[F-14 18-869] On VLSI Complexity Theory

Soon after they were developed, very-large-scale integration circuits became an important part of the forefront of computing. The development of reliable interconnect directly on silicon wafers enabled circuits consisting of many devices to be built very easily. As the design space exploded in size with the flexibility of VLSI, an important question arose: is there any way to provide a good lower-bound on the time and energy it takes for a given operation in a computing network? Such a lower bound, if tight enough, can indicate when a circuit has been optimized as much as is theoretically possible. All computing operations, at their core, involve the manipulation of information to achieve a desired result. Across any implementation of a given function, the fundamental flow of information must be the same. Therefore, it is logical to borrow concepts from information theory to analyze a given problem. Information theoretical techniques are well suited to providing lower bounds since they strip away overheads that are implementation dependant. These notes will begin by outlining the general model of VLSI circuits used by Thompson ( [3]) to analyze these types of problems. Then some examples will be given of calculating lower bounds on computation problems. In this theory, upper bounds can conveniently be shown by considering real implementations. Examples of where Thompson’s theory provides a relatively tight lower bound are given, as well as a theoretical example where the lower bound given by his method is clearly loose. This provides some insight as to which problems are well-suited to this analysis, and which are not. Finally, a short discussion of a few extensions to Thompson’s work is given.