The Power of Negation in Higher-Order Datalog

2025-07-27

AI Summary

This paper investigates the expressive power of higher-order Datalog¬ (with negation) under the well-founded semantics and stable model semantics. To achieve this, the authors introduce key higher-order logic programming techniques—including existentially quantified predicate variables, partial application of relations, and relation enumeration—thereby circumventing the need for a predefined database ordering. They establish that, under the well-founded semantics, (k+1)-order Datalog¬ precisely captures k-EXP; under the stable model semantics, it exactly characterizes both co-(k-NEXP) and k-NEXP. These results reveal a fine-grained trade-off between logical order and nondeterminism, and yield complete expressiveness hierarchies for both semantics: each order corresponds bijectively to a canonical complexity class, and the hierarchy is strict with no collapse across levels. This work constitutes a fundamental advance in the theory of expressive power for higher-order logic programs.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We investigate the expressive power of Higher-Order Datalog$^ eg$ under both the well-founded and the stable model semantics, establishing tight connections with complexity classes. We prove that under the well-founded semantics, for all $kgeq 1$, $(k+1)$-Order Datalog$^ eg$ captures k-EXP, a result that holds without explicit ordering of the input database. The proof of this fact can be performed either by using the powerful existential predicate variables of the language or by using partially applied relations and relation enumeration. Furthermore, we demonstrate that this expressive power is retained within a stratified fragment of the language. Under the stable model semantics, we show that $(k+1)$-Order Datalog$^ eg$ captures co-(k-NEXP) using cautious reasoning and k-NEXP using brave reasoning, again with analogous results for the stratified fragment augmented with choice rules. Our results establish a hierarchy of expressive power, highlighting an interesting trade-off between order and non-determinism in the context of higher-order logic programming: increasing the order of programs under the well-founded semantics can surpass the expressive power of lower-order programs under the stable model semantics.

Problem

• Study expressive power of Higher-Order Datalog with negation
• Compare well-founded and stable model semantics complexity
• Establish hierarchy between program order and non-determinism

Innovation

• Higher-Order Datalog with negation captures complexity classes
• Uses well-founded and stable model semantics for analysis
• Stratified fragment retains expressive power with choice rules