Abstract Let Polk be the set of all functions of k-valued logic representable by a polynomial modulo k, and let Int (Polk) be the family of all closed classes (with respect to superposition) in the partial k-valued logic containing Polk and consisting only of functions extendable to some function from Polk. Previously the author showed that if k is the product of two different primes, then the family Int (Polk) consists of 7 closed classes. In this paper, it is proved that if k has at least 3 different prime divisors, then the family Int (Polk) contains an infinitely decreasing (with respect to inclusion) chain of different closed classes.
{"title":"On closed classes in partial k-valued logic that contain all polynomials","authors":"V. Alekseev","doi":"10.1515/dma-2021-0020","DOIUrl":"https://doi.org/10.1515/dma-2021-0020","url":null,"abstract":"Abstract Let Polk be the set of all functions of k-valued logic representable by a polynomial modulo k, and let Int (Polk) be the family of all closed classes (with respect to superposition) in the partial k-valued logic containing Polk and consisting only of functions extendable to some function from Polk. Previously the author showed that if k is the product of two different primes, then the family Int (Polk) consists of 7 closed classes. In this paper, it is proved that if k has at least 3 different prime divisors, then the family Int (Polk) contains an infinitely decreasing (with respect to inclusion) chain of different closed classes.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42312171","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 The probabilistic characteristics of the graph of k-fold iteration of uniform random mapping are studied. Formulas for the distribution of the length of the aperiodicity segment of an arbitrary vertex with some restrictions are calculated. We obtain exact expressions for the probabilities that two arbitrary vertices belong to the same connected component, that an arbitrary vertex belongs to the preimage set of another vertex and that there exists a collision in the considered graph.
{"title":"Collisions and incidence of vertices and components in the graph of k-fold iteration of the uniform random mapping","authors":"V. O. Mironkin","doi":"10.1515/dma-2021-0023","DOIUrl":"https://doi.org/10.1515/dma-2021-0023","url":null,"abstract":"Abstract The probabilistic characteristics of the graph of k-fold iteration of uniform random mapping are studied. Formulas for the distribution of the length of the aperiodicity segment of an arbitrary vertex with some restrictions are calculated. We obtain exact expressions for the probabilities that two arbitrary vertices belong to the same connected component, that an arbitrary vertex belongs to the preimage set of another vertex and that there exists a collision in the considered graph.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48654805","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 use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j. We also provide the asymptotics for compositions and words which have no alphabetic points, as the size tends to infinity. This is achieved by the construction of upper and lower bounds which converge to each other, and in the latter case by probabilistic arguments.
{"title":"Alphabetic points in compositions and words","authors":"M. Archibald, A. Blecher, A. Knopfmacher","doi":"10.1515/dma-2021-0021","DOIUrl":"https://doi.org/10.1515/dma-2021-0021","url":null,"abstract":"Abstract We use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j. We also provide the asymptotics for compositions and words which have no alphabetic points, as the size tends to infinity. This is achieved by the construction of upper and lower bounds which converge to each other, and in the latter case by probabilistic arguments.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45452622","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}
{"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":null,"pages":null},"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 A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.
{"title":"Local limit theorems for generalized scheme of allocation of particles into ordered cells","authors":"A. N. Timashev","doi":"10.1515/dma-2021-0026","DOIUrl":"https://doi.org/10.1515/dma-2021-0026","url":null,"abstract":"Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46045268","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}
Pub Date : 2021-08-01DOI: 10.1515/dma-2021-frontmatter4
{"title":"Frontmatter","authors":"","doi":"10.1515/dma-2021-frontmatter4","DOIUrl":"https://doi.org/10.1515/dma-2021-frontmatter4","url":null,"abstract":"","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47119490","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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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