Double Pushout Research Articles

Algebraic properties and transformations of monographs

Monographs are graph-like structures with directed edges of unlimited length that are freely adjacent to each other. The standard nodes are represented as edges of length zero. They can be drawn in a way consistent with standard graphs and many others, like E-graphs or ∞-graphs. The category of monographs share many properties with the categories of graph structures (algebras of monadic many-sorted signatures, equivalent to presheaf toposes), except that there is no terminal monograph. It is universal in the sense that its slice categories (or categories of typed monographs) are equivalent to the categories of graph structures. Type monographs thus emerge as a natural way of specifying graph structures. A detailed analysis of single and double pushout transformations of monographs is provided, and a notion of attributed typed monographs generalizing typed attributed E-graphs is analyzed w.r.t. attribute-preserving transformations.

Fundamentals of compositional rewriting theory

A foundational theory of compositional categorical rewriting theory is presented, based on a collection of fibration-like properties that collectively induce and intrinsically structure the large collection of lemmata used in the proofs of theorems such as concurrency and associativity. The resulting highly generic proofs of these theorems are given. It is noteworthy that the proof of the concurrency theorem takes only a few lines and, while that of associativity remains somewhat longer, it would be unreadably long if written directly in terms of the basic lemmata. In essence, our framework improves the readability and ease of comprehension of these proofs by exposing latent modularity. A curated list of known instances of our framework is used to conclude the paper with a detailed discussion of the conditions under which the Double Pushout and Sesqui-Pushout semantics of graph transformation are compositional.

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

We study how to relate well-known hypergraph grammars based on the double pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL (called HL-grammars). It turns out that DPO rules can be naturally encoded by types of HL using methods similar to those used by Kanazawa for multiplicative-exponential linear logic. In order to generalize his reasonings we extend the hypergraph Lambek calculus by adding the exponential modality, which results in a new calculus HMEL0; then we prove that any DPO grammar can be converted into an equivalent HMEL0-grammar. We also define the conjunctive Kleene star, which behaves similarly to this exponential modality, and establish a similar result. If we add neither the exponential modality nor the conjunctive Kleene star to HL, then we can still use the same encoding and show that any DPO grammar with a linear restriction on the length of derivations can be converted into an equivalent HL-grammar.

Comparison of Atom Maps

The computation of reliable, chemically correct atom maps from educt/product pairs has turned out to be a difficult problem in cheminformatics because the chemically correct solution is not necessarily an optimal solution for combinatorial formulations such as maximum common subgraph problems. As a consequence, competing models have been devised and compared in extensive benchmarking studies. Due to isomorphisms among products and educts it is not immediately obvious, however, when two atom maps for a given educt/product pairs are the same. We formalize here the equivalence of atom maps and show that equivalence of atom maps is in turn equivalent to the isomorphism of labeled auxiliary graphs. In particular, we demonstrate that Fujita's Imaginary Transition State can be used for this purpose. Numerical experiments show that practical feasibility. Generalizations to the equivalence of subgraph matches, double pushout graph transformation rules, and mechanisms of multi-step reactions are discussed briefly.

String diagram rewrite theory II: Rewriting with symmetric monoidal structure

AbstractSymmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

Compositionality of Rewriting Rules with Conditions

We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon M-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.

A structural approach to graph transformation based on symmetric Petri nets

Graph Transformation Systems (GTS) and Petri Nets (PN) are two central, theoretically sound, formal models for concurrent or distributed systems. A lot of papers have focused on the relationship between GTS and PN. It is generally accepted that PN are instances of GTS due to the lack of ability to adapt or reconfigure their structure. In this paper, which extends a recent one, we reverse this perspective by presenting a formal definition of GTS in terms of Symmetric Nets (SN). SN are a standard High-Level PN formalism featuring a compact syntax which highlights behavioral symmetries of systems. The major strength lies in the possibility of using well-established tools to edit/efficiently analyze (SN-based) graph transformation models. We follow a structural approach, based on a newly implemented symbolic calculus, to validate Graph Transformation Rules (GTR) and formally verify GTS properties.In the second part of the paper, we supply a semantic characterization of the SN-based graph transformation model, by carrying out a thorough comparison with the classical, algebraic double-pushout (DPO) approach. We constructively and formally show how to translate any DPO rule to a corresponding, elementary or composite, SN rule. We treat both injective and non-injective DPO rules/rule matches. The comparison shows that the SN approach is, in some sense, a generalization of the DPO.

Combining Graph Transformations and Semigroups for Isotopic Labeling Design.

