The Algebraic Theory of Interaction Nets

The theory of interaction nets, invented by Lafont, is re-examined from the algebraic hypergraph rewriting perspective. Supersimple nets are defined and discussed, and some related classes of nets, the polysimple and monosimple classes, are defined and investigated. Their static properties are established, and the invariants that need to be preserved by rewriting are investigated in detail. It is shown that in the general case, context-specific information may be used to ensure that rules actually preserve the characteristics of the rewritten net. Subordinate agents, which like logical constants for falsity may be introduced only in non-void contexts, are presented, and the ramifications of the theory in their presence are investigated, relating it to the simple and semisimple classes of Lafont. Under suitable conditions, describable in purely combinatorial terms, net rewriting systems possess Church-Rosser and Strong Normalisation properties usually associated with rewriting systems derived from logic. This leads to the reformulation of the proof nets of multiplicative linear logic including constants in an essentially logic-free way.