Asymptotic truth-value laws in many-valued logics

TL;DR

This paper extends zero-one laws to many-valued logics with finite and some infinite truth-value sets, analyzing the likelihood of truth-values in finite models and their computational complexity.

Contribution

It generalizes Fagin's zero-one law to many-valued logics, including ukasiewicz logic, and characterizes the complexity of determining almost sure truth-values.

Findings

Generalized zero-one laws for finite and some infinite-valued logics.

PSPACE-completeness of deciding almost sure truth-values.

Complete description of possible almost sure truth-values for some logics.

Abstract

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. We obtain generalizations of Fagin's classical zero-one law for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including \L ukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp's generalization of Fagin's result. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete, and for some logics we may describe completely the set of truth-values that can be taken by sentences almost surely.