ALinear LogicViewofGamma Style Computations as Proof Searches

Using the methodology of abstract logic programming in linear logic, we establish a correct and complete translation between the language Nabla and rst order linear logic. Nabla is a modiication of the coordination language Gamma with parallel and sequential composition. Nabla, without modifying Gamma basic computational model, is amenable to this kind of analysis, at the price of a weaker expressive power. The translation is correct and complete in the sense that we establish a two way correspondence between computations in Nabla and the search for proofs in a suitable fragment of rst order linear logic. Moreover, the translation is not an encoding, meaning that to the algebraic structure of Nabla programs is assigned logical meaning through a non-trivial use of linear logic connec-tives, as opposed to merely reeecting their operational behavior through a simulation into terms of the logic. In this way we hope that the connection established between the two formalisms can compensate for the diminished expressive power of Nabla with a powerful analysis tool, which could lead both to theoretical and practical improvements in semantic foundations of Gamma-style languages and in the design of eecient implementations of their interpreters. The main diiculty has been to deal with sequential composition of programs, and to smoothly integrate its logical treatment in a recursive framework. An intermediate step is the deenition of the language SMR, by which it is possible to specify in a very intuitive way Nabla operational semantics, and to prove that this speciication is actually equivalent to the SOS-style one derived from Gamma semantics.