A proof theoretical approach to communication

The paper investigates a concurrent computation model, chi calculus, in which communications resemble cut eliminations for classical proofs. The algebraic properties of the model are studied. Its relationship to sequential computation is illustrated by showing that it incorporates the operational semantics of the call-by-name lambda calculus. Practi- cally the model has pi calculus as a submodel. 1 Communication as Cut Elimination Concurrent computation is currently an open-ended issue. The situation is in contrast with sequential computation whose operational semantics is formalized by, among others, the -calculus ((2)). In retrospect, the -calculus can be seen as a fallout of proof theory. Curry-Howard's proposition-as-type principle allows one to code up constructive proofs as typed terms. At the core of the construc- tive logic is the minimal logic, whose type theoretical formulation gives rise to, roughly, the simply typed -calculus. Now the untyped -calculus is obtained from the simply typed -calculus by removing all the typing information. In recent years, classical proofs have been investigated in a computational set- ting. Girard proposed proof nets ((4)) as term representations of classical linear proofs. These classical terms are typed. The conclusion of a proof derivation is the type of the proof net corresponding to that proof derivation. The computations of these terms are cut eliminations modeled by rewritings of graphs. As the terms are typed, cuts happen between nodes of correlated types. Abramsky's proof- as-process interpretation ((1,3)) relates proof nets to processes. At operational level, this interpretation is supported by a cut-elimination-as-communication paradigm. It looks like a type-erasing interpretation similar to the one found in a constructive world. This paper investigates a concurrent computation model obtained by revers- ing the roles of proofs and processes in Abramsky's paradigm. That is to say that we regard communications as cut eliminations. The way to arrive at such a model of communication echoes that in the sequential world. First we take the multiplicative linear logic as the 'minimal logic' in a classical framework. There is nothing canonical about this choice. As the typed classical terms we take the