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See detailSatisfaction, Restriction and Amalgamation of Constraints in the Framework of M-Adhesive Categories
Schölzel, Hanna; Ehrig, Hartmut; Maximova, Maria et al

in Electronic Proceedings in Theoretical Computer Science [=EPTCS] (2012), 93

Application conditions for rules and constraints for graphs are well-known in the theory of graph transformation and have been extended already to M-adhesive transformation systems. According to the ... [more ▼]

Application conditions for rules and constraints for graphs are well-known in the theory of graph transformation and have been extended already to M-adhesive transformation systems. According to the literature we distinguish between two kinds of satisfaction for constraints, called general and initial satisfaction of constraints, where initial satisfaction is defined for constraints over an initial object of the base category. Unfortunately, the standard definition of general satisfaction is not compatible with negation in contrast to initial satisfaction. Based on the well-known restriction of objects along type morphisms, we study in this paper restriction and amalgamation of application conditions and constraints together with their solutions. In our main result, we show compatibility of initial satisfaction for positive constraints with restriction and amalgamation, while general satisfaction fails in general. Our main result is based on the compatibility of composition via pushouts with restriction, which is ensured by the horizontal van Kampen property in addition to the vertical one that is generally satisfied in M-adhesive categories [less ▲]

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See detailLow- and High-Level Petri Nets with Individual Tokens
Modica, Tony; Gabriel, Karsten; Ehrig, Hartmut et al

Report (2010)

In this article, we present a new variant of Petri nets with markings called Petri nets with individual tokens, together with rule-based transformation following the double pushout approach. The most ... [more ▼]

In this article, we present a new variant of Petri nets with markings called Petri nets with individual tokens, together with rule-based transformation following the double pushout approach. The most important change to former Petri net transformation approaches is that the marking of a net is no longer a collective set of tokens, but each each has an own identity leading to the concept of Petri nets with individual tokens. This allows us to formulate rules that can change the marking of a net arbitrarily without necessarily manipulating the structure. As a first main result that depends on nets with individual markings we show the equivalence of transition firing steps and the application of firing-simulating rules. We define categories of low-level and of algebraic high-level nets with individual tokens, called PTI nets and AHLI nets, respectively and relate them with each other and their collective counterparts by functors. To be able to use the properties and analysis results of \MCALM-adhesive HLR systems (formerly know as weak adhesive high-level replacement systems) we show in further main results that both categories of PTI nets and AHLI nets are \MCALM-adhesive categories. By showing how to construct initial pushouts we also give necessary and sufficient conditions for the applicability of transformation rules in these categories, known as gluing condition in the literature. [less ▲]

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