The double pushout approach for graph transformation naturally allows an abstraction level of biochemical systems in which individual atoms of molecules can be traced automatically within chemical reaction networks. Aiming at a mathematical rigorous approach for isotope labeling design, we convert chemical reaction networks (represented as directed hypergraphs) into transformation semigroups. Symmetries within molecules correspond to permutations, whereas (not necessarily invertible) chemical reactions define the transformations of the semigroup. An approach for the automatic inference of informative labeling of atoms is presented, which allows to distinguish the activity of different pathway alternatives within reaction networks. To illustrate our approaches, we apply them to the reaction network of glycolysis, which is an important and well-understood process that allows for different alternatives to convert glucose into pyruvate.

Semantics-Preserving DPO-Based Term Graph Rewriting

Term graph rewriting is important as "conceptual implementation" of the execution of functional programs, and of data-flow optimisations in compilers. One way to define term graph transformation rule application is via the well-established and intuitively accessible double-pushout (DPO) approach; we present a new result proving semantics preservation for such DPO-based term graph rewriting.

Rule-based transformation of graph rewriting rules: Towards higher-order graph grammars

Higher-order functions – the ones which manipulate other functions – have a fundamental role in Computer Science, specially in areas such as functional programming and theory of computation. Graph transformation – the rule-based modification of graphs – is also an important approach used in many contexts, as for instance, the definition of syntax and semantics of visual languages and the modelling and analysis of concurrent and non-deterministic systems. Although the algebraic double-pushout (DPO) approach to graph transformation has a vast theory, it does not present a notion of higher-order transformation, i.e., transformation of graph rewriting rules themselves. Higher-order would be required, for instance, to ease the modelling of dynamic systems through graph transformation systems, and to describe model transformations that modify graph rewriting rules. The main contribution of this work is a notion of double-pushout transformation for graph rewriting rules, where the result of the rewriting is a valid graph rule. We use DPO diagrams in the category of graph spans to describe rule transformations, and negative application conditions (NACs) in the second-order rules to enforce preservation of the rule structure. We define a model named second-order graph grammars, capable of first- and second-order derivations. Finally, we discuss alternative ways of defining second-order transformations.

A sound and complete theory of graph transformations for service programming with sessions and pipelines

Graph transformation techniques, the Double-Pushout (DPO) approach in particular, have been successfully applied in the modeling of concurrent systems. In this area, a research thread has addressed the definition of concurrent semantics for process calculi. In this paper, we propose a theory of graph transformations for service programming with sophisticated features such as sessions and pipelines. Through graph representation of CaSPiS, a recently proposed process calculus, we show how graph transformations can cope with advanced features of service-oriented computing, such as several logical notions of scoping together with the interplay between linking and containment. We first exploit a graph algebra and set up a graph model that supports graph transformations in the DPO approach. Then, we show how to represent CaSPiS processes as hierarchical graphs in the graph model and their behaviors as graph transformation rules. Finally, we provide the soundness and completeness results of these rules with respect to the reduction semantics of CaSPiS.

Open-graphs and monoidal theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits ‘enough adhesivity’ from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are ‘type safe’ in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discrete and finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parameterised by graphical signatures, which are similar to the monoidal signatures of Joyal and Street, and define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs and more general monoidal theories. Thus, open-graphs give us a tool for mechanised reasoning in monoidal categories.

Adhesivity with Partial Maps instead of Spans

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks that started over thirty years ago for the category of graphs. Adhesive categories - of which graphs are the “archetypal” example - are defined by a single property of pushouts along monos that implies essential lemmas and central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that a strictly weaker condition on pushouts suffices to obtain essentially the same results: it suffices to require pushouts to be hereditary, i.e. they have to remain pushouts when they are embedded into the associated category of partial maps. This fact however is not the only reason to introduce partial map adhesive categories as categories with pushouts along monos (of a certain stable class) that are hereditary. There are two equally important motivations: first, there is an application relevant example category that cannot be captured by the more established variations of adhesive categories; second, partial map adhesive categories are “conceptually similar” to adhesive categories as the latter can be characterized as those categories with pushout along monos that remain bi-pushouts when they are embedded into the associated bi-category of spans. Thus, adhesivity with partial maps instead of spans appears to be a natural candidate for a general rewriting framework.

Specification and Verification of Model Transformations

Model transformations are a key concept within model driven development and there is an enormous need for suitable formal analysis techniques for model transformations, in particular with respect to behavioural equivalence of source models and their corresponding target models. For this reason, we discuss the general challenges that arise for the specification and verification of model transformations and present suitable formal techniques that are based on graph transformation. In this context, triple graph grammars show many benefits for the specification process, e.g. modelers can work on an intuitive level of abstraction and there are formal results for syntactical correctness, completeness and efficient execution. In order to verify model transformations with respect to behavioural equivalence we apply well-studied techniques based on the double pushout approach with borrowed context, for which the model transformations specified by triple graph transformation rules are flattened to plain (in-situ) graph transformation rules. The potential and adequateness of the presented techniques are demonstrated by an intuitive example, for which we show the correctness of the model transformation with respect to bisimilarity of source and target models.

