: The structure of High-level nets is studied from an algebraic and a logical point of view using Algebraic nets as an example. First the category of Algebraic nets is defined and the semantics given through an unfolding construction. Other kinds of Highlevel net formalisms are then presented. It is shown that nets given in these formalisms can be transformed into equivalent Algebraic nets. Then the semantics of nets in terms of universal constructions is discussed. A definition of Algebraic nets in terms of structured transition systems is proposed. The semantics of the Algebraic net is then given as a free completion of this structured transition system to a category. As an alternative also a sheaf semantics of nets is examined. Here the semantics of the net arises as a limit of a diagram of sheaves. Next Algebraic nets are characterized as encodings of special morphisms called foldings. Each algebraic net gives rise to a surjective morphism between Petri nets and conversely each sur...

Well know categories of Petri nets lack coproducts and some restrictions on nets, morphisms or initial markings are required in order to guarantee the existence of colimits. Categories of Petri nets equipped with a set of initial markings (instead of a single initial marking) are introduced. It is shown that the proposed categories of nets are complete and cocomplete. Moreover, interpretations of limits and colimits are adequate for expressing semantics of concurrent systems. Examples of structuring and modeling of behavior of nets using categorial constructions based on limits and colimits are provided.

A categorial framework for structured graph based systems with or without distinguished nodes or labeling on both arcs and nodes is proposed. Requirements for the existence of limits and colimits in the resulting categories are set. In this context, unrestricted and bicomplete categories of graph based systems such as Petri Nets, Labeled Transition Systems, Nonsequential Automata, etc., are easily defined. Then it is shown how limits and colimits can be interpreted as structuring and anticipatory properties of systems. The proposed framework called duo-internalization generalizes the notion of internal graphs allowing that nodes and arc may be objects from different categories. The results about limits and colimits of (reflexive) duo-internal (labeled) graphs (with distinguished nodes) are, for our knowledge, new.

Graph transformation systems have been introduced for the formal specication of software systems. States are thereby modeled as graphs, and computations as graph derivations according to the rules of the specication. Operations on graph derivations provide means to reason about the distribution and composition of computations. In this paper we discuss the development of an algebra of graph derivations as a descriptive model of graph transformation systems. For that purpose we use a categorical three level approach for the construction of models of computations based on structured transition systems. Categorically the algebra of graph derivations can then be characterized as a free double category with nite horizontal colimits.

: The semantics of Petri Nets are discussed within the "Objects are sheaves" paradigm. Transitions and places are represented as sheaves and nets are represented as diagrams of sheaves. Both an interleaving semantics, and a non-interleaving semantics are shown to arise as the limit of the sheaf diagram representing the net. This work was supported by the Information Technology Promotion Agency, Japan, as part of the R & D of Basic Technology for Future Industries "New Models of Software Architecture" project sponsored by NEDO (New Energy and Industrial Technology Developments Organization). Printing: TKK Monistamo; Otaniemi 1993 Helsinki University of Technology Phone: 90 +358--0 4511 Department of Computer Science Digital Systems Laboratory Telex: 125 161 htkk sf Otaniemi, Otakaari 1 Telefax: +358--0--465 077 SF--02150 ESPOO, FINLAND E-mail: lab@hutds.hut.fi -- 1 -- 1 Introduction Sheaf theory is a mathematical tool that has been successfully applied to the solution of difficult m...

A set of courseware tools based on Automata Theory and Category Theory is defined in order to create a semi-automated system for developing Web Courses. Courses are automata with output and links between pages are automata transitions (not HTML source) and thus, reusing the instructional materials is straightforward. In this context, the categorial product allows creation and synchronization of browsing paths, the categorial coproduct results in a special kind of disjoint union of courses, the functorial operation of restriction (fibration technique) allows the author to select specific parts of the course and thus, automatically removing corresponding links between content pages, and the functorial operation of relabeling (cofibration technique) allows relabeling and encapsulation of transitions. Also, limits and colimits (in general) have meaningful interpretations.

A categorical semantic domain is constructed for Petri nets which satisfies the diagonal compositionality requirement, i.e., Petri nets are equipped with a hierarchical specification mechanism (vertical compositionality) that distributes through net combinators (horizontal compositionality). The abstraction mechanism is based on graph transformations (single pushout approach). A finitely bicomplete category of partial Petri nets and partial morphisms is introduced.

Inspired by Meseguer and Montanari's "Petri Nets are Monoids", we propose that a reification of a Petri net is a special kind of net morphism were the target object is enriched with all conceivable sequential and concurrent computations. Then it is proven that while reification of nets satisfies the vertical compositionality requirement (i.e., reifications compose), it lacks the horizontal compositionality requirement (i.e., reification does not distribute over parallel composition). To achieve both requirements, a new categorial semantic domain based on labeled transition systems with full concurrency, called nonsequential automata, is constructed. Again, a class of morphisms stands for reification and, in this framework, the diagonal compositionality requirement (i.e., both vertical and horizontal) is achieved. Adjunctions between both models are provided extending the approach of Winskel and Nielsen. The steps of abstraction involved in moving between models show that nonsequential automata are more concrete than Petri nets. Moreover, categories of Petri Nets are isomorphic to subcategories of nonsequential automata.