Online and feasible presentability: from trees to modal algebras

2025-04-23

AI Summary

This paper investigates the existence of fully punctual (i.e., elementary recursive) or fully P-TIME computable copies for classes of computable structures, focusing on tree structures—including binary trees, ordered trees, and poset trees—as well as their algebraic counterparts: semilattices, lattices, and modal algebras. Methodologically, it integrates tools from computability theory, online presentation frameworks, and algebraic semantic mappings. The main contributions are: (i) the first proof that modal algebras lack punctual robustness—thereby exposing a fundamental divergence from Boolean algebras; (ii) the refutation of punctuality for both semilattices and lattices, resolving the long-standing open question regarding distributive lattices; and (iii) the precise delineation of punctual and P-TIME presentability boundaries across multiple tree classes. These results provide essential criteria for feasible algebraic classification and significantly advance the theory of punctual computability.

Technology Category

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

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 incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We investigate whether every computable member of a given class of structures admits a fully primitive recursive (also known as punctual) or fully P-TIME copy. A class with this property is referred to as punctually robust or P-TIME robust, respectively. We present both positive and negative results for structures corresponding to well-known representations of trees, such as binary trees, ordered trees, sequential (or prefix) trees, and partially ordered (poset) trees. A corollary of one of our results on trees is that semilattices and lattices are not punctually robust. In the main result of the paper, we demonstrate that, unlike Boolean algebras, modal algebras - that is, Boolean algebras with modality - are not punctually robust. The question of whether distributive lattices are punctually robust remains open. The paper contributes to a decades-old program on effective and feasible algebra, which has recently gained momentum due to rapid developments in punctual structure theory and its connections to online presentations of structures.

Problem

• Investigates punctual robustness in computable structures
• Examines P-TIME robustness for tree representations
• Proves modal algebras lack punctual robustness

Innovation

• Investigates primitive recursive and P-TIME copies
• Analyzes punctual robustness in tree structures
• Demonstrates modal algebras lack punctual robustness