Lazy evaluation is a powerful tool that enables better compositionality and potentially better performance in functional programming, but it is challenging to analyze its computation cost. Existing works either require manually annotating sharing, or rely on separation logic to reason about heaps of mutable cells. In this paper, we propose a bidirectional demand semantics that allows for extrinsic reasoning about the computation cost of lazy programs without relying on special program logics. To show the effectiveness of our approach, we apply the demand semantics to a variety of case studies including insertion sort, selection sort, Okasaki’s banker’s queue, and the implicit queue. We formally prove that the banker’s queue and the implicit queue are both amortized and persistent using the Rocq Prover (formerly known as Coq). We also propose the reverse physicist’s method, a novel variant of the classical physicist’s method, which enables mechanized, modular and compositional reasoning about amortization and persistence with the demand semantics.

This paper introduces the architecture and initial algorithms for Pixel-Planes 5, a heterogeneous multi-computer designed both for high-speed polygon and sphere rendering (1M Phong-shaded triangles/second) and for supporting algorithm and application research in interactive 3D graphics. Techniques are described for volume rendering at multiple frames per second, font generation directly from conic spline descriptions, and rapid calculation of radiosity form-factors. The hardware consists of up to 32 math-oriented processors, up to 16 rendering units, and a conventional 1280 × 1024-pixel frame buffer, interconnected by a 5 gigabit ring network. Each rendering unit consists of a 128 × 128-pixel array of processors-with-memory with parallel quadratic expression evaluation for every pixel. Implemented on 1.6 micron CMOS chips designed to run at 40MHz, this array has 208 bits/pixel on-chip and is connected to a video RAM memory system that provides 4,096 bits of off-chip memory. Rendering units can be independently reasigned to any part of the screen or to non-screen-oriented computation. As of April 1989, both hardware and software are still under construction, with initial system operation scheduled for fall 1989.