{"title":"最小等效DNF问题和最短隐含问题","authors":"C. Umans","doi":"10.1109/SFCS.1998.743506","DOIUrl":null,"url":null,"abstract":"We prove that the Minimum Equivalent DNF problem is /spl Sigma//sub 2//sup p/-complete, resolving a conjecture due to L.J. Stockmeyer (1976). The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the shortest implicant problem that may be of independent interest. When the input is a formula, the shortest implicant problem is /spl Sigma//sub 2//sup p/-complete, and /spl Sigma//sub 2//sup p/-hard to approximate to within an n/sup 1/2-/spl epsiv// factor. When the input is a circuit, approximation is /spl Sigma//sub 2//sup p/-hard to within an n/sup 1-/spl epsiv// factor. However, when the input is a DNF formula, the shortest implicant problem cannot be /spl Sigma//sub 2//sup p/-complete unless /spl Sigma//sub 2//sup p/=NP[log/sup 2/n]/sup NP/.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"142 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"151","resultStr":"{\"title\":\"The minimum equivalent DNF problem and shortest implicants\",\"authors\":\"C. Umans\",\"doi\":\"10.1109/SFCS.1998.743506\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that the Minimum Equivalent DNF problem is /spl Sigma//sub 2//sup p/-complete, resolving a conjecture due to L.J. Stockmeyer (1976). The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the shortest implicant problem that may be of independent interest. When the input is a formula, the shortest implicant problem is /spl Sigma//sub 2//sup p/-complete, and /spl Sigma//sub 2//sup p/-hard to approximate to within an n/sup 1/2-/spl epsiv// factor. When the input is a circuit, approximation is /spl Sigma//sub 2//sup p/-hard to within an n/sup 1-/spl epsiv// factor. However, when the input is a DNF formula, the shortest implicant problem cannot be /spl Sigma//sub 2//sup p/-complete unless /spl Sigma//sub 2//sup p/=NP[log/sup 2/n]/sup NP/.\",\"PeriodicalId\":228145,\"journal\":{\"name\":\"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)\",\"volume\":\"142 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1998-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"151\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/SFCS.1998.743506\",\"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 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/SFCS.1998.743506","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The minimum equivalent DNF problem and shortest implicants
We prove that the Minimum Equivalent DNF problem is /spl Sigma//sub 2//sup p/-complete, resolving a conjecture due to L.J. Stockmeyer (1976). The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the shortest implicant problem that may be of independent interest. When the input is a formula, the shortest implicant problem is /spl Sigma//sub 2//sup p/-complete, and /spl Sigma//sub 2//sup p/-hard to approximate to within an n/sup 1/2-/spl epsiv// factor. When the input is a circuit, approximation is /spl Sigma//sub 2//sup p/-hard to within an n/sup 1-/spl epsiv// factor. However, when the input is a DNF formula, the shortest implicant problem cannot be /spl Sigma//sub 2//sup p/-complete unless /spl Sigma//sub 2//sup p/=NP[log/sup 2/n]/sup NP/.