this paper. Multiset-signatures and-algebras In an algebraic system net a token on a place is an element of a domain of some algebra. Then, the multitude of tokens on a particular place can be represented as a multiset. In order to talk about the multitude of tokens one a place within the algebraical framework, we include the multiset over the corresponding sort in the algebra. To this end, we introduce multiset-signatures as particular signatures. For a signature

High-level replacement systems are an axiomatic categorical framework based on double-pushouts in order to unify replacement systems like grammars for different kinds of graphs and relational structures or other types of structures like algebraic specifications. Parallel high-level replacement systems are introduced to formalize parallel rewriting of these high-level structures. On one hand this concept generalizes and extends parallel graph grammars presented so far in the algebraic approach by allowing other structures than graphs, on the other hand the kinds of replacement introduced for high-level replacement systems are extended by different types of parallel replacement which are compared to each other in different parallel replacement theorems. An abstract version of a window-based graph editor and movement of objects in configuration spaces are presented as examples of parallel high-level replacement systems. Contents 1 Introduction 2 2 Basic concepts of high-level replacement...

: This work discusses three aspects of net theory: compositionality of nets, analysis of nets and high-level nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In category theory compositionality is represented by colimits. We show how a high-level net can be mapped into a low-level net that represents its behaviour. This construction is functorial and preserves colimits, giving a compositional semantics for these high-level nets. Using this semantics we propose some proof rules for compositional reasoning with high-level nets. Linear logic is a recently discovered logic that has many interesting properties. From a net theoretic point of view its interest lies in the fact that it is able to express resources in an analogous manner to nets. Several interpretations of Petri nets in terms of linear logic exist. In this work we study a model theoretic inter...