Pumping Lemmata for Recognizable Weighted Languages over Artinian Semirings

TL;DR

This paper introduces a unified pumping lemma applicable to all Artinian semirings, extending previous results for recognizable weighted languages over various semirings, including fields and finite semirings.

Contribution

It provides a generalization of existing pumping lemmata to all Artinian semirings, covering most known cases for recognizable weighted languages.

Findings

Unified pumping lemma for all Artinian semirings

Includes fields and finite semirings as special cases

Enables new techniques for language recognition proofs

Abstract

Pumping lemmata are the main tool to prove that a certain language does not belong to a class of languages like the recognizable languages or the context-free languages. Essentially two pumping lemmata exist for the recognizable weighted languages: the classical one for the Boolean semiring (i.e., the unweighted case), which can be generalized to zero-sum free semirings, and the one for fields. A joint generalization of these two pumping lemmata is provided that applies to all Artinian semirings, over which all finitely generated semimodules have a finite bound on the length of chains of strictly increasing subsemimodules. Since Artinian rings are exactly those that satisfy the Descending Chain Condition, the Artinian semirings include all fields and naturally also all finite semirings (like the Boolean semiring). The new pumping lemma thus covers most previously known pumping lemmata for recognizable weighted languages.