The algebra of recursive graph transformation language UnCAL: complete axiomatisation and iteration categorical semantics

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 λG-calculus, which shows an interesting connection to lazy functional programming.

Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories

“…Our recent work. In [Ham15,HMA], the author and collaborators gave algebraic and categorical semantics of a graph transformation language UnCAL [BFS00, HHI + 10] using iteration theories [B É93]. The graph data of UnCAL corresponds to cyclic sharing trees of type CTree in the present paper, where graphs are treated modulo bisimulation.…”

“…This was a key to obtain SN and CR ∼ of FOLDr. We also hope that it will be useful for further optimisation such as the fold fusion based on semantics as done in [HMA,§4.3]. The general study of graph rewriting was also important for our study at the foundational level.…”