We present a hybrid methodology for the formal verification of arithmetic RTL designs that combines sequential logic equivalence checking with interactive theorem proving in a two-step process. First, an intermediate model of the design is extracted by hand and coded in Restricted Algorithmic C, a simple C subset augmented by the C++ register class templates of Algorithmic C, which provide the bit manipulation features of Verilog. The model is designed to mirror the RTL microarchitecture closely enough to allow efficient equivalence checking, but sufficiently abstract to be amenable to formal analysis. The model is then automatically translated to the logic of the ACL2 theorem prover, which is used to establish correctness with respect to an architectural specification. As an illustration, we describe the modeling and proof of correctness of a chained multiply-add module, designed to test techniques for area and power reduction and intended for implementation in future Arm graphics nrocessors.

There has been much work done on designing cellular scheduling algorithms. These algorithms are set up in the manner of a direct mechanism in which the queues reveal their states (backlog), and the scheduler chooses an allocation. In policies such as longest queue first (LQF), scheduling can be shown to yield short queue lengths. However, these algorithms are reliant on truthful declarations of state. Our goal in this article is to determine if a Vickrey–Clarke–Groves (VCG)-type mechanism (a second price auction) that elicits truthful value will also possess the desired LQF-like behavior in a system of dynamically evolving queues. Our approach is to use the concept of a mean field game under which at each time instant, a queue chooses its bid as a best response to its belief that the bids of the others will be drawn independently from a common bid distribution. If this best response is itself a sample from the belief bid distribution, a mean field equilibrium (MFE) is said to exist. We show the existence of the MFE and find it using its structure that the results of the allocation policy are the same as LQF. Thus, the desired LQF-like behavior arises naturally under the second-price auction mechanism. We also present results on the accuracy of the model as the number of agents (queues) becomes large. Finally, using simulations with a large number of queues, we show that computation of the MFE is straightforward.