How to delete categorically — Two pushout complement constructions

In category theory, most set-theoretic constructions–union, intersection, etc.–have direct categorical counterparts. But up to now, there is no direct construction of a deletion operation like the set-theoretic complement. In rule-based transformation systems, deletion of parts of a given object is one of the main tasks. In the double pushout approach to algebraic graph transformation, the construction of pushout complements is used in order to locally delete structures from graphs. But in general categories, even if they have pushouts, pushout complements do not necessarily exist or are unique. In this paper, two different constructions for pushout complements are given and compared. Both constructions are based on certain universal constructions in the sense of category theory. More specifically, one uses initial pushouts while the other one uses quasi-coproduct complements. These constructions are applied to examples in the categories of graphs and simple graphs.

Structural Decomposition of Reactions of Graph-Like Objects

Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation.

Resource-Bound Quantification for Graph Transformation

Graph transformation has been used to model concurrent systems in software engineering, as well as in biochemistry and life sciences. The application of a transformation rule can be characterised algebraically as construction of a double-pushout (DPO) diagram in the category of graphs. We show how intuitionistic linear logic can be extended with resource-bound quantification, allowing for an implicit handling of the DPO conditions, and how resource logic can be used to reason about graph transformation systems.

Integrated Structure and Semantics for Reo Connectors and Petri Nets

In this paper, we present an integrated structural and behavioral model of Reo connectors and Petri nets, allowing a direct comparison of the two concurrency models. For this purpose, we introduce a notion of connectors which consist of a number of interconnected, user-defined primitives with fixed behavior. While the structure of connectors resembles hypergraphs, their semantics is given in terms of so-called port automata. We define both models in a categorical setting where composition operations can be elegantly defined and integrated. Specifically, we formalize structural gluings of connectors as pushouts, and joins of port automata as pullbacks. We then define a semantical functor from the connector to the port automata category which preserves this composition. We further show how to encode Reo connectors and Petri nets into this model and indicate applications to dynamic reconfigurations modeled using double pushout graph transformation.

Dynamic Reorganization Mechanisms in Multi-Agent Organizational Structure with Hierarchical Graph Transformation

如何动态适应环境是基于组织计算的多Agent系统的关键研究内容之一.组织结构的动态重组为多Agent系统柔性地实现组织目标提供了有效途径.结合Agent组织结构特点,给出了一种描述组织结构的社会结构、角色指定和Agent协调的单根节点层次图模型.通过单根节点和层次化地维护组织结构内元素的拓扑关系,有效地降低了大规模Agent组织重组问题的复杂性;扩展DPO(double-pushout)代数图变换,形式定义了Agent组织结构的重组过程.单根节点层次图描述了重组过程中给定时刻的组织结构状态,图变换规则序列定义了组织结构的变化过程.Agent组织重组和图匹配算法实验结果表明,该层次图变换方法有效地刻画了多Agent组织动态重组过程,并支持图形化重组过程要素设计和大规模Agent组织的重组计算.;The self-adaptation to the environmental changes is one of the key issues of organization-based multi-agent systems. Dynamic reorganization of organizational structures provides an effective approach for multi-agent systems to realize organizational objectives flexibly. Based on the structural characteristics of agent organizations, this paper presents a single-rooted hierarchical graph model describing social structure, role enactment and agent coordination of the organizational structures. This model decreases effectively the complexity of reorganization for large-scale agent organizations by maintaining their structural elements based on the single- rooted and hierarchical graph approach. It formalizes the reorganization process of agent organizational structures by extending the algebraic graph transformation with the DPO (double-pushout) approach. In this formal specification, the single-rooted hierarchical graphs characterize different states of organizational structures and the derivation sequences of transformation rules formulate the transition process of organizational structures. Finally, the experimental results on reorganization simulation and matching algorithm of organization transformation rules indicate that this hierarchical graph transformation approach defines formally the reorganization process of agent organizations, and supports the graph-based design of organizational elements during the reorganization process and the reorganization computation of large-scale agent organizations.

Towards Algebraic High-Level Systems as Weak Adhesive HLR Categories

Adhesive high-level replacement (HLR) systems have been recently established as a suitable categorical framework for double pushout transformations based on weak adhesive HLR categories. Among different types of graphs and graph-like structures, various kinds of Petri nets and algebraic high-level (AHL) nets are interesting instantiations of adhesive HLR systems. AHL nets combine algebraic specifications with Petri nets to allow the modeling of data, data flow and data changes within the net.For the development and analysis of reconfigurable systems, not only AHL schemas based on an algebraic specification and AHL nets using an additional algebra should be considered, but also AHL systems which additionally include markings of nets.In this paper, we summarize the results for different kinds of AHL schemas and nets, and extend these results to AHL systems. The category of markings is introduced, which allows a general construction combining AHL nets with possible markings leading under certain properties to a weak adhesive HLR category.