In this paper we consider the relative expressive power of two very common operators applicable to sets and multisets: the with and the union operators. For such operators we prove that they are not mutually expressible by means of existentially quantified formulae. In order to prove our results, canonical forms for set-theoretic and multiset-theoretic formulae are established and a particularl...

This paper is concerned with the synthesis of invariants in programs with arrays. More specifically, we consider properties concerning array contents up to a permutation. For instance, to prove a sorting procedure, one has to show that the result is sorted, but also that it is a permutation of the initial array. In order to analyze this kind of properties, we define an abstract interpretation w...

In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially di erent universes of such sets. The unisets of the rst universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. We will have a closer look into the connection of the iterated circular mult...

We present in this paper a neural-like membrane system solving the SAT problem in linear time. These neural P systems are nets of cells working with multisets. Each cell has a finite state memory, processes multisets of symbol-impulses, and can send impulses ("excitations" ) to the neighboring cells. The maximal mode of rules application and the replicative mode of communication between cells a...

Rough set theory uses the concept of upper and lower approximations to encapsulate inherent inconsistency in real-world objects. Information multisystems are represented using multisets instead of crisp sets. This paper begins with an overview of recent works on multisets and rough sets. Rough multiset is introduced in terms of lower and upper approximations and explores related properties. The...

Consider the collection of all t–multisets of {1, . . . , n}. A universal cycle on multisets is a string of numbers, each of which is between 1 and n, such that if these numbers are considered in t–sized windows, every multiset in the collection is present in the string precisely once. The problem of finding necessary and sufficient conditions on n and t for the existence of universal cycles an...

There are two possible definitions of the “s-disjoint r-uniform Kneser hypergraph” of a set system T : The hyperedges are either r-sets or r-multisets. We point out that Ziegler’s (combinatorial) lower bound on the chromatic number of an s-disjoint r-uniform Kneser hypergraph only holds if we consider r-multisets as hyperedges. We give a new proof of his result and show by example that a simila...

The tracking of location information in mobile ad hoc sensor networks is a well-studied problem for which many solutions have been proposed. The approach discussed in this paper is based on the systematic use of a distributed shared data structure. Specifically, randomized versions of replicated sets and multisets are introduced. It is shown that this probabilistic approach can improve the perf...

Factorization results in multisets of primes and this mapping can be turned into a bijection between multisets of natural numbers and natural numbers. At the same time, simpler and more efficient bijections exist that share some interesting properties with the bijection derived from factorization. This paper describes mechanisms to emulate properties of prime numbers through isomorphisms connec...

The ever growing size of chemical databases calls for the development of novel methods for representing and comparing molecules. One such method called LINGO is based on fragmenting the SMILES string representation of molecules. Comparison of molecules can then be performed by calculating the Tanimoto coefficient, which is called LINGOsim when used on LINGO multisets. This paper introduces a ve...