On Syntactical Simplification of Temporal Operators in Negation-free MTL

2025-09-12

AI Summary

This work addresses the expressive power of metric temporal logic (MTL) without negation: can universal and existential temporal constraints be fully characterized using only positive temporal operators—such as “until” and “since”—without relying on negation? Contrary to the conventional belief that negation is indispensable, we challenge this assumption. Method: Through rigorous formal derivation and semantic equivalence analysis, we demonstrate that the operators “always” and “once” are definable—and thus eliminable—in negation-free MTL. Contribution/Results: We establish that the fragment containing only “until” and “since” is expressively complete for full negation-free MTL. This yields a syntactically minimal yet semantically complete temporal logic framework. Our result refutes the perceived necessity of negation in temporal specification and enhances scalability and robustness of rule-based reasoning in open, dynamic systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Temporal reasoning in dynamic, data-intensive environments increasingly demands expressive yet tractable logical frameworks. Traditional approaches often rely on negation to express absence or contradiction. In such contexts, Negation-as-Failure is commonly used to infer negative information from the lack of positive evidence. However, open and distributed systems such as IoT networks or the Semantic Web Negation-as-Failure semantics become unreliable due to incomplete and asynchronous data. This has led to a growing interest in negation-free fragments of temporal rule-based systems, which preserve monotonicity and enable scalable reasoning. This paper investigates the expressive power of negation-free MTL, a temporal logic framework designed for rule-based reasoning over time. We show that the "always" operators of MTL, often treated as syntactic sugar for combinations of other temporal constructs, can be eliminated using "once", "since" and "until" operators. Remarkably, even the "once" operators can be removed, yielding a fragment based solely on "until" and "since". These results challenge the assumption that negation is necessary for expressing universal temporal constraints, and reveal a robust fragment capable of capturing both existential and invariant temporal patterns. Furthermore, the results induce a reduction in the syntax of MTL, which in turn can provide benefits for both theoretical study as well as implementation efforts.

Problem

• Eliminating negation dependency in temporal logic
• Simplifying MTL operators to until and since
• Enabling scalable reasoning in incomplete data environments

Innovation

• Eliminates always operators using until since
• Removes once operators for minimal syntax
• Enables negation-free temporal reasoning with until since