Despite their vast deployment, distributed memory multiprocessors (DMM) still remain dicult to program, this is why portable and ecient languages are denitely needed. For the time being, parallelization tools like PVM provides a widely portable solution for low-level (explicit communications) programming. But PVM programs (C or Fortran with send and receive for communications) are hard to write and debug because of their undeterministic behaviors and rudimentary communication support. We propose an intermediate-level functional language called Caml Flight, which integrate a programming technique used by PVM programmers and featured in MPI. Right now, Caml Flight allows to write deterministic SPMD programs more easily. A long term goal is to be act as a target language for compilers of high-level languages. 1

this paper we describe a new, categorical approach to normalization in typed -

Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambda-calculus terms. We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the pre-logical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.

Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterises isomorphism in the typed lambda calculus, where the constant for one and the operations of product and exponentiation respectively correspond to the unit type and the product and arrow type constructors. This paper studies isomorphisms in typed lambda calculi with empty and sum types from this viewpoint. We close an open problem by establishing that the theory of type isomorphisms in the presence of product, arrow, and sum types (with or without the unit type) is not finitely axiomatisable. Further, we observe that for type theories with arrow, empty and sum types the correspondence between isomorphism and arithmetic equality generally breaks down, but that it still holds in some particular cases including that of type isomorphism with the empty type and equality with zero. 1