Single Pushout Research Articles

Partial pullback complement rewriting along admissible matches

In this article, we first give two equivalent criteria for the existence of pullback complements along a universal morphism in a category in terms of the existence of pullbacks and co-universality of certain morphisms. Then we introduce the notion of admissible morphism and we characterize the existence of pullback complements in a partial morphism category along an admissible morphism in terms of the existence of pullback complements in its base category; when the base category is a certain admissible category. At the end we make a comparison between partial pullback complement rewriting and sesqui-pushout, double-pushout and single pushout rewritings, for an admissible match.

Single pushout rewriting in comprehensive systems of graph-like structures

The elegance of the single-pushout (SPO) approach to graph transformations arises from substituting total morphisms by partial ones in the underlying category. SPO's applicability depends on the durability of pushouts after this transition. There is a wide range of work on the question when pushouts exist in categories with partial morphisms starting with the pioneering work of Löwe and Kennaway and ending with an essential characterisation in terms of an exactness property (for the interplay between pullbacks and pushouts) and an adjointness condition (w.r.t. inverse image functions) by Hayman and Heindel.Triple graphs and graph diagrams are frameworks to synchronise two or more updatable data sources by means of internal mappings, which identify common sub-structures. Comprehensive systems generalise these frameworks, treating the network of data sources and their structural inter-relations as a homogeneous comprehensive artefact, in which partial maps identify commonalities. Although this inherent partiality produces amplified complexity, we can show that Heindel's characterisation still yields existence of pushouts in the category of comprehensive systems and reflective partial morphisms and thus enables computing by typed SPO graph transformation.

Attributed Hierarchical Port Graphs and Applications

We present attributed hierarchical port graphs (AHP) as an extension of port graphs that aims at facilitating the design of modular port graph models for complex systems. AHP consist of a number of interconnected layers, where each layer defines a port graph whose nodes may link to layers further down the hierarchy; attributes are used to store user-defined data as well as visualisation and run-time system parameters. We also generalise the notion of strategic port graph rewriting (a particular kind of graph transformation system, where port graph rewriting rules are controlled by user-defined strategies) to deal with AHP following the Single Push-out approach. We outline examples of application in two areas: functional programming and financial modelling.

An Algebraic Approach for Inferring and Using Symmetries in Rule-based Models

Symmetries arise naturally in rule-based models, and under various forms. Besides automorphisms between site graphs, which are usually built within the semantics, symmetries can take the form of pairs of sites having the same capabilities of interactions, of some protein variants behaving exactly the same way, or of some linear, planar, or 3D molecular complexes which could be seen modulo permutations of their axis and/or mirror-image symmetries.In this paper, we propose a unifying handling of symmetries in Kappa. We follow an algebraic approach, that is based on the single pushout semantics of Kappa. We model classes of symmetries as finite groups of transformations between site graphs, which are compatible with the notion of embedding (that is to say that it is always possible to restrict a symmetry that is applied with the image of an embedding to the domain of this embedding) and we provide some assumptions that ensure that symmetries are compatible with pushouts. Then, we characterise when a set of rules is symmetric with respect to a group of symmetries and, in such a case, we give sufficient conditions so that this group of symmetries induces a forward bisimulation and/or a backward bisimulation over the population semantics.

Partial pullback complement rewriting

In this article, we first characterize the existence of certain pullback complements in a category, in terms of the existence of exponentials in a corresponding slice category and also in terms of the existence of pullbacks in the corresponding partial morphism category. We also prove that certain pullback complements exist in a weak adhesive VK category if and only if they exist in the corresponding partial morphism category.We then propose a new graph rewriting technique, called the partial pullback complement rewriting, that uses pullback complements in a partial morphism category. We finally compare this with the double-pullback, single pushout, double-pushout and sesqui-pushout techniques, proving that in certain categories and under certain conditions, the partial pullback complement rewriting, implies and/or is implied by the above mentioned rewritings. Some illustrative examples are also given.

Processes and unfoldings: concurrent computations in adhesive categories

We generalise both the notion of a non-sequential process and the unfolding construction (which was 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, that is, 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 ω-chains of monos. Compared with 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.

Applications of typed lambda-terms to categorical attributed graph transformations

This paper deals with modal transformation based on attributed graph rewriting. Our contribution investigates f single pushout approach for applying the rewrite rules. The computation of graph attributes is obtained through the use of lambda-terms in a typed lambda calculus with inductive types. In this paper we present solutions to cope with single pushout construction for the graph structure and the computations functions. As the rewrite systen use inductiv types, the expressive power of attribute computations increases and appears to be more efficient than the one based on sigma-algebras. Some examples showing the interest of our computation approach are described in this paper.

Typed lambda-terms in categorical attributed graph transformation

This paper deals with model transformation based on attributed graph rewriting. Our contribution investigates a single pushout approach for applying the rewrite rules. The computation of graph attributes is obtained through the use of typed lambda-calculus with inductive types. In this paper we present solutions to cope with single pushout construction for the graph structure and the computations functions. As this rewrite system uses inductive types, the expressiveness of attribute computations is facilitated and appears more efficient than the one based on Sigma-algebras. Some examples showing the interest of our computation approach are described in this paper.

