The Algebra of Recursive Graph Transformation Language UnCAL: Complete Axiomatisation and Iteration Categorical Semantics

TL;DR

This paper establishes a complete algebraic and categorical foundation for the graph transformation language UnCAL, enhancing its theoretical understanding and practical applicability through new axioms, semantics, and models.

Contribution

It provides the first complete axiomatisation and categorical semantics for UnCAL, clarifying its algebraic structure and computational mechanisms.

Findings

A complete set of axioms for UnCAL's bisimulation semantics.

A categorical model of UnCAL using lambdaG-calculus.

Characterization of structural recursion on graphs.

Abstract

The aim of this paper is to provide mathematical foundations of a graph transformation language, called UnCAL, using categorical semantics of type theory and fixed points. About twenty years ago, Buneman et al. developed a graph database query language UnQL on the top of a functional meta-language UnCAL for describing and manipulating graphs. Recently, the functional programming community has shown renewed interest in UnCAL, because it provides an efficient graph transformation language which is useful for various applications, such as bidirectional computation. In order to make UnCAL more flexible and fruitful for further extensions and applications, in this paper, we give a more conceptual understanding of UnCAL using categorical semantics. Our general interest of this paper is to clarify what is the algebra of UnCAL. Thus, we give an equational axiomatisation and categorical semantics of UnCAL, both of which are new. We show that the axiomatisation is complete for the original bisimulation semantics of UnCAL. Moreover, we provide a clean characterisation of the computation mechanism of UnCAL called "structural recursion on graphs" using our categorical semantics. We show a concrete model of UnCAL given by the lambdaG-calculus, which shows an interesting connection to lazy functional programming.