{"title":"半环的乐趣:滥用线性代数的功能珍珠","authors":"Stephen Dolan","doi":"10.1145/2500365.2500613","DOIUrl":null,"url":null,"abstract":"Describing a problem using classical linear algebra is a very well-known problem-solving technique. If your question can be formulated as a question about real or complex matrices, then the answer can often be found by standard techniques. It's less well-known that very similar techniques still apply where instead of real or complex numbers we have a closed semiring, which is a structure with some analogue of addition and multiplication that need not support subtraction or division. We define a typeclass in Haskell for describing closed semirings, and implement a few functions for manipulating matrices and polynomials over them. We then show how these functions can be used to calculate transitive closures, find shortest or longest or widest paths in a graph, analyse the data flow of imperative programs, optimally pack knapsacks, and perform discrete event simulations, all by just providing an appropriate underlying closed semiring.","PeriodicalId":20504,"journal":{"name":"Proceedings of the 18th ACM SIGPLAN international conference on Functional programming","volume":"2015 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2013-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"33","resultStr":"{\"title\":\"Fun with semirings: a functional pearl on the abuse of linear algebra\",\"authors\":\"Stephen Dolan\",\"doi\":\"10.1145/2500365.2500613\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Describing a problem using classical linear algebra is a very well-known problem-solving technique. If your question can be formulated as a question about real or complex matrices, then the answer can often be found by standard techniques. It's less well-known that very similar techniques still apply where instead of real or complex numbers we have a closed semiring, which is a structure with some analogue of addition and multiplication that need not support subtraction or division. We define a typeclass in Haskell for describing closed semirings, and implement a few functions for manipulating matrices and polynomials over them. We then show how these functions can be used to calculate transitive closures, find shortest or longest or widest paths in a graph, analyse the data flow of imperative programs, optimally pack knapsacks, and perform discrete event simulations, all by just providing an appropriate underlying closed semiring.\",\"PeriodicalId\":20504,\"journal\":{\"name\":\"Proceedings of the 18th ACM SIGPLAN international conference on Functional programming\",\"volume\":\"2015 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2013-09-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"33\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 18th ACM SIGPLAN international conference on Functional programming\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/2500365.2500613\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 18th ACM SIGPLAN international conference on Functional programming","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2500365.2500613","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Fun with semirings: a functional pearl on the abuse of linear algebra
Describing a problem using classical linear algebra is a very well-known problem-solving technique. If your question can be formulated as a question about real or complex matrices, then the answer can often be found by standard techniques. It's less well-known that very similar techniques still apply where instead of real or complex numbers we have a closed semiring, which is a structure with some analogue of addition and multiplication that need not support subtraction or division. We define a typeclass in Haskell for describing closed semirings, and implement a few functions for manipulating matrices and polynomials over them. We then show how these functions can be used to calculate transitive closures, find shortest or longest or widest paths in a graph, analyse the data flow of imperative programs, optimally pack knapsacks, and perform discrete event simulations, all by just providing an appropriate underlying closed semiring.