The Addition of Temporal Neighborhood in Logic of Prefixes and Sub-Intervals

Concetti Chiave

The addition of temporal neighborhood modality increases the complexity and expressiveness of logic.

Sintesi

This article explores the complexity of satisfiability problems in interval temporal logics, focusing on the logic BDhom. It introduces a spatial representation called homogeneous compass structures to simplify proofs. The relationship between BDhom and generalized ∗-free regular expressions is discussed, showing that BDhom belongs to ExpSpace. The content also delves into the concept of atoms in BD formulas and their temporal behavior functions.
Structure:

Introduction to Interval Temporal Logics (ITLs)
Complexity Analysis: Satisfiability Problems in ITLs
Logic BD of Prefixes and Infixes under Homogeneity Assumption
Homogeneous Compass Structures for Spatial Representation
Relationship with Generalized ∗-Free Regular Expressions

Highlights:

Stockmeyer's result on non-elementary lower bound for regular expressions.
Decidable fragments of HS and CDT identified by restricting interval relations.
Model checking problem for ITLs under homogeneity assumption studied.
Logic BDhom shown to be ExpSpace-complete with temporal neighborhood modality A.

Statistiche

A classic result by Stockmeyer states that the emptiness problem for generalized ∗-free regular expressions is non-elementarily decidable (tower-complete) for unbounded nesting of negation.
Such a problem can be easily turned into the satisfiability problem for the logic C of the chop modality, over finite linear orders, under the homogeneity assumption.

Citazioni

"The close relationships between formal languages and ITLs have been already pointed out..." - [MS13a]
"Under such an assumption, full HS has a decidable satisfiability problem..." - [MMM+16]

Approfondimenti chiave tratti da

The addition of temporal neighborhood makes the logic of prefixes and sub-intervals EXPSPACE-complete

by L. Bozzelli,... alle arxiv.org 03-22-2024

https://arxiv.org/pdf/2202.07881.pdf

Domande più approfondite

How do restricted forms of generalized ∗-free regular expressions impact the complexity analysis?

Restricted forms of generalized ∗-free regular expressions have a significant impact on the complexity analysis by allowing for a more manageable translation into logical formalisms. By restricting the expressions to use only prefixes and infixes, we can simplify the mapping process into logic formulas. This restriction avoids the need for complex operators like concatenation (represented by modality C in this case), which can lead to non-elementary complexity in the translation process. As a result, working with these restricted forms makes it easier to analyze and determine properties such as satisfiability within a more reasonable computational space.

What are potential implications of using homogeneous compass structures in other areas?

Homogeneous compass structures offer a spatial representation that simplifies modeling and analysis tasks in various domains beyond interval temporal logics. Some potential implications include:

Modeling Complex Systems: Homogeneous compass structures can be used to represent relationships between entities or events in complex systems where intervals play a crucial role.
Spatial Reasoning: The structured nature of homogeneous compass structures can aid in spatial reasoning tasks, such as navigation systems, robotics, or geographical information systems.
Data Visualization: These structures could be utilized for visualizing data patterns over intervals or time periods, providing insights into trends and correlations.
Knowledge Representation: In artificial intelligence applications, homogeneous compass structures could serve as a basis for representing knowledge about temporal relationships among entities.

Overall, the use of homogeneous compass structures outside their original context offers opportunities for enhanced modeling capabilities and improved understanding of temporal phenomena across different fields.

How does the introduction of temporal neighborhood affect existing models beyond ExpSpace completeness?

The introduction of temporal neighborhood modality A has broader implications beyond just ExpSpace completeness:

Increased Expressiveness: The addition of A expands the expressiveness of existing models by introducing new ways to define relationships between intervals based on Allen's relations like Meets.
Complexity Impact: Introducing A may lead to higher computational complexities than just ExpSpace completeness due to its ability to capture more intricate interval-based interactions.
Enhanced Modeling Capabilities: Temporal neighborhoods allow for finer-grained specifications regarding how intervals relate temporally, enabling more detailed modeling scenarios compared to previous models without this modality.
Applications Across Domains: The inclusion of temporal neighborhoods opens up possibilities for applications requiring precise timing constraints or nuanced temporal dependencies among events or states.

In summary, incorporating the temporal neighborhood modality extends not only computational considerations but also enriches model semantics and widens applicability across diverse domains needing sophisticated interval-based reasoning capabilities.