A Higher-Order Calculus for Graph Transformation

Abstract

This paper presents a formalism for defining higher-order systems based on the notion of graph transformation (by rewriting or interaction). The syntax is inspired by the Combinatory Reduction Systems of Klop. The rewrite rules can be used to define first-order systems, such as graph or term-graph rewriting systems, Lafont's interaction nets, the interaction systems of Asperti and Laneve, the non-deterministic nets of Alexiev, or a process calculus. They can also be used to specify higher-order systems such as hierarchical graphs and proof nets of Linear Logic, or to specify the operational semantics of graph-based languages.

Attributed Hierarchical Port Graphs and Applications

“…Boxes are represented by extra nodes requiring additional rules for book-keeping. To palliate this problem, several extensions of interaction nets have been proposed (see e.g., [1,2,22]). In [2], a representation of intuitionistic logic proofs (or λ -terms) is given by means of higher-order port-graphs (HOPG).…”

Extending the Interaction Nets Calculus by Generic Rules

“…We based our extensions on the improved lightweight calculus, which was introduced in [7]. A different approach to higher-order computation in the interaction calculus can be found in [4].…”

Higher-order port-graph rewriting

“…Higher-order extensions have been defined for more restricted formalisms: Term rewriting has been extended with higher-order features, with formalisms such as Combinatory Reduction Systems [11] and Nominal Rewriting Systems [9] amongst others. Higher-order graph rewriting theories have been defined in [10,13] via textual calculi instead of graphical formalisms. The examples that motivate the higher-order extension of port-graphs presented in this paper come from the graphical representation of proofs in intuitionistic logic given in [1].…”