Bidirectional transformations provide a novel mechanism for synchronizing and maintaining the consistency of information between input and output. Despite many promising results on bidirectional transformations, these have been limited to the context of relational or XML (tree-like) databases. We challenge the problem of bidirectional transformations within the context of graphs, by proposing a formal definition of a well-behaved bidirectional semantics for UnCAL, i.e., a graph algebra for the known UnQL graph query language. The key to our successful formalization is full utilization of both the recursive and bulk semantics of structural recursion on graphs. We carefully refine the existing forward evaluation of structural recursion so that it can produce sufficient trace information for later backward evaluation. We use the trace information for backward evaluation to reflect in-place updates and deletions on the view to the source, and adopt the universal resolving algorithm for inverse computation and the narrowing technique to tackle the difficult problem with insertion. We prove our bidirectional evaluation is well-behaved. Our current implementation is available online and confirms the usefulness of our approach with nontrivial applications.

Bidirectional transformations provide a novel mechanism for synchronizing and maintaining the consistency of information between input and output. Despite many promising results on bidirectional transformations, these have been limited to the context of relational or XML (tree-like) databases. We challenge the problem of bidirectional transformations within the context of graphs, by proposing a formal definition of a well-behaved bidirectional semantics for UnCAL, i.e., a graph algebra for the known UnQL graph query language. The key to our successful formalization is full utilization of both the recursive and bulk semantics of structural recursion on graphs. We carefully refine the existing forward evaluation of structural recursion so that it can produce sufficient trace information for later backward evaluation. We use the trace information for backward evaluation to reflect in-place updates and deletions on the view to the source, and adopt the universal resolving algorithm for inverse computation and the narrowing technique to tackle the difficult problem with insertion. We prove our bidirectional evaluation is well-behaved. Our current implementation is available online and confirms the usefulness of our approach with nontrivial applications.

Bidirectional model transformation is a key technology in Model-Driven Engineering (MDE), when two models that can change over time have to be kept constantly consistent with each other. While several model transformation tools include at least a partial support to bidirectionality, it is not clear how these bidirectional capabilities relate to each other and to similar classical problems in computer science, from the view-update problem in databases to bidirectional graph transformations. This paper tries to clarify and visualize the space of design choices for bidirectional transformations from an MDE point of view, in the form of a feature model. The selected list of existing approaches are characterized by mapping them to the feature model. Then the feature model is used to highlight some unexplored research lines in bidirectional transformations.

We introduce a lambda calculus λ T FG for transformations of finite graphs by generalizing and extending an existing calculus UnCAL. Whereas UnCAL can treat only unordered graphs, λ T FG can treat a variety of graph models: directed edge-labeled graphs whose branch styles are represented by monads T . For example, λ T FG can treat unordered graphs, ordered graphs, weighted graphs, probability graphs, and so on, by using the powerset monad, list monad, multiset monad, probability monad, respectively. In λ T FG , graphs are considered as extension of tree data structures, i.e. as infinite (regular) trees, so the semantics is given with bisimilarity.A remarkable feature of UnCAL and λ T FG is structural recursion for graphs, which gives a systematic programming basis like that for trees. Despite the non-well-foundedness of graphs, by suitably restricting the structural recursion, UnCAL and λ T FG ensures that there is a termination property and that all transformations preserve the finiteness of the graphs. The structural recursion is defined in a "divide-and-aggregate" way; "aggregation" is done by connecting graphs with ε-edges, which are similar to the ε-transitions of automata. We give a suitable general definition of bisimilarity, taking account of ε-edges; then we show that the structural recursion is well defined with respect to the bisimilarity.

Model transformation plays an important role in model-driven software development that aims to introduce significant efficiencies and rigor to the theory and practice of software development. Although models may have different notations and representations, they are basically graphs, and model transformations are thus nothing but graph transformations. Despite a large amount of theoretical work and a lot of experience with research prototypes on graph-based model transformations, it remains an open issue how to compose model transformations. In this paper, we report our first attempt at a compositional framework for graph-based model transformations using the graph querying language UnQL. The main idea of UnQL is that graph queries are fully captured by structural recursion that is suitable for efficient composition. We show that the idea can be applied to graph-based model transformations. We have implemented a prototype of the framework and tested it with several nontrivial examples. Our new framework supports systematic development of model transformation "in the large" with the advantage that it can automatically remove inefficiencies arising from their composition.

Structural recursion, in the form of, for example, folds on lists and catamorphisms on algebraic data structures including trees, plays an important role in functional programming, by providing a systematic way for constructing and manipulating functional programs. It is, however, a challenge to define structural recursions for graph data structures, the most ubiquitous sort of data in computing. This is because unlike lists and trees, graphs are essentially not inductive and cannot be formalized as an initial algebra in general. In this paper, we borrow from the database community the idea of structural recursion on how to restrict recursions on infinite unordered regular trees so that they preserve the finiteness property and become terminating, which are desirable properties for query languages. We propose a new graph transformation language called λFG for transforming and querying ordered graphs, based on the well-defined bisimulation relation on ordered graphs with special ε-edges. The language λFG is a higher order graph transformation language that extends the simply typed lambda calculus with graph constructors and more powerful structural recursions, which is extended for transformations on the sibling dimension. It not only gives a general framework for manipulating graphs and reasoning about them, but also provides a solution to the open problem of how to define a structural recursion on ordered graphs, with the help of the bisimilarity for ordered graphs with ε-edges.

Abstract-Bidirectional model transformation is useful for maintaining consistency between two models, and has many potential applications in software development including model synchronization, round-trip engineering, and software evolution. Despite these attractive uses, the lack of a practical tool support for systematic development of well-behaved and efficient bidirectional model transformation prevents it from being widely used. In this paper, we solve this problem by proposing an integrated framework called GRoundTram, which is carefully designed and implemented for compositional development of well-behaved and efficient bidirectional model transformations. GRoundTram is built upon a well-founded bidirectional framework, and is equipped with a user-friendly language for coding bidirectional model transformation, a new tool for validating both models and bidirectional model transformations, an optimization mechanism for improving efficiency, and a powerful debugging environment for testing bidirectional behavior. GRoundTram has been used by people of other groups and their results show its usefulness in practice.

Abstract. Buneman et al. proposed a graph algebra called UnCAL (Unstructured CALculus) for compositional graph transformations based on structural recursion, and we have recently applied to model transformations. The compositional nature of the algebra greatly enhances the modularity of transformations. However, intermediate results generated between composed transformations cause overhead. Buneman et al. proposed fusion rules that eliminate the intermediate results, but auxiliary rewriting rules that enable the actual application of the fusion rules are not apparent so far. UnCAL graph model includes the concept of markers, which correspond to recursive function call in the structural recursion. We have found that there are many optimization opportunities at rewriting level based on static analysis, especially focusing on markers. The analysis can safely eliminate redundant function calls. Performance evaluation shows its practical effectiveness for non-trivial examples in model transformations.