{"title":"具有(迭代)构成的魏赫劳赫网格的等式理论","authors":"Cécilia Pradic","doi":"arxiv-2408.14999","DOIUrl":null,"url":null,"abstract":"We study the equational theory of the Weihrauch lattice with composition and\niterations, meaning the collection of equations between terms built from\nvariables, the lattice operations $\\sqcup$, $\\sqcap$, the composition operator\n$\\star$ and its iteration $(-)^\\diamond$ , which are true however we substitute\n(slightly extended) Weihrauch degrees for the variables. We characterize them\nusing B\\\"uchi games on finite graphs and give a complete axiomatization that\nderives them. The term signature and the axiomatization are reminiscent of\nKleene algebras, except that we additionally have meets and the lattice\noperations do not fully distributes over composition. The game characterization\nalso implies that it is decidable whether an equation is universally valid. We\ngive some complexity bounds; in particular, the problem is Pspace-hard in\ngeneral and we conjecture that it is solvable in Pspace.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The equational theory of the Weihrauch lattice with (iterated) composition\",\"authors\":\"Cécilia Pradic\",\"doi\":\"arxiv-2408.14999\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the equational theory of the Weihrauch lattice with composition and\\niterations, meaning the collection of equations between terms built from\\nvariables, the lattice operations $\\\\sqcup$, $\\\\sqcap$, the composition operator\\n$\\\\star$ and its iteration $(-)^\\\\diamond$ , which are true however we substitute\\n(slightly extended) Weihrauch degrees for the variables. We characterize them\\nusing B\\\\\\\"uchi games on finite graphs and give a complete axiomatization that\\nderives them. The term signature and the axiomatization are reminiscent of\\nKleene algebras, except that we additionally have meets and the lattice\\noperations do not fully distributes over composition. The game characterization\\nalso implies that it is decidable whether an equation is universally valid. We\\ngive some complexity bounds; in particular, the problem is Pspace-hard in\\ngeneral and we conjecture that it is solvable in Pspace.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.14999\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.14999","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The equational theory of the Weihrauch lattice with (iterated) composition
We study the equational theory of the Weihrauch lattice with composition and
iterations, meaning the collection of equations between terms built from
variables, the lattice operations $\sqcup$, $\sqcap$, the composition operator
$\star$ and its iteration $(-)^\diamond$ , which are true however we substitute
(slightly extended) Weihrauch degrees for the variables. We characterize them
using B\"uchi games on finite graphs and give a complete axiomatization that
derives them. The term signature and the axiomatization are reminiscent of
Kleene algebras, except that we additionally have meets and the lattice
operations do not fully distributes over composition. The game characterization
also implies that it is decidable whether an equation is universally valid. We
give some complexity bounds; in particular, the problem is Pspace-hard in
general and we conjecture that it is solvable in Pspace.