From Historical Puzzles to Grammatical Constraints: Circular Partitions, Generalized Run-Length Encodings, and Polynomial-Time Decidability

TL;DR

This paper explores circular partition algorithms inspired by historical puzzles, introduces generalized run-length encodings, and proves polynomial-time decidability of certain language existence problems using formal language theory.

Contribution

It presents new algorithms for circular partitioning, introduces generalized run-length encodings, and establishes polynomial-time decidability for a language existence problem involving block constraints.

Findings

Algorithms for balanced circular partitions

Explicit formulas for generalized run-length encodings

Decidability of language existence in polynomial time

Abstract

Motivated by a historical combinatorial problem that resembles the well-known Josephus problem, we investigate circular partition algorithms and formulate problems in deterministic finite automata with practical algorithms. The historical problem involves arranging individuals on a circle and eliminating every k-th person until a desired group remains. We analyze both removal and non-removal approaches to circular partitioning, establishing conditions for balanced partitions and providing explicit algorithms. We introduce generalized run-length encodings over partitioned alphabets to capture alternating letter patterns, computing their cardinalities using Stirling numbers of the second kind. Connecting these combinatorial structures to formal language theory, we formulate an existence problem: given a context-free grammar over a dictionary and block-pattern constraints on letters, does a valid sentence exist? We prove decidability in polynomial time by showing block languages are regular and applying standard parsing techniques. Complete algorithms with complexity analysis are provided and validated through implementation on both historical and synthetic instances.