a research program which investigates the intersection of deductive reasoning with explicit quantitative capabilities. These capabilities encompass probabilistic reasoning, counting and counting quantifiers, and similar systems. The need to have a combined reasoning system that enables a unified way of reasoning with quantities has always been recognized in modern logic, as proposals of probabilistic logic reasoning are present since the work of Boole [1854]. Equally ubiquitous is the need to deal with cardinality restrictions on finite sets. More recently, a well-founded probabilistic theory has been developed for non-classical settings as well, such as probabilistic reasoning over Lukasiewicz infinitely-valued logic.

We show that there is a common way to deal with these several deductive quantitative capabilities, involving a framework based on Linear Algebra and Linear Programming. The distinction between classical and non-classical reasoning on the one hand, and probabilistic and cardinality reasoning on the other hand, comes from the different family of algebras employed. The quantitative logic systems also allow for the introduction of inconsistency measurements, which quantify the degree of inconsistency of a given quantitative logic theory, following some basic principles of inconsistency measurements.

On the practical level, we aim at exploring quantitative logic systems in which the complexity of reasoning is "only NP-complete". We provide open-source implementations for solvers operating over those systems and study some notable empirical properties, such as the present of a phase transition.

Quantitative Logic Reasoning -- Combining Logical Reasoning with Probabilities and Counting (Invited Talk)