Jean-Baptiste Joinet

The present report is a, somewhat lengthy, addendum to [4], where the elimination of cuts from derivations in sequent calculus for classical logic was studied 'from the point of view of linear logic'. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives. The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions. We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative. We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e., by only one specific combination, and structural rules

The present report is a, somewhat lengthy, addendum to "A new deconstructive logic: linear logic," where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.We give a detailed description of the simulation of tq-reductions by means of reductions of the interpre- tation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e., by only one specific combination, and structural rules

We examine 'weak-distributivity' as a rewriting rule $??$ defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proof-nets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for $??$: starting from a set of simple generators (proof-nets which are a n-ary ⊗ of &-ized axioms), any mono-conclusion MLL proof-net can be reached by $??$ rewriting (up to ⊗ and & associativity and commutativity)

The main concern of this paper is the design of a noetherian and confluent normalization for LK 2. The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD, delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' to classical logic ; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic using appropriate embeddings ; it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ brings to the fore the latter's defects for these `deconstructive purposes'.

The main concern of this paper is the design of a noetherian and confluent normalization for $\mathbf{LK}^2$. The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's $\mathbf{LC}$ and Parigot's $\lambda\mu, \mathbf{FD}$, delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic using appropriate embeddings ; it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding $\mathbf{LK}$ into $\mathbf{LJ}$ brings to the fore the latter's defects for these `deconstructive purposes'.