Multiset

Abstract: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes.

Abstract: We present a simple, exact algorithm for identifying in a multiset the items with frequency more than a threshold θ. The algorithm requires two passes, linear time, and space 1/θ. The first pass is an on-line algorithm, generalizing a well-known algorithm for finding a majority element, for identifying a set of at most 1/θ items that includes, possibly among others, all items with frequency greater than θ.

Abstract: Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic,inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a natural class of models of concurrent processes. The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets orpomsets. Just as sets of strings form languages, so do sets of pomsets form processes. We introduce a number of operations useful for specifying concurrent processes and demonstrate their utility on some basic examples. Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components. The general benefits of the approach are that it is conceptually straightforward, involves fewer artificial constructs than many competing models of concurrency, yet is applicable to a considerably wider range of types of systems, including systems with buses and ethernets, analog systems, and real-time systems.

Abstract: A common tool for proving the termination of programs is the well-founded set, a set ordered in such a way as to admit no infinite descending sequences. The basic approach is to find a termination function that maps the values of the program variables into some well-founded set, such that the value of the termination function is repeatedly reduced throughout the computation. All too often, the termination functions required are difficult to find and are of a complexity out of proportion to the program under consideration.Multisets (bags) over a given well-founded set S are sets that admit multiple occurrences of elements taken from S. The given ordering on S induces an ordering on the finite multisets over S. This multiset ordering is shown to be well-founded. The multiset ordering enables the use of relatively simple and intuitive termination functions in otherwise difficult termination proofs. In particular, the multiset ordering is used to prove the termination of production systems, programs defined in terms of sets of rewriting rules.