Diagonal Compositionality of Partial Petri Nets

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). Therefore, a net may be hierarchically specified before or after the joint behavior of component parts (nets) in order to obtain the same resulting system. The abstraction mechanism proposed is based on graph transformations using the so-called single pushout approach. A finitely complete and cocomplete category of partial marked Petri nets and partial morphisms is introduced and it is claimed that, with respect to partial morphisms, “Petri nets are semi-groups” . Classes of transformations stand for hierarchical specification mechanism, named reification, where part of a net (usually a transition) is replaced by another (possible complex) net. The composition of reifications (i.e., composition of pushouts) is defined, leading to a category of nets and reifications which is also complete and cocomplete. Since the reification operation composes, the vertical compositionality requirement of Petri nets is achieved. Then, it is proven that the reification also satisfies the horizontal compositionality requirement, i.e., the reification of nets distributes through the parallel composition. A specification grammar and the induced subcategory of nets and reifications stand for a system specification and all possible levels of system abstractions and their relationship, respectively. Also, a generalization of the top-down approach for concurrent systems is introduced.

DEFINING OPERATIONAL BEHAVIOR OF OBJECT SPECIFICATIONS BY ATTRIBUTED GRAPH TRANSFORMATIONS

A single pushout approach to the transformation of attributed partial graphs based on categories of partial algebras and partial morphisms is introduced. A sufficient condition for pushouts in these categories is presented. As the synchronization mechanism we use amalgamation of rules and show how synchronization can be minimized. We point out how the results obtained can be employed in order to define an operational semantics for object specification languages.

Relational graph rewritings

This note presents a new formalization of graph rewritings which generalizes traditional graph rewritings. Relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory of binary relations). Single pushout approaches to graph rewritings proposed by Raoult and Kennaway are compared with our rewritings of relational (labeled) graph. Moreover, a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given.

Concurrent Derivations as Single Pushout Graph Grammar Processes

Algebraic graph transformations visually support intuition, have a strong theoretical basis, and provide a formal, implementation independent basis for the description of discretely evolving computational systems and their formal and tractable analysis. Graph grammar models of concurrent systems (petri nets, actor systems) have inspired corresponding semantics developments. Recently this led to the introduction of partial orders of concurrent derivations (concurrent computations). A concurrent derivation (CDer) abstracts from the (sequential) order of rule applications in the sequential derivation and thus can be considered as a concurrent process. Complementary, a morphism between two concurrent derivations expresses that the first is a computational approximation of the second. In this paper we newly introduce non-deterministic concurrent derivations (CTrees) as classes of concurrently equivalent sequential derivation trees. Due to the fact that also infinite computations are represented by CTrees, the category of all CTrees of a given graph grammar has a final object (the concurrent counterpart of the whole sequential tree of the given grammar) which is approximated by all other CTrees. We show that (syntactical) morphisms between two graph grammars induce corresponding adjunction between the corresponding (semantic) categories of CDers and CTrees respectively.

Effect of Coating Thickness on Interfacial Shear Behavior of Zirconia-Coated Sapphire Fibers in a Polycrystalline Alumina Matrix

ABSTRACTThe effect of zirconia (ZrO2) interfacial coatings on the interfacial shear behavior in sapphire reinforced alumina was examined in this study. Zirconia coatings of thicknesses ranging from 0.15 to 1.45 μm were applied to single crystal sapphire (Saphikon) fibers using a particulate loaded sol dipping technique. After calcining at 1100°C in air, the coated fibers were incorporated into a polycrystalline alumina matrix via hot pressing. Interfacial shear strength and sliding behavior of the coated fibers was examined using thin-slice indentation fiber pushout and pushback techniques. In all cases, debonding and sliding occurred at the interface between the fibers and the coating. The coatings exhibited a dense microstructure and led to a higher interfacial shear strength (> 240 MPa) and interfacial sliding stress (>75 MPa) relative to previous studies on the effect of a porous interphase on interfacial properties [1]. The interfacial shear strength decreased with increasing fiber coating thickness (from 389 ± 59 to 241 ± 43 MPa for 0.15 to 1.45 μm thick coatings, respectively). Sliding behavior exhibited load modulation with increasing displacement during fiber sliding which is characteristic of fiber roughnessinduced “stick-slip”. No effect of fiber coating thickness on the interfacial sliding stress was observed for single pushout or pushback events. Heat treatment at 1550°C in air coarsened the fiber surface roughness, resulting in significantly higher interfacial shear strengths (>30%) and interfacial sliding stresses (>60%) relative to the coated fibers in an as-hot-pressed condition. Interfacial sliding resistance decreased significantly after the first sliding cycle. Evidence of substantial “stick-slip” behavior was eliminated from the load displacement plots after one pushout/pushback cycle; however, the pushback plots exhibited evidence of fiber reseating, followed by a decreasing trend in load with increasing displacement of the fiber back to the original position. These results directly support the interphase fragmentation scenario proposed for interface fatigue in ceramic composites.The high interfacial shear strengths and sliding stresses measured in this study, as well as the potentially strength degrading surface reconstruction observed on the coated fibers after hot pressing and heat treatment, indicate that dense zirconia coatings are not suitable candidates for optimizing composite toughness and strength in the sapphire fiber reinforced alumina system.

On graph rewritings

The purpose of the present paper is twofold: Firstly, show that it is possible to rewrite graphs in a way equivalent to, and in fact slightly more powerful than that of Ehrig, Pfender and Schneider (1973), which has, since then, been developed mainly by the Berlin school. Our method consists in using a single push-out of partial morphisms and is described in Section 3. Section 1 is devoted to the elementary definitions concerning graphs and related terms. Section 2 contains the set-theoretic prerequisites for the sequel but the proofs have been moved into an appendix, for easier reading. Secondly, we indicate in Section 4 why this method is not really fit for rewriting graphs that represent collapsed terms (i.e., sharing common subterms) and we introduce pushouts of total functions, which are not morphisms everywhere on their domain. This method is connected to the classical rewriting of the corresponding terms. The adequacy of these new rewriting rules is then tested to prove a local confluence criterion a la Knuth-Bendix (1970) in Section 5, the proof of which turns out to be very short.