# Universal (and Existential) Nulls

## TL;DR

This paper introduces a novel finite representation called Star Cylinders for handling universal nulls in databases, establishing their algebraic properties and translating queries efficiently, thus advancing incomplete information research.

## Contribution

It develops the Cylindric Star Algebra for universal nulls and demonstrates polynomial-time query translation and evaluation methods involving universal and existential nulls.

## Key findings

- Star Cylinders effectively represent universal nulls.
- Query translation between first-order calculus and cylindric star-algebra is polynomial.
- Evaluation of certain answers remains polynomial, but general reasoning is coNP-hard.

## Abstract

Incomplete Information research is quite mature when it comes to so called {\em existential nulls}, where an existential null is a value stored in the database, representing an unknown object. For some reason {\em universal nulls}, that is, values representing {\em all} possible objects, have received almost no attention. We remedy the situation in this paper, by showing that a suitable finite representation mechanism, called {\em Star Cylinders}, handling universal nulls can be developed based on the {\em Cylindric Set Algebra} of Henkin, Monk and Tarski. We provide a finitary version of the cylindric set algebra, called {\em Cylindric Star Algebra}, and show that our star-cylinders are closed under this algebra. Moreover, we show that any {\em First Order Relational Calculus} query over databases containing universal nulls can be translated into an equivalent expression in our cylindric star-algebra, and vice versa, in time polynomial in the size of the database. The representation mechanism is then extended to {\em Naive Star Cylinders}, which are star-cylinders allowing existential nulls in addition to universal nulls. For positive queries (with universal quantification), the well known naive evaluation technique can still be applied on the existential nulls, thereby allowing polynomial time evaluation of certain answers on databases containing both universal and existential nulls. If precise answers are required, certain answer evaluation with universal and existential nulls remains in coNP. Note that the problem is coNP-hard, already for positive existential queries and databases with only existential nulls. If inequalities $\neg(x_i\approx x_j)$ are allowed, reasoning over existential databases is known to be $\Pi^p_2$-complete, and it remains in $\Pi^p_2$ when universal nulls and full first order queries are allowed.