Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Digital systems, linear algebra, introduction to formal proof / 1 / c / Solution

All requester indices in the following expressions are interpreted modulo four. Let $h_{\ell ,j}$ mean that requester $j$ is the first asserted request after last recipient $\ell$:An empty logical conjunction is true. Decode the remembered index and combine the four priority orders:Exactly one $w_j$ is high when any request is high. Write $L$ as bits $L_1{L}_0$. The next-state function and Moore machine outputs areThus an idle step preserves the priority memory. The $w_j$ are combinational selection signals, not the current-cycle grants: exposing them directly as outputs would change the design into a Mealy machine. If a requester remains continuously asserted, each intervening successful grant advances the circular order, so it receives service after at most four grant decisions.