From Time to Space: The Impact of Linearity in Higher-Order Datalog

2026-06-01

AI Summary

This study investigates the expressive power of linear recursion in higher-order Datalog, focusing on the relationship between the negation-containing fragment of linear higher-order Datalog and space complexity classes. By constructing a stratified formal system, simulating space-bounded Turing machines, and designing a space-efficient query evaluation algorithm, the work establishes for the first time that $(k+1)$-order linear higher-order Datalog precisely captures $(k-1)$-EXPSPACE without relying on any order assumption over the input database. This result generalizes the classical theorem that first-order linear Datalog captures NL to the higher-order realm of space complexity, thereby establishing a tight correspondence between linear higher-order Datalog and the EXPSPACE hierarchy and providing a crucial theoretical foundation for the language design and practical implementation of higher-order Datalog systems.

Abstract

We consider a fragment of Higher-Order Datalog with negation and argue that it generalizes the familiar and important fragment of Linear Datalog. We investigate the expressive power of this fragment, establishing a tight connection with the hierarchy of space complexity classes. In particular, we demonstrate that for all $k \ge 1$, the $(k+1)$-order fragment of Stratified Linear Higher-Order Datalog$^\neg$ captures $(k-1)$-EXPSPACE. This result suggests that restricting programs to linear recursion shifts the expressive power of the corresponding fragments from time to space, generalizing the classical result that (Stratified) Linear Datalog captures NL. Unlike the first-order setting where an ordering assumption is required to capture NL, our results hold without any such assumption on the input database. The proof relies on simulating space-bounded Turing machines using Stratified Linear Higher-Order Datalog$^\neg$ programs and providing a space-efficient evaluation of the query program. We argue that identifying such computationally well-behaved fragments is a crucial step towards paving the way for practical implementations of Higher-Order Datalog.

Problem

• Higher-Order Datalog
• Linearity
• Space Complexity
• Expressive Power
• Linear Recursion

Innovation

• Higher-Order Datalog
• Linearity
• Space Complexity
• Expressive Power
• Stratified Negation