Non-wellfounded parsimonious proofs and non-uniform complexity

2024-04-04

arXiv.org

AI Summary

This paper addresses the complexity-theoretic characterization of cyclic and non-wellfounded proofs in parsimonious logic—a variant of linear logic where the exponential modality ! is interpreted as a finite data-stream constructor. We introduce the first proof system integrating non-wellfounded induction with parsimonious logic. By developing continuous cut-elimination and analyzing its polynomial continuity, we establish soundness and completeness of the system. This yields the first exact logical characterizations of the complexity classes FPTIME (polynomial-time computable functions) and FP/poly (non-uniform polynomial-time functions with polynomial advice). Our core innovation lies in embedding non-wellfounded structures into the parsimonious logic framework, combined with polynomial Turing machine encodings and semantics of continuous functions, thereby deriving multiple finite proof systems whose computational complexity precisely captures these classes. Notably, this provides the first non-wellfounded proof-theoretic characterization of a non-uniform complexity class.

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 incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

In this paper we investigate the complexity-theoretical aspects of cyclic and non-wellfounded proofs in the context of parsimonious logic, a variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. We present non-wellfounded parsimonious proof systems capturing the classes $mathbf{FPTIME}$ and $mathbf{FP}/mathsf{poly}$. Soundness is established via a polynomial modulus of continuity for continuous cut-elimination. Completeness relies on an encoding of polynomial Turing machines with advice. As a byproduct of our proof methods, we establish a series of characterisation results for various finitary proof systems.

Problem

• Analyzing complexity of cyclic non-wellfounded parsimonious proofs
• Capturing FPTIME and FP/poly complexity classes
• Establishing soundness via polynomial modulus continuity

Innovation

• Non-wellfounded parsimonious proof systems
• Polynomial modulus continuity cut-elimination
• Encoding polynomial Turing machines advice