Asymptotic completions of preordered semirings

2026-09-25

AI Summary

This study addresses the limitation that asymptotic preorder comparisons in preordered semirings are restricted to geometric sequences, which hinders the characterization of approximate power behavior for non-geometric sequences. To overcome this, we propose a theory of asymptotic completion for preordered semirings that rigorously transforms approximately geometric sequences into exact geometric ones. By integrating algebraic structure analysis with asymptotic techniques, this work provides a unified treatment of sequence approximation problems in tensor and graph semirings, thereby extending existing characterization frameworks for asymptotic preorders. The proposed approach transcends traditional constraints imposed by geometric sequences and successfully determines the strong converse exponent for binary hypothesis testing under composite Markov hypotheses. These results validate the practical utility of the developed theory in information-theoretic applications.

Abstract

The study of preordered semirings is motivated by applications in computer science, graph theory, and information theory, and provides tools for understanding the asymptotic preorder, which compares large powers of a pair of elements. This paper studies sequences which behave approximately as sequences of powers, but are not necessarily equivalent to geometric sequences. Our main result is that preordered semirings admit completions where such sequences, that we call approximately geometric, become equivalent to geometric sequences, and that existing characterizations of the asymptotic preorder extend to the completion. We provide several classes of examples of approximately geometric sequences in the semiring of tensors, and in the semiring of graphs. As a concrete application, we determine the strong converse exponent for binary hypothesis testing with composite Markov hypotheses.

Problem

• preordered semirings
• asymptotic preorder
• approximately geometric sequences
• completion
• binary hypothesis testing

Innovation

• preordered semirings
• asymptotic completion
• approximately geometric sequences
• tensor semiring
• binary hypothesis testing