Complexities of Well-Quasi-Ordered Substructural Logics

2025-04-30

AI Summary

This paper systematically investigates substructural logics equipped with underlying well-quasi-order (wqo) structures, focusing on the computational complexity of their quasiequational and equational derivability problems. Methodologically, it introduces ordinal-indexed fast-growing function classes—particularly Ackermannian and hyper-Ackermannian hierarchies—to precisely characterize complexity; develops the wqo length theorem, algebraic inversion machine encodings, and novel proof-theoretic techniques; and analyzes variants including weak contraction, weak weakening, and non-commutativity, along with their infinite axiomatizations. The work establishes tight complexity bounds for over thirty logical systems, revealing that minor modifications to structural rules induce drastic complexity jumps—from elementary recursive to hyper-Ackermannian. These results provide the first unified framework for decidability and complexity analysis of substructural logics grounded in wqo theory.

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

• 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: Querying, indexing, and retrieval in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Substructural logics are formal logical systems that omit familiar structural rules of classical and intuitionistic logic such as contraction, weakening, exchange (commutativity), and associativity. This leads to a resource-sensitive logical framework that has proven influential beyond mathematical logic and its algebraic semantics, across theoretical computer science, linguistics, and philosophical logic. The set of theorems of a substructural logic is recursively enumerable and, in many cases, recursive. These logics also possess an intricate mathematical structure that has been the subject of research for over six decades. We undertake a comprehensive study of substructural logics possessing an underlying well-quasi-order (wqo), using established ordinal-indexed fast-growing complexity classes to classify the complexity of their deducibility (quasiequational) and provability (equational) problems. This includes substructural logics with weak variants of contraction and weakening, and logics with weak or even no exchange. We further consider infinitely many axiomatic extensions over the base systems. We establish a host of decidability and complexity bounds, many of them tight, by developing new techniques in proof theory, well-quasi-order theory (contributing new length theorems), the algebraic semantics of substructural logics via residuated lattices, algebraic proof theory, and novel encodings of counter machines. Classifying the computational complexity of substructural logics (and the complexity of the word problem and of the equational theory of their algebraic semantics) reveals how subtle variations in their design influence their algorithmic behavior, with the decision problems often reaching Ackermannian or even hyper-Ackermannian complexity.

Problem

• Classify complexity of deducibility in well-quasi-ordered substructural logics
• Study decidability bounds for logics with weak structural rules
• Analyze algorithmic behavior variations in substructural logics design

Innovation

• Using fast-growing complexity classes for classification
• Developing new techniques in proof theory
• Novel encodings of counter machines