bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

Process algebra is the study of distributed or parallel syst em by algebraic means. Originating in computer science, process algebra has been extended in re ce t years to encompass not just discrete-event systems, but also continuously evolving ph enomena, resulting in so-called hybrid process algebras. A hybrid process algebra can be used for th e specification, simulation, control and verification of embedded systems in combination with the ir environment, and for any dynamic system in general. As the vehicle of our exposition, we use th e hybrid process algebra χ (Chi). The syntax and semantics of χ are discussed, and it is explained how equational reasoning simplifies tool implementations for simulation and verification. A bot tle filling line example is introduced to illustrate system analysis by means of equational reasonin g.

Process algebra

Text algorithms

Publisher Summary This chapter discusses the algorithms for solving string-matching problems that have proven useful for text-editing and text-processing applications. String pattern matching is an important problem that occurs in many areas of science and information processing. In computing, it occurs naturally as part of data processing, text editing, term rewriting, lexical analysis, and information retrieval. Many text editors and programming languages have facilities for matching strings. In biology, string-matching problems arise in the analysis of nucleic acids and protein sequences, and in the investigation of molecular phylogeny. String matching is also one of the central and most widely studied problems in theoretical computer science. The simplest form of the problem is to locate an occurrence of a keyword as a substring in a sequence of characters, which is called the input string. For example, the input string queueing contains the keyword ueuei as a substring. Even for this problem, several innovative, theoretically interesting algorithms have been devised that run significantly faster than the obvious brute-force method.

Algorithms for finding patterns in strings

Petri nets are widely used to model concurrent systems. However, their composition and abstraction mechanisms are inadequate: we solve this problem in a satisfactory way. We start by remarking that place/transition Petri nets can be viewed as ordinary, directed graphs equipped with two algebraic operations corresponding to parallell and sequential composition of transitions. A distributive law between the two operations captures a basic fact about concurrency. New morphisms are defined, mapping single, atomic transitions into whole computations, thus relating system descriptions at different levels of abstraction. Categories equipped with products and coproducts (corresponding to parallel and nondeterministic compositions) are introduced for Petri nets with and without initial markings. Petri net duality is expressed as a duality functor, and several new invariants are introduced. A tensor product is defined on nets, and their category is proved to be symmetric monoidal closed. This construction is generalized to a large class of algebraic theories on graphs. These results provide a formal basis for expressing the semantics of concurrent languages in terms of Petri nets. They also provide a new understanding of concurrency in terms of algebraic structures over graphs and categories that should apply to other models besides Petri nets and thus contribute to the conceptual unification of concurrency.

Petri nets are moniods

An O ( n log n ) algorithm for finding all repetitions in a string

Reaction systems are a formal framework for investigating processes carried out by biochemical reactions. This paper is an introduction to reaction systems. It provides basic notions together with the underlying intuition and motivation as well as two examples (a binary counter and transition systems) of "programming" with reaction systems. It also provides a tour of some research themes.

A tour of reaction systems

A square is an immediately repeated nonempty string, eg, aa, abab, abcabc This paper presents a new O(n log n) algorithm to determine whether a string of length n has a substring which is a square The algorithm is not as general as some previous algorithms for finding all squares [1,7,8,13], but it does have a simplicity which the others lack Also, for a fixed alphabet of size k, the algorithm can be improved by a factor of log k (n), yielding an O(n) algorithm for determining whether a string contains a square

Linear Time Recognition of Squarefree Strings

Reaction systems are a formal model of interactions between biochemical reactions. They consist of sets of reactions, where each reaction is classified by its set of reactants (needed for the reaction to take place), its set of inhibitors (each of which prevents the reaction from taking place), and its set of products (produced when the reaction takes place) – the set of reactants and inhibitors form the resources of the reaction. Each reaction system defines a (transition) function on its set of states. (States here are subsets of an a priori given set of biochemical entities.) In this paper we investigate properties of functions defined by reaction systems. In particular, we investigate how the power of defining functions depends on available resources, and we demonstrate that with small resources one can define functions exhibiting complex behavior.

Functions defined by reaction systems

Reaction systems are a functional model of interactions between biochemical reactions. They define functions on finite sets (over a common finite domain). In this paper, we investigate combinatorial properties of functions defined by reaction systems. In particular, we provide analytical approximations of combinatorial properties of random reaction systems, with a focus on the probability of whether a system lives or dies. Based on these results, we can create parameterized random reaction systems that rarely die. We also empirically analyze the length of time before such a system enters cyclic behavior, and find that the time is related to the behavior of completely random functions on a smaller domain.

Michael G. Main

Papers

An O ( n log n ) algorithm for finding all repetitions in a string

A tour of reaction systems

Linear Time Recognition of Squarefree Strings

Functions defined by reaction systems

Combinatorics of life and death for reaction systems