The paper is mainly concerned with the extension of proof-nets to additives, for which the best known solution is presented. It proposes two cut-elimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional proof-theory deals with cut-elimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However old-fashioned proof-theory, which is concerned with the ritual question: “is-that-theory-consistent? ” never really cared. The situation changed when subtle algorithmic aspects of cut-elimination became prominent: typically

The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗-algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponential-free conclusions. 1

: 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...

This paper represents an extended version of [16]. In the next section we describe partially additive categories. In section 3 we present the execution of pseudoalgorithms in the framework of the partially additive categories as a kind of iteration of endomorphisms. Moreover, we consider deadlock free pseudoalgorithms as to be characterized by a summability condition; concepts about structural aspects are organized via type discipline and type compatible dynamics.