Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.04377

Monodromy Computation Model

Xiaoyang Liu
View original source ↗Read source document ↗Export metadata ↓

Abstract

A closed path can return to its starting point while analytic continuation reaches another branch. We construct a finite computational calculus in which retained logarithmic branches supply memory, analytic observations select operations, and descent through a covering exposes deeper memory. Three branch coordinates suffice, and every finite observation uses the same cubic power covering. Each program defines a globally real analytic return map whose integer-lattice execution is realized by actual path lifting. Our first theorem simulates every finite-alphabet, finite-control Post--Turing machine, preserving complete configurations, halting, and divergence. Our second theorem identifies the partial numeric functions of the calculus, under fixed primitive-recursive input and output maps, with exactly the partial recursive functions. Deleting descent leaves a Forward language with a syntax-derived finite observation quotient and a primitive-recursive halting predicate. This yields a strict separation within the calculus and identifies access to deeper memory as its decisive additional operation. The constructions and semantic correspondences are formalized in Lean 4 with Mathlib.

Explore related subjects

cs.CCcs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xiaoyang Liu. 2026-11-03. Monodromy Computation Model. https://arxiv.org/abs/2611.04377

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

NP-Hardness and Additive Inapproximability of Deterministic Communication Complexity

Given the full truth table of a Boolean function whose input is split between two players, we study the complexity of determining the minimum worst-case number of bits they must communicate to compute it. This is a basic meta-complexity question, posed by Yao in 1979. We prove NP-hardness of approximation within every fixed additive error for deterministic communication complexity, under deterministic polynomial-time reductions. The result holds both in the standard model and in Yao's original model, where the players alternate after every bit and only the receiver of the last bit must determine the answer. In particular, it resolves the exact computation question under Yao's original conventions. Earlier versions of this same paper established exact NP-hardness for the standard model, independently and concurrently with Hirahara, Ilango, and Loff. In both models, a polynomial-time estimator with any fixed additive error would yield a polynomial-time $3$-SAT algorithm, deterministic or bounded-error randomized according to the estimator. For the standard model, we also give quantitative implications for errors that grow with the number $T$ of table entries. A bounded-error randomized polynomial-time estimator with additive error $c\log T/\log\log T$, for any fixed $0\le c<1/8$, yields a bounded-error randomized $3$-SAT algorithm on $N$ variables and $O(N)$ clauses running in time $2^{O(N^α)}$ for every fixed $8c<α<1$. A deterministic polynomial-time estimator with additive error $c_*\log\log T$, for an effective absolute constant $c_*>0$, yields a deterministic algorithm running in time $2^{O((\log N)^2)}$ on such formulas.

cs.CC↗
arXiv2026

On the average-case complexity landscape for Tensor-Isomorphism-complete problems over finite fields

In Grochow and Qiao (\emph{SIAM J. Comput.}, 2023), the complexity class Tensor Isomorphism (TI) was introduced and isomorphism problems for groups, algebras, and polynomials were shown to be TI-complete. In this paper, we study average-case algorithms for several TI-complete problems over finite fields, including algebra isomorphism, matrix code conjugacy, and $4$-tensor isomorphism. Our main results reveal a rich landscape of average-case complexities of Tensor Isomorphism problems, despite their equivalence in the worst-case setting. Over the finite field of order $q$, we devise (1) average-case \emph{polynomial-time} algorithms for algebra isomorphism and matrix code conjugacy that succeed for $1/Θ(q)$ fraction of inputs; (2) an average-case \emph{polynomial-time} algorithm for the $4$-tensor isomorphism that succeeds for $1/q^{Θ(1)}$ fraction of inputs; and (3) an average-case \emph{polynomial-time} algorithm for the orthogonal $3$-tensor isomorphism that succeed for $1-1/\poly(n)$ fraction of $n\times n\times n$ tensors. Our main technical contribution is to introduce the spectral properties of random matrices into algorithms for TI-complete problems. This leads to not only new algorithms but also new questions in random matrix theory over finite fields. To settle these questions, we need to extend both the generating function approach as in Neumann and Praeger (\emph{J. London Math. Soc.}, 1998) and the character sum method of Gorodetsky and Rodgers (\emph{Trans. Amer. Math. Soc.}, 2021).

cs.CC↗
arXiv2026

Constant-factor approximation of MinCostCSP with a conservative majority polymorphism

For a relational structure A, the Minimum Cost Constraint Satisfaction Problem is the following problem denoted by MinCostCSP(A): Given an instance of CSP(A) with rational costs on variable-value pairs, find a solution to the instance minimizing the sum of the chosen costs. For the exact minimization, a classification of MinCostCSP(A) in terms of A was established by Takhanov [STACS'10]. We focus on constant-factor approximations of MinCostCSP(A). DeHaan, Huang, and Lee recently showed that if A fails to admit a conservative near-unanimity polymorphism then MinCostCSP(A) is not constant-factor approximable [APPROX'25]. We provide a significant step towards a full classification, by proving a dichotomy for structures A admitting a conservative majority (also known as 3-near-unanimity) polymorphism. Our dichotomy criterion is not in terms of an algebraic condition on A but we show that this is unavoidable. We include a simple argument proving that no such condition exists.

cs.CC↗