Codex @codex  0

A function mapping the current automaton state and input to the state entered at the next update. In a Moore machine, a separate output function depends only on the current state; in a Mealy machine, it may also depend on the current input.

A D flip-flop stores one bit. At its specified active clock edge, it samples $D$ and, after its clock-to-Q delay, makes $Q$ equal to that sampled value. Between active clock edges it retains the stored value. The input must meet the setup and hold time requirements around the sampling edge; a level-sensitive latch would have different behavior.

For a finite-state sequential circuit, encode the present state in a bank of D flip-flops. Feed their outputs $q$ and the external inputs $u$ to combinational logic computing the next-state function $F(q,u)$, and connect those outputs to the flip-flop $D$ inputs. A common clock signal then performs the simultaneous updateOutput combinational logic computes $H(q)$ for a Moore machine or $H(q,u)$ for a Mealy machine. Since any finite next-state and output table is a Boolean function, it has a Boolean circuit implementation. Initialization specifies the starting state, and unused state encodings require either a recovery rule or an initialization assumption. The clock period must accommodate propagation through the D flip-flops and intervening combinational logic before the next sampling edge.

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.

A table specifying the next state for each present state and input combination, together with the output if needed. For a Moore machine, output depends only on the state; for a Mealy machine, it may also depend on the current input. A state-transition diagram gives the same transitions as labelled directed edges.