ON DATABASE QUERIES OVER A WEAKLY O-MINIMAL DOMAIN WITH A SMALL COUNTABLE SPECTRUM

Abstract

In the relational database model introduced by E.F. Codd, the database state is understood as a finite set of relationships between elements. The names of the relationships and their arnts (locations) are fixed and called the database schema. The individual information stored in the relationships of this schema is called the state of the database. Although relational databases were devised for finite sets of data, it is often convenient to assume that there is an infinite domain. We consider relational databases organized over an ordered domain with some additional relations – a typical example is the set of rational numbers together with the relation of linear order and binary operation of addition. If the first-order predicate logic language is used as the query language, then queries can use both database relationships and domain relationships, with variables changing throughout the domain. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving permutations – such queries are called order-generic. It was discovered that for some domains order-generic FO queries fail to express more than pure order queries. Here we prove the collapse result theorem over a weakly o-minimal domain having convexity rank 1 and a small countable spectrum.