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 ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

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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