Stage 5 — Instruction Read RAF & Reductions

Degree 19

Rounds $\log_2 K_{\text{instr}} + \log_2 T$

$\displaystyle\sum$ over: $X_k \in \{0,1\}^{\log_2 K_{\text{instr}}}, X_t \in \{0,1\}^{\log_2 T}$

Opening point $(r_{K_{\text{instr}}}^{(5)}, r_{\text{cycle}}^{(5)})$

\textcolor{BurntOrange}{\textsf{LookupOutput}}(r_{\text{cycle}}^{(2)}) + \gamma \cdot \textcolor{BurntOrange}{\textsf{LeftLookupOperand}}(r_{\text{cycle}}^{(2)}) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{RightLookupOperand}}(r_{\text{cycle}}^{(2)})

\widetilde{\text{eq}}(r_{\text{cycle}}^{(2)}, X_t) \cdot \prod_{i=0}^{15} \textcolor{BurntOrange}{\textsf{InstructionRa}(i)}(X_k, X_t) \cdot \left(\sum_{j=0}^{N_tables-1} T_j(X_k) \cdot \textcolor{BurntOrange}{\textsf{TableFlag}(j)}(X_t) + \gamma \cdot \left(\left(1 - \textcolor{BurntOrange}{\textsf{InstructionRafFlag}}(X_t)\right) \cdot \left(\textsf{LeftOp}(X_k) + \gamma \cdot \textsf{RightOp}(X_k)\right) + \textcolor{BurntOrange}{\textsf{InstructionRafFlag}}(X_t) \cdot \gamma \cdot \textsf{unmap}(X_k)\right)\right)

$\textsf{InstructionRa}(i)(r_{K_{\text{instr}}^{(i)}}^{(5)}, r_{\text{cycle}}^{(5)}) \text{ for } i=0,\ldots,d_v-1$
$\textsf{TableFlag}(j)(r_{\text{cycle}}^{(5)}) \text{ for } j=0,\ldots,N_{\text{tables}}-1$
$\textcolor{BurntOrange}{\textsf{InstructionRafFlag}}(r_{\text{cycle}}^{(5)})$

Degree 2

Rounds $\log_2 T$

$\displaystyle\sum$ over: $X_t \in \{0,1\}^{\log_2 T}$

Opening point $(r_{\text{cycle}}^{(5)})$

\textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, r_{\text{cycle}}^{(1)}) + \gamma \cdot \textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, r_{\text{cycle}}^{(2)}) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, r_{\text{cycle}}^{(4)})

\left(\widetilde{\text{eq}}(r_{\text{cycle}}^{(1)}, X_t) + \gamma \cdot \widetilde{\text{eq}}(r_{\text{cycle}}^{(2)}, X_t) + \gamma^{2} \cdot \widetilde{\text{eq}}(r_{\text{cycle}}^{(4)}, X_t)\right) \cdot \textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, X_t)

$\textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, r_{\text{cycle}}^{(5)})$

Degree 3

Rounds $\log_2 T$

$\displaystyle\sum$ over: $X_t \in \{0,1\}^{\log_2 T}$

Opening point $(r_{\text{cycle}}^{(5)})$

\textcolor{BurntOrange}{\textsf{RegistersVal}}(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(4)})

\textcolor{ForestGreen}{\textsf{RdInc}}(X_t) \cdot \textcolor{BurntOrange}{\textsf{RdWa}}(r_{K_{\text{reg}}}^{(4)}, X_t) \cdot \widetilde{\textsf{LT}}(X_t, r_{\text{cycle}}^{(4)})

$\textcolor{ForestGreen}{\textsf{RdInc}}(r_{\text{cycle}}^{(5)})$
$\textcolor{BurntOrange}{\textsf{RdWa}}(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(5)})$