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 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

www.csd.uwo.ca / ∼ watt Abstract. Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any particular problem the domain can be given explicitly, but when dealing with a family of problems given in terms of symbolic parameters, matters become more difficult. This article shows how hybrid sets, that is multisets allowing negative multiplicity, may be used to express symbolic domain decompositions in an efficient, elegant and uniform way, simplifying both computation and reasoning. We apply this theory to the arithmetic of piecewise functions and symbolic matrices and show how certain operations may be reduced from exponential to linear complexity. 1

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)

Given a universe of discourse U, a multiset can be thought of as a function M from U to the natural numbers N. In this paper, we define a hybrid set to be any function from the universe U to the integers Z. These sets are called hybrid since they contain elements with either a positive or negative multiplicity. Our goal is to use these hybrid sets as if they were multisets in order to adequately generalize certain combinatorial facts which are true classically only for nonnegative integers. However, the definition above does not tell us much about these hybrid sets; we must define operations on them which provide us with the needed combinatorial structure. In section 1, we will define an analog of a set which can contain either a positive or negative number of elements. We will allow sums to be calculated over an arbitrary hybrid set and in particular over “improper ” intervals. This will lead us in section 2 to the calculation of the complementary symmetric functions comp n(H) whose argument is a hybrid set of variables which is at the same time a generalization of elementary and complete symmetric functions. In particular, as we will see in section 2.2, the complementary symmetric function generalizes the following two classic results concerning sequences of polynomials with persistant roots, that is to say,