ACalculus of Order and Interaction

System MV is a simple, propositional linear calculus that deals with the commutative as well as the non-commutative composition of structures. The multiplicative fragment of linear logic is a special case of MV , and the tensor rule does not suffer from unnecessary non-determinism in context partitioning as it does in the sequent calculus of linear logic. System MV is obtained by breaking the symmetry in a formal system that is symmetric in the top-down vs. bottom-up building of derivations. A cut elimination theorem is the direct consequence of this break in the symmetry. The formal systems presented in this paper act on structures intermediate between proof nets and sequents. Connectives disappear, giving place to structural relations between atoms: as a result, there are no logical rules in MV , all the rules are structural. Structures are not decomposed in their main connectives, rather they freely move and interleave respecting order. Interaction happens when structures that are dual through negation come into close communication. The rules in MV result from enforcing a small set of basic conservation laws, making the design of the calculus as little arbitrary as possible, and indeed very natural in the perspective of the concurrent management of information. The calculus of structures can naturally embrace classical and linear logic, and allows to prove cut elimination properties on atomic versions of the cut rule.