Probability Models of Distributed Proof Generation for zk-SNARK-Based Blockchains

The paper is devoted to the investigation of the distributed proof generation process, which makes use of recursive zk-SNARKs. Such distributed proof generation, where recursive zk-SNARK-proofs are organized in perfect Mercle trees, was for the first time proposed in Latus consensus protocol for zk-SNARKs-based sidechains. We consider two models of a such proof generation process: the simplified one, where all proofs are independent (like one level of tree), and its natural generation, where proofs are organized in partially ordered set (poset), according to tree structure. Using discrete Markov chains for modeling of corresponding proof generation process, we obtained the recurrent formulas for the expectation and variance of the number of steps needed to generate a certain number of independent proofs by a given number of provers. We asymptotically represent the expectation as a function of the one variable n/m, where n is the number of provers m is the number of proofs (leaves of tree). Using results obtained, we give numerical recommendation about the number of transactions, which should be included in the current block, idepending on the network parameters, such as time slot duration, number of provers, time needed for proof generation, etc.

Application To Lipschitzian and Integral Systems Via Quadruple Coincidence Point in Fuzzy Metric Spaces

Without a partially ordered set, in this manuscript, we investigate quadruple coincidence point (QCP) results for commuting mapping in the setting of fuzzy metric spaces (FMSs). Furthermore, some relevant ndings are presented to generalize some of the previous results in this direction. Ultimately, non-trivial examples and applications to nd a unique solution for Lipschitzian and integral quadruple systems are provided to support and strengthen our theoretical results.

$m$-submultisets and $m$-permutations of multisets elements

The article is devoted to two classical combinatorial problems on multisets, which in the existing literature are given unjustifiably little space. Namely, the calculation of the number of all submultisets of power $m$ for an arbitrary multiset and the number of $m$-permutations of such multisets. The first problem is closely related to the width of a partially ordered set of all submultisets of a multiset with the inclusion $\subseteq$. The article contains some important classes of multisets. Combinatorial proofs of problems on the number of $m$-submultisets and $m$-permutations of multiset elements are considered. In the article, on the basis of the generatrix method, economical algorithms for calculating $m$-submultisets and $m$-permutations of multiset elements are constructed. The paper also provides a brief overview of the results that are related to this area of research.

On a poset of quantum exact promise problems

Two of the most well-known quantum algorithms, those introduced by Deutsch–Jozsa and Bernstein–Vazirani, can solve promise problems with just one function query, showing an oracular separation with deterministic classical algorithms. In this work, we generalise those methods to study a family of quantum algorithms that can, with just one query, exactly solve promise problems stated over Boolean functions. We also show that these problems can be naturally ordered, inducing a partially ordered set of promise problems. We study the properties of such a poset, showing that the Deutsch–Jozsa and Bernstein–Vazirani problems are, in a certain sense, extremal problems in it, determining some of its automorphisms and proving that it is connected. We also prove that, for the problems in the poset, the corresponding classical query complexities can take any value between 1 and $$2^{n-1}+1$$ 2 n - 1 + 1 .

On invariants of polynomial functions, II

Let P be a finite partially ordered set. In our previous paper, we defined the sectional geometric genus gi(P) of P and studied gi(P). In this paper, by using this sectional geometric genus of P, we will give a criterion about the case in which P has no order.

Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory

For a partially ordered set (A,\le ) , let {G}_{A} be the simple, undirected graph with vertex set A such that two vertices a\ne b\in A are adjacent if either a\le b or b\le a . We call {G}_{A} the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G={G}_{A} . For a class {\mathcal{C}} of simple, undirected graphs and n, m\ge 1 , we define the Ramsey number { {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to {\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in {\mathcal{C}} consisting of r vertices contains either a complete n-clique {K}_{n} or an independent set consisting of m vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.