Structural recursion for querying ordered graphs

Abstract

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.

“Bidirectionalization for free” for monomorphic transformations

“…For example, in a certain model of graphs in which we identify graphs with infinite trees, the contract operation may convert a finite graph to an infinite one and does not terminate in such a case without careful treatment [41,42]. Our framework is applicable even to such model of graphs, as long as the parametricity and the totality assumption, including the termination of a graph transformation, are kept (Section 4).…”

A parameterized graph transformation calculus for finite graphs with monadic branches

“…List-graphs are ordered graphs [14], where the branches are ordered. An example of an ordered graph is shown in Figure 1.…”

“…Our use of ε-edge is just in an implementation of λ T FG for efficiency and for defining structural recursion; so we shall suppose that what users of λ T FG can observe is just finite-width graphs, and graphs whose ε-elimination have infinite widths are regarded as errors. (For any finite List-graph G, it is decidable whether the ε-elimination of G keeps finite width or not; see [14,2].) Therefore, the finite powerset monad and the other monads in Example 2 are more suitable for practical purposes; however, they do not have iteration operators.…”

“…In previous work [14], the authors studied how to modify the graph model of UnCAL so that we can treat ordered graphs in a similar way to UnCAL. There it is found that ε-edges of ordered graphs may produce a subtle problem of infinite width on branches; this problem does not occur in the case of unordered graphs.…”

“…In Section 4 we discuss related work, which includes more detail comparison between λ T FG and the calculus in [14]. Our contributions are summarized as follows:…”