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The Complexity of Minimal Positive Realization

Zhaobo Liu · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.27962 · cited by 0 works (OpenAlex) · open-access PDF

Japanese

We prove that, given a strictly proper transfer matrix with rational coefficients and an order bound encoded in unary, deciding whether a positive realization exists within that bound is complete for the existential theory of the real. We further prove hardness for stable single-input single-output transfer functions that admit positive realizations, even when the bound equals the McMillan degree. 我々は、理論的に合理的な転送マトリックスとユニアリスケートでコードされた順序の結びつきを考慮して、その境界内で肯定的な実現が存在するかどうかを決定することは、現実の現実の存在理論にとって完全である。

Machine translation (Meta M2M100); the English original follows.

Abstract

We study the complexity of minimal positive realization for discrete-time systems. We prove that, given a strictly proper transfer matrix with rational coefficients and an order bound encoded in unary, deciding whether a positive realization exists within that bound is complete for the existential theory of the reals. We further prove hardness for stable single-input single-output transfer functions that admit positive realizations, even when the bound equals the McMillan degree.

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Topics: Realization (probability), Mathematics, Upper and lower bounds, Existential quantification, Transfer (computing), Discrete mathematics

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