Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39941

Block-exchange descent for insertion: commuting projections and minimum masks

Haobo Ma·Wenlin Zhang·Charles E. Hughes
View original source ↗Read source document ↗Export metadata ↓

Abstract

For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.

Explore related subjects

cs.FLmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haobo Ma, Wenlin Zhang, Charles E. Hughes. 2026-10-29. Block-exchange descent for insertion: commuting projections and minimum masks. https://arxiv.org/abs/2610.39941

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

KEEP EXPLORING

Related papers

arXiv2026

Tree Equivalence of Higher-Order Recursion Schemes Is Undecidable

Higher-order recursion schemes (HORS) are finite, simply typed recursive programs that generate potentially infinite trees. Their monadic second-order model-checking problem is decidable. We prove that tree equivalence of deterministic HORS is undecidable and, more precisely, $Π^0_1$-complete under total computable many-one reductions. The lower bound holds for safe, productive schemes of order at most four over the fixed ranked alphabet $Σ_2=\{e:0,d:2\}$, consisting of one nullary and one binary symbol. Adapting the counter-guard witness of Boiret, Piorkowski, and Schmude, we generate all candidate executions of a two-counter machine by producing both continuations at every counter test. An integer weight is nonzero exactly on legal prefixes. Functions at fixed simple types represent the polynomial of growing degree. A shift recurrence updates them without storing a separate step counter, allowing the witness to be implemented by safe applicative HORS rules. At candidate halts, the intermediate schemes generate unary observation subtrees whose lengths encode the two natural-number components of the signed weight. Invalid candidates give matching observations; a legal halt produces a finite discrepancy. Fixed constructor contexts encode these intermediate trees over $Σ_2$. For the upper bound, a finite monotone semantics decides head convergence and supports an algorithm computing exact tree prefixes at any prescribed finite depth, including positions labelled $\bot$.

cs.FL↗
arXiv2026

Exact Avoid-or-Compress Extrema for Dense Reachable Images: Defects Four and Five

Let $S$ be a reachable image of a strongly connected synchronizing automaton and $\varnothing\ne A\subsetneq S$. We determine the maximum shortest avoid-or-compress length over binary and arbitrary finite alphabets for $m=n-|S|=4$ and $m=5$. The exact values are $\min(n-|A|,11)$ and $\min(n-|A|,14)$, respectively. Uniform binary constructions attain both bounds. For the upper bounds, short-pair graphs and actual-image constraints reduce all parameters to finite necessary domains. Exhaustive checks exclude the domains, apart from a single relaxation eliminated by a multipartite overlap lemma. The largest calculation covers 3,206,988 domains.

cs.FL↗
arXiv2026

Reachability in Five-Dimensional Vector Addition Systems Is PSPACE-Complete

Vector addition systems (VAS) are a fundamental model of concurrent computation in which transitions update nonnegative integer counters by adding integer vectors, provided all counters remain nonnegative. Closely related to Petri nets, VAS are the single-state counterpart of vector addition systems with states (VASS), which use finitely many states to restrict the available transitions. A classical simulation of $d$-VASS by $(d+3)$-VAS connects their reachability problems, but the complexity-theoretic picture of fixed-dimensional VAS remains largely unclear. While reachability in $2$-VASS is PSPACE-complete under binary encoding, the best known upper bound for $5$-VAS was only elementary. We prove that every reachable $5$-VAS instance has a reachability witness of length polynomial in its unary size. Consequently, $5$-VAS reachability is NL-complete under unary encoding and PSPACE-complete under binary encoding. We also establish a polynomial factorization of the reachability relation of every $d$-VAS with $d \ge 3$. For $5$-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed $d\ge3$, it also yields an $\mathsf F_d$ upper bound for reachability in $(d+3)$-VAS.

cs.FL↗