{"title":"Tight length theorems for multiset extensions of Higman’s lemma","authors":"Vitor Greati , Revantha Ramanayake","doi":"10.1016/j.tcs.2025.115546","DOIUrl":null,"url":null,"abstract":"<div><div>A well-quasi-ordered (wqo) set generalizes the notion of well-foundedness and is a powerful tool for analyzing the complexity of computational problems through upper bounds on the length of controlled bad sequences, known as length theorems. The finitary multiset extension of a wqo-set induces an ordering on finite multisets over elements of that set, where one multiset precedes another if there exists an injective mapping between their elements that preserves the original ordering. In this work, we refine existing length theorems for the finitary multiset extension of Higman’s ordering over finite alphabets, and we establish a matching lower bound. As a corollary, we obtain tighter length bounds for the majoring extension of Higman’s ordering over finite alphabets. We demonstrate the application of our results in the complexity analysis of noncommutative hypersequent logics.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1057 ","pages":"Article 115546"},"PeriodicalIF":1.0000,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397525004840","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
A well-quasi-ordered (wqo) set generalizes the notion of well-foundedness and is a powerful tool for analyzing the complexity of computational problems through upper bounds on the length of controlled bad sequences, known as length theorems. The finitary multiset extension of a wqo-set induces an ordering on finite multisets over elements of that set, where one multiset precedes another if there exists an injective mapping between their elements that preserves the original ordering. In this work, we refine existing length theorems for the finitary multiset extension of Higman’s ordering over finite alphabets, and we establish a matching lower bound. As a corollary, we obtain tighter length bounds for the majoring extension of Higman’s ordering over finite alphabets. We demonstrate the application of our results in the complexity analysis of noncommutative hypersequent logics.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.