{"title":"Medial strongly dependent n-ary operations","authors":"A. V. Cheremushkin","doi":"10.1515/dma-2021-0022","DOIUrl":"https://doi.org/10.1515/dma-2021-0022","url":null,"abstract":"Abstract We prove an analogue of Toyoda–Belousov theorem on the structure of medial n-quasigroups for the case of strongly dependent n-ary operations.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"251 - 258"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42079310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the class Fn, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class Fn, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis {x&y,x‾} $ {x&y,overline x} $ (or in the basis {x∨y,x‾}) $ {xvee y,overline x}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.
{"title":"Implementation complexity of Boolean functions with a small number of ones","authors":"N. P. Redkin","doi":"10.1515/dma-2021-0024","DOIUrl":"https://doi.org/10.1515/dma-2021-0024","url":null,"abstract":"Abstract We consider the class Fn, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class Fn, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis {x&y,x‾} $ {x&y,overline x} $ (or in the basis {x∨y,x‾}) $ {xvee y,overline x}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"271 - 279"},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42341160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM(n) of learning of monotone Boolean functions equals Cn⌊n/2⌋ $begin{array}{} displaystyle C_n^{lfloor n/2rfloor} end{array}$ + Cn⌊n/2⌋+1 $begin{array}{} displaystyle C_n^{lfloor n/2rfloor+1} end{array}$ (φM(n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.
{"title":"Learning of monotone functions with single error correction","authors":"S. Selezneva, Yongqing Liu","doi":"10.1515/dma-2021-0017","DOIUrl":"https://doi.org/10.1515/dma-2021-0017","url":null,"abstract":"Abstract Learning of monotone functions is a well-known problem. Results obtained by V. K. Korobkov and G. Hansel imply that the complexity φM(n) of learning of monotone Boolean functions equals Cn⌊n/2⌋ $begin{array}{} displaystyle C_n^{lfloor n/2rfloor} end{array}$ + Cn⌊n/2⌋+1 $begin{array}{} displaystyle C_n^{lfloor n/2rfloor+1} end{array}$ (φM(n) denotes the least number of queries on the value of an unknown monotone function on a given input sufficient to identify an arbitrary n-ary monotone function). In our paper we consider learning of monotone functions in the case when the teacher is allowed to return an incorrect response to at most one query on the value of an unknown function so that it is still possible to correctly identify the function. We show that learning complexity in case of the possibility of a single error is equal to the complexity in the situation when all responses are correct.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"193 - 205"},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49212931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the problem of synthesis of irredundant two-pole contact circuits which implement n-place Boolean functions and allow short single fault detection or diagnostic tests of closures of at most k contacts. We prove that the Shannon function of the length of a fault detection test is equal to n for any n and k, and that the Shannon function of the length of a diagnostic test is majorized by n + k(n − 2) for n ⩾ 2.
{"title":"Bounds on Shannon functions of lengths of contact closure tests for contact circuits","authors":"K. A. Popkov","doi":"10.1515/dma-2021-0015","DOIUrl":"https://doi.org/10.1515/dma-2021-0015","url":null,"abstract":"Abstract We consider the problem of synthesis of irredundant two-pole contact circuits which implement n-place Boolean functions and allow short single fault detection or diagnostic tests of closures of at most k contacts. We prove that the Shannon function of the length of a fault detection test is equal to n for any n and k, and that the Shannon function of the length of a diagnostic test is majorized by n + k(n − 2) for n ⩾ 2.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"165 - 178"},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49638077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables.
{"title":"Convex algebras of probability distributions induced by finite associative rings","authors":"A. Yashunsky","doi":"10.1515/dma-2021-0019","DOIUrl":"https://doi.org/10.1515/dma-2021-0019","url":null,"abstract":"Abstract We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"223 - 230"},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/dma-2021-0019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43618396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract On the set Pk∗ $begin{array}{} displaystyle P_k^* end{array}$ of partial functions of the k-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any k ⩾ 2, the number of implicative closed classes in Pk∗ $begin{array}{} displaystyle P_k^* end{array}$ is finite. For any k ⩾ 2, in Pk∗ $begin{array}{} displaystyle P_k^* end{array}$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in P3∗ $begin{array}{} displaystyle P_3^* end{array}$.
{"title":"On the action of the implicative closure operator on the set of partial functions of the multivalued logic","authors":"S. Marchenkov","doi":"10.1515/dma-2021-0014","DOIUrl":"https://doi.org/10.1515/dma-2021-0014","url":null,"abstract":"Abstract On the set Pk∗ $begin{array}{} displaystyle P_k^* end{array}$ of partial functions of the k-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any k ⩾ 2, the number of implicative closed classes in Pk∗ $begin{array}{} displaystyle P_k^* end{array}$ is finite. For any k ⩾ 2, in Pk∗ $begin{array}{} displaystyle P_k^* end{array}$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in P3∗ $begin{array}{} displaystyle P_3^* end{array}$.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"155 - 164"},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/dma-2021-0014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41993872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the problem of A-completeness in the class of linear automata such that the sets of inputs, outputs and states are Cartesian products of dyadic rationals; systems checked for completeness are comprised of a variable finite set and a fixed additional set. We obtain conditions of A-completeness in terms of maximal subclasses in the cases when the additional set is the set of all unary automata and when the additional set consists of the adder.
{"title":"Conditions of A-completeness for linear automata over dyadic rationals","authors":"Dmitriy V. Ronzhin","doi":"10.1515/dma-2021-0016","DOIUrl":"https://doi.org/10.1515/dma-2021-0016","url":null,"abstract":"Abstract We consider the problem of A-completeness in the class of linear automata such that the sets of inputs, outputs and states are Cartesian products of dyadic rationals; systems checked for completeness are comprised of a variable finite set and a fixed additional set. We obtain conditions of A-completeness in terms of maximal subclasses in the cases when the additional set is the set of all unary automata and when the additional set consists of the adder.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"179 - 192"},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41366856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions EX < 0 and EXeX > 0.
{"title":"Multitype weakly subcritical branching processes in random environment","authors":"V. Vatutin, E. Dyakonova","doi":"10.1515/dma-2021-0018","DOIUrl":"https://doi.org/10.1515/dma-2021-0018","url":null,"abstract":"Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions EX < 0 and EXeX > 0.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"207 - 222"},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48069323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A complete description of trees with maximal possible number of maximum independent sets among all n-vertex trees with exactly l leaves is obtained. For all values of the parameters n and l the extremal tree is unique and is the result of merging the endpoints of l simple paths.
{"title":"Trees with a given number of leaves and the maximal number of maximum independent sets","authors":"D. S. Taletskii, D. Malyshev","doi":"10.1515/dma-2021-0012","DOIUrl":"https://doi.org/10.1515/dma-2021-0012","url":null,"abstract":"Abstract A complete description of trees with maximal possible number of maximum independent sets among all n-vertex trees with exactly l leaves is obtained. For all values of the parameters n and l the extremal tree is unique and is the result of merging the endpoints of l simple paths.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"31 1","pages":"135 - 144"},"PeriodicalIF":0.5,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44756109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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