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

On the definition of parallel independence in the algebraic approaches to graph transformation

Parallel independence between transformation steps is a basic and well-understood notion of the algebraic approaches to graph transformation, and typically guarantees that the two steps can be applied in any order obtaining the same resulting graph, up to isomorphism. The concept has been redefined for several algebraic approaches as variations of a classical “algebraic” condition, requiring that each matching morphism factorizes through the context graphs of the other transformation step. However, looking at some classical papers on the double-pushout approach, one finds that the original definition of parallel independence was formulated in set-theoretical terms, requiring that the intersection of the images of the two left-hand sides in the host graph is contained in the intersection of the two interface graphs. The relationship between this definition and the standard algebraic one is discussed in this position paper, both in the case of left-linear and non-left-linear rules

A Category Theoretical Approach to the Concurrent Semantics of Rewriting: Adhesive Categories and Related Concepts

This thesis studies formal semantics for a family of rewriting formalisms that have arisen as category theoretical abstractions of the so-called algebraic approaches to graph rewriting. The latter in turn generalize and combine features of term rewriting and Petri nets. Two salient features of (the abstract versions of) graph rewriting are a suitable class of categories which captures the structure of the objects of rewriting, and a notion of independence or concurrency of rewriting steps – as in the theory of Petri nets. Category theoretical abstractions of graph rewriting such as double pushout rewriting encapsulate the complex details of the structures that are to be rewritten by considering them as objects of a suitable abstract category, for example an adhesive one. The main difficulty of the development of appropriate categorical frameworks is the identification of the essential properties of the category of graphs which allow to develop the theory of graph rewriting in an abstract framework. The motivations for such an endeavor are twofold: to arrive at a succint description of the fundamental principles of rewriting systems in general, and to apply well-established verification and analysis techniques of the theory of Petri nets (and also term rewriting systems) to a wide range of distributed and concurrent systems in which states have a "graph-like" structure. The contributions of this thesis thus can be considered as two sides of the same coin: on the one side, concepts and results for Petri nets (and graph grammars) are generalized to an abstract category theoretical setting; on the other side, suitable classes of "graph-like" categories which capture the essential properties of the category of graphs are identified. Two central results are the following: first, (concatenable) processes are faithful partial order representations of equivalence classes of system runs which only differ w.r.t. the rescheduling of causally independent events; second, the unfolding of a system is established as the canonical partial order representation of all possible events (following the work of Winskel). Weakly ω-adhesive categories are introduced as the theoretical foundation for the corresponding formal theorems about processes and unfoldings. The main result states that an unfolding procedure for systems which are given as single pushout grammars in weakly ω-adhesive categories exists and can be characetrised as a right adjoint functor from a category of grammars to the subcategory of occurrence grammars. This result specializes to and improves upon existing results concerning the coreflective semantics of the unfolding of graph grammars and Petri nets (under an individual token interpretation). Moreover, the unfolding procedure is in principle usable as the starting point for static analysis techniques such as McMillan’s finite complete prefix method. Finally, the adequacy of weakly ω-adhesive categories as a categorical framework is argued for by providing a comparison with the notion of topos, which is a standard abstraction of the categories of sets (and graphs)

Rule Algebras for Adhesive Categories

We demonstrate that the most well-known approach to rewriting graphical structures, the Double-Pushout (DPO) approach, possesses a notion of sequential compositions of rules along an overlap that is associative in a natural sense. Notably, our results hold in the general setting of $\mathcal{M}$-adhesive categories. This observation complements the classical Concurrency Theorem of DPO rewriting. We then proceed to define rule algebras in both settings, where the most general categories permissible are the finitary (or finitary restrictions of) $\mathcal{M}$-adhesive categories with $\mathcal{M}$-effective unions. If in addition a given such category possess an $\mathcal{M}$-initial object, the resulting rule algebra is unital (in addition to being associative). We demonstrate that in this setting a canonical representation of the rule algebras is obtainable, which opens the possibility of applying the concept to define and compute the evolution of statistical moments of observables in stochastic DPO rewriting systems

Reactive Systems over Cospans

The theory of reactive systems, introduced by Leifer and Milner and previously extended by the authors, allows the derivation of well-behaved labelled transition systems (LTS) for semantic models with an underlying reduction semantics. The derivation procedure requires the presence of certain colimits (or, more usually and generally, bicolimits) which need to be constructed separately within each model. In this paper, we offer a general construction of such bicolimits in a class of bicategories of cospans. The construction sheds light on as well as extends Ehrig and Konig’s rewriting via borrowed contexts and opens the way to a unified treatment of several applications

Justified sequences in string diagrams: A comparison between two approaches to concurrent game semantics

We compare two approaches to concurrent game semantics, one by Tsukada and Ong for a simply-typed λ-calculus and the other by the authors and collaborators for CCS and the π-calculus. The two approaches are obviously related, as they both define strategies as sheaves for the Grothendieck topology induced by embedding ‘views’ into ‘plays’. However, despite this superficial similarity, the notions of views and plays differ significantly: the former is based on standard justified sequences, the latter uses string diagrams. In this paper, we relate both approaches at the level of plays. Specifically, we design a notion of play (resp. view) for the simply-typed λ-calculus, based on string diagrams as in our previous work, into which we fully embed Tsukada and Ong's plays (resp. views). We further provide a categorical explanation of why both notions yield essentially the same model, thus demonstrating that the difference is a matter of presentation. In passing, we introduce an abstract framework for producing sheaf models based on string diagrams, which unifies our present and previous models

Orchestrating cancer cell migration: Quantatitive analysis of protrusion, adhesion and contraction dynamics regulated by epidermal growth factor and collagen

Cell migration plays an important role in cancer metastasis. Traditional diagnostic methods often involve obtaining tissue biopsies and examining the morphology of the cells and the molecular composition of the microenvironment in static microscopy images. A link between dynamic cellular processes and static microenvironmental inputs must be made. This connection is often made qualitatively with a lack of quantitative information. Therefore, the aims of this work are to investigate how subcelluar dynamics of cell migration such as protrusion and adhesion are quantitatively modulated under different environmental inputs such as epidermal growth factor (EGF) and collagen. There are two major subcellular processes of migration, protrusion and adhesion. Protrusion is a dynamic movement of the cell edge and adhesion is mediated through macromolecular complexes called focal adhesions (FA). EGF concentration is an input that regulates FA and protrusion dynamics, whereas cell speed is an output that integrates information determined by inputs such as EGF. Several FA signatures and protrusion waves are associated with fast migration, but not necessarily with EGF. This suggests that other factors like contractility or extracellular matrix (ECM) might alter protrusion and FA for fast migration. Because fast migrating cells are usually invasive and cause metastasis, I designed a high-throughput method to identify the fast cells for determining what differences in cell properties such as protein expression level lead to the cell-to-cell variability. As mentioned above, contractility and ECM adhesivity are other inputs that affect migration. Although their effects on migration may be similar, upstream responses may vary. For example, both increasing adhesivity and decreasing contractility decreased migration speed, but their impact on protrusion and adhesion were distinct. Adhesivity affects migration not only on uniform substrates, but also under contact guidance. Both increasing adhesivity and the number of lines a cell contacted resulted in decreased directionality with more protrusion waves, which suggest that adhesivity and line spacing drive the effciency of contact guidance through the presence of protrusion waves. In summary, quantification of protrusion and FA properties might provide signatures that relate short timescale dynamics to long timescale migrational properties, making them ideal measurements for cancer diagnosis

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