# Complexity Bounds for Relational Algebra over Document Spanners

## TL;DR

This paper studies the computational complexity of evaluating relational algebra queries over document spanners, identifying intractability in generalized cases and proposing syntactic constraints for efficient evaluation.

## Contribution

It introduces new complexity bounds for relational algebra over regex-based document spanners, especially under schemaless semantics and with difference operators, and proposes automaton-based evaluation techniques.

## Key findings

- Intractability of join and difference operations under schemaless semantics.
- Polynomial delay evaluation achievable with syntactic constraints.
- Automaton-based compilation method for efficient query evaluation.

## Abstract

We investigate the complexity of evaluating queries in Relational Algebra (RA) over the relations extracted by regex formulas (i.e., regular expressions with capture variables) over text documents. Such queries, also known as the regular document spanners, were shown to have an evaluation with polynomial delay for every positive RA expression (i.e., consisting of only natural joins, projections and unions); here, the RA expression is fixed and the input consists of both the regex formulas and the document. In this work, we explore the implication of two fundamental generalizations. The first is adopting the "schemaless" semantics for spanners, as proposed and studied by Maturana et al. The second is going beyond the positive RA to allowing the difference operator. We show that each of the two generalizations introduces computational hardness: it is intractable to compute the natural join of two regex formulas under the schemaless semantics, and the difference between two regex formulas under both the ordinary and schemaless semantics. Nevertheless, we propose and analyze syntactic constraints, on the RA expression and the regex formulas at hand, such that the expressive power is fully preserved and, yet, evaluation can be done with polynomial delay. Unlike the previous work on RA over regex formulas, our technique is not (and provably cannot be) based on the static compilation of regex formulas, but rather on an ad-hoc compilation into an automaton that incorporates both the query and the document. This approach also allows us to include black-box extractors in the RA expression.