On Single-Pushout Rewriting of Partial Algebras

We introduce Single-Pushout Rewriting for arbitrary partial algebras. Thus, we give up the usual restriction to graph structures, which are algebraic categories with unary operators only. By this generalisation, we obtain an integrated and straightforward treatment of graphical structures (objects) and attributes (data). We lose co-completeness of the underlying category. Therefore, a rule is no longer applicable at any match. We characterise the new application condition and make constructive use of it in some practical examples

Satisfaction, Restriction and Amalgamation of Constraints in the Framework of M-Adhesive Categories

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.Comment: In Proceedings ACCAT 2012, arXiv:1208.430

M-adhesive transformation systems with nested application conditions. Part 1: parallelism, concurrency and amalgamation

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church–Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of M-adhesive categories, where M-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules

Formalization of Petri Nets with Individual Tokens as Basis for DPO Net Transformations

Reconfigurable place/transition systems are Petri nets with initial markings and a set of rules which allow the modification of the net structure during runtime. They have been successfully used in different areas like mobile ad-hoc networks. In most of these applications the modification of net markings during runtime is an important issue. This requires the analysis of the interaction between firing and rule-based modification. For place/transition systems this analysis has been started explicitly without using the general theory of M-adhesive transformation systems, because firing cannot be expressed by rule-based transformations for P/T systems in this framework. This problem is solved in this paper using the new approach of P/T nets with individual tokens. In our main results we show that on one hand this new approach allows to express firing by transformation via suitable transition rules. On the other hand transformations of P/T nets with individual tokens can be shown to be an instance ofM-adhesive transformation systems, such that several well-known results, like the local Church-Rosser theorem, can be applied. This avoids a separate conflict analysis of token firing and transformations. Moreover, we compare the behavior of P/T nets with individual tokens with that of classical P/T nets. Our new approach is also motivated and demonstrated by a network scenario modeling a distributed communication system

Finitary M-adhesive categories

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Finitary M-adhesive categories are M-adhesive categories with finite objects only, where M-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of M-subobjects. In this paper, we show that in finitary M-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for M-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary M-adhesive categories have a unique ε'-M factorisation and initial pushouts, and the existence of an M-initial object implies we also have finite coproducts and a unique ε' -M pair factorisation. Moreover, we can show that the finitary restriction of each M-adhesive category is a finitary M-adhesive category, and finitarity is preserved under functor and comma category constructions based on M-adhesive categories. This means that all the classical results are also valid for corresponding finitary M-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-M-adhesive categories

Abstract representation theory of Dynkin quivers of type A

We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields. These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result. Finally, using abstract representation theory of linearly oriented $A_n$-quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable $\infty$-categories

A Unifying Theory for Graph Transformation

The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO

Processes and unfoldings: concurrent computations in adhesive categories

We generalise both the notion of non-sequential process and the unfolding construction (previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, i.e., the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars. As the unfolding represents potentially infinite computations, we need to work in adhesive categories with "well-behaved" colimits of omega-chains of monos. Compared to previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism

Adolescent Health

The report responds to the request of numerous Members of Congress to review the physical, emotional, and behavioral health status of contemporary American adolescents, including adolescents in groups who might be more likely to be in special need of health-related interventions: adolescents living in poverty, adolescents from racial and ethnic minority groups, Native American adolescents, and adolescents in rural areas