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GP (for Graph Programs) is an experimental nondeterministic programming language for solving problems on graphs and graph-like structures. The language is based on graph transformation rules, allowing visual programming at a high level of abstraction. In particular, GP frees programmers from dealing with low-level data structures. In this paper, we present a Hoare-style proof system for verifying the partial correctness of (a subset of) graph programs. The pre- and post-conditions of the calculus are nested graph conditions with expressions, a formalism for specifying both structural graph properties and properties of labels. We show that our proof system is sound with respect to GP's operational semantics and give examples of its use.

It is shown that it is undecidable in general whether a graph rewriting system (in the “double pushout approach”) is terminating. The proof is by a reduction of the Post Correspondence Problem. It is also argued that there is no straightforward reduction of the halting problem for Turing machines or of the termination problem for string rewriting systems to the present problem.

Jungle evaluation is proposed as a new graph rewriting approach to the evaluation of functional expressions and, in particular, of algebraically specified operations. Jungles – being intuitively forests of coalesced trees with shared substructures – are certain acyclic hypergraphs (or equivalently, bipartite graphs) the nodes and edges of which are labeled with the sorts and operation symbols of a signature. Jungles are manipulated and evaluated by the application of jungle rewrite rules, which generalize equations or, more exactly, term rewrite rules. Indeed, jungle evaluation turns out to be a compromise between term rewriting and graph rewriting displaying some favorable properties: the inefficiency of term rewriting is partly avoided while the possibility of structural induction is maintained, and a good part of the existing graph grammar theory is applicable so that there is some hope that the rich theory of term rewriting is not lost forever without a substitute.