Word Subtraction Games

Yusuke Inoue; Hiroki Inazu

Home/Archive/arXiv

arXiv · 2611.09157

Word Subtraction Games

Yusuke Inoue·Hiroki Inazu

View original source ↗Read source document ↗Export metadata ↓

Abstract

This note is a self-contained text on word subtraction games, combinatorial games in which players alternately remove a nonempty suffix belonging to a prescribed subtraction set from the current word. It collects and develops basic results on these games from the viewpoints of combinatorial game theory and formal language theory. We focus on the relationship between subtraction sets and the resulting languages of winning and losing positions. The topics include finite, regular, and context-free subtraction sets, as well as several variants and further developments.

Explore related subjects

cs.FL

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yusuke Inoue, Hiroki Inazu. 2026-11-06. Word Subtraction Games. https://arxiv.org/abs/2611.09157

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

KEEP EXPLORING

Related papers

arXiv2026

Fast and General Automatic Differentiation for Finite-State Methods

We propose a new method, that we coined the ``morphism-trick'', to integrate custom implementations of vector-Jacobian products in automatic differentiation softwares, applicable to a wide range of semiring-based computations. Our approach leads to efficient and semiring-agnostic implementations of the backward pass of dynamic programming algorithms. For the particular case of finite-state methods, we introduce an algorithm that computes and differentiates the $\oplus$-sum of all paths' weight of a finite-state automaton. Results show that, with minimal effort from the user, our novel library allows computing the gradient of a function w.r.t. to the weights of a finite state automaton orders of magnitude faster than state-of-the-art automatic differentiation systems. Implementations are made available via an open-source library distributed under a permissive license.

cs.FL↗

arXiv2026

Shortest Synchronizing Words on Product Automata

We consider some operations on finite automata graphs, in particular various product, union and merge operations, and investigate how they relate to the parameter~$\mathrm{rt}$, the \emph{length of a shortest synchronizing word}. We derive some lower and upper bounds for the operations that we study. We also examine several classes of automata (for instance, partially ordered, commutative or idempotent automata) that are preserved under these operations and show in which way the parameter $\mathrm{rt}$ behaves in these automata classes. Also, we consider automata classes that can be constructed by combining automata from better known classes; in particular, these new classes of automata will also satisfy the bound of the famous Černý conjecture. This also motivates us to study even more closure properties. In particular, we show that certain automata classes related to synchronization form pseudo-varieties. We also commence a similar study concerning the length of shortest homing sequences of Mealy automata.

cs.FL↗

arXiv2026

A quadratic lower bound for 2DFAs against one-way liveness

We show that every two-way deterministic finite automaton (2DFA) that solves one-way liveness on height h has Omega(h^2) states. This implies a quadratic lower bound for converting one-way nondeterministic finite automata to 2DFAs, which asymptotically matches Chrobak's well-known lower bound for this conversion on unary languages. In contrast to Chrobak's simple proof, which relies on a 2DFA's inability to differentiate between any two sufficiently distant locations in a unary input, our argument can be applied to inputs over any alphabet and is structured around a main lemma that is general enough to potentially be reused elsewhere.

cs.FL↗