Abstract The chemical reaction metaphor describes computation in terms of a chemical solution in which molecules interact freely according to reaction rules. Chemical models use the multiset as their basic data structure. Computation proceeds by rewritings of the multiset which consume elements according to reaction conditions and produce new elements according to specific transformation rules. Since the introduction of Gamma in the mid-eighties, many other chemical formalisms have been proposed such as the Cham, the P-systems and various higher-order extensions. The main objective of this paper is to identify a basic calculus containing the very essence of the chemical paradigm and from which extensions can be derived and compared to existing chemical models. 1

This paper describes a multirelation semantics for a fragment of the Extended Entity-Relationship schemata formalism based on the bicategorical definition of multirelations. The approach we follow is elementary---we try to introduce as few notions as possible. We claim that bicategorical algebra handles gracefully multirelations and their operations, and that multirelations are essential in a number of applications; moreover, the bicategorical composition of multirelations turns out to correspond to natural joins. From the formal semantics we derive an algorithm that can establish statically the possibility of building parallel ownership paths of weak entities. The ideas described in this paper have been implemented in a free tool, ERW, which lets users edit sets and multirelations instantiating an EER schema via a sophisticated web interface.

In this paper we define in an axiomatic way scalar and fuzzy cardinalities of finite crisp and fuzzy multisets, and we obtain explicit descriptions for them.

This paper is an attempt to summarize most things that are related to multiset theory. We begin by describing multisets and the operations between them. Then we present hybrid sets and their operations. We continue with a categorical approach to multisets. Next, we present fuzzy multisets and their operations. Finally, we present partially ordered multisets. 1

Gamma is a programming model where computation can be seen as chemical reactions between data represented as molecules floating in a chemical solution. This model can be formalized as associative, commutative, conditional rewritings of multisets where rewrite rules and multisets represent chemical reactions and solutions, respectively. In this article, we generalize the notion of multiset used by Gamma and present applications through various programming examples. First, multisets are generalized to include rewrite rules which become first-class citizen. This extension is formalized by the γ-calculus, a chemical model that summarizes in a few rules the essence of higher-order chemical programming. By extending the γ-calculus with constants, operators, types and expressive patterns, we build a higher-order chemical programming language called HOCL. Finally, multisets are further generalized by allowing elements to have infinite and negative multiplicities. Semantics, implementation and applications of this extension are considered.

sets and graph models of set and multiset theories

Publication interne n ˚ 1762 — Novembre 2005 — 26 pages Abstract: Gamma is a programming model where computation can be seen as chemical reactions between data represented as molecules floating in a chemical solution. This model can be formalized as associative, commutative, conditional rewritings of multisets where rewrite rules and multisets represent chemical reactions and solutions, respectively. In this article, we generalize the notion of multiset used by Gamma and present applications through various programming examples. First, multisets are generalized to include rewrite rules which become first-class citizen. This extension is formalized by the γ-calculus, a chemical model that summarizes in a few rules the essence of higherorder chemical programming. By extending the γ-calculus with constants, operators, types and expressive patterns, we build a higher-order chemical programming language called HOCL. Finally, multisets are further generalized by allowing elements to have infinite and negative multiplicities. Semantics, implementation and applications of this extension are considered. Key-words: multisets, chemical programming model, rewriting, higher-order, infinite and negative multiplicities (Résumé: tsvp)

Abstract. We present novel methods to compute changes to materialized views in logic databases like those used by rule-based reasoners. Such reasoners have to address the problem of changing axioms in the presence of materializations of derived atoms. Existing approaches have drawbacks: some require to generate and evaluate large transformed programs that are in Datalog ¬ while the source program is in Datalog and significantly smaller; some recompute the whole extension of a predicate even if only a small part of this extension is affected by the change. The methods presented in this article overcome these drawbacks and derive additional information useful also for explanation, at the price of an adaptation of the semi-naïve forward chaining. 1

In this paper an attempt is made to extend the concept of topological spaces in the context of multisets (mset, for short). The paper begins with basic definitions and operations on msets. The mset space [X] w is the collection of msets whose elements are from X such that no element in the mset occurs more than finite number (w) of times. Different types of collections of msets such as power msets, power whole msets and power full msets which are submsets of the mset space and operations under such collections are defined. The notion of M-topological space and the concept of open msets are introduced. More precisely, an M-topology is defined as a set of msets as points. Furthermore the notions of basis, sub basis, closed sets, closure and interior in topological spaces are extended to M-topological spaces and many related theorems have been proved. The paper concludes with the definition of continuous mset functions and related properties, in particular the comparison of discrete topology and discrete M-topology are established.