Typed Intermediate Languages for Shape Analysis

We introduce S2, a typed intermediate language for vectors, based on a 2level type-theory, which distinguishes between compile-time and run-time. The paper shows how S2 can be used to extract useful information from programs written in the Nested Sequence Calculus NSC, an idealized high-level parallel calculus for nested sequences. We study two translations from NSC to S2. The most interesting shows that shape analysis (in the sense of Jay) can be handled at compile-time. Good intermediate languages are an important prerequisite for program analysis and optimization, the main purpose of such languages is to make as explicit as possible the information that is only implicit in source programs (see [18]). A common features of such intermediate languages is an aggressive use of types to incorporate additional information, e.g.: binding times (see [18]), boxed/unboxed values (see [20]), effects (see [23]). In particular, among the ML community the use of types in intermediate languages has been advocated for the TIL compiler (see [9]) and for region inference (see [24, 1]). In areas like parallel programming, where efficiency is a paramount issue, good intermediate languages are even more critical to bridge the gap between high-level languages (e.g. NESL) and efficient implementations on a variety of architectures (see [2, 3, 22]). However, in this area of computing intermediate languages (e.g. VCODE) have not made significant use of types, yet. This paper proposes a typed intermediate language S2 for vector languages, and shows how it may be used to extract useful information from programs written in the Nested Sequence Calculus NSC (see [22]), an idealized vector language very closed to NESL. For an efficient compilation of NSC (and similar languages) on parallel machines it is very important to know in advance the size of vectors (more generally the shape of data structures). We study two translations from the NSC to S2. The most interesting one separates what can be computed at compile-time from what must be computed at run-time, in particular array bound-checking can be done at compile-time (provided the while-loop of NSC is replaced by a for-loop). Section 1 introduces the two-level calculus S2 and outlines its categorical semantics. Section 2 summarizes the high-level language NSC, outlines two translations from NSC in S2 and the main results about them. Sections 3 and 4 give the syntactic details of the translations. Appendix A gives a formal description of S2 and defines auxiliary notation and notational conventions used in the paper.