| \section{LILY / compiling complete certificates} |
| \label{sec:compiler} |
|
|
| \subsection{Why the original single-sensor greedy rule does not transfer} |
| AELHYEUM's sensor objective is a multiplicative reduction of pairwise overlap. The exact probability of obtaining a semantic certificate can have increasing returns. If $C=\{1,2\}$ is a minimal forbidden support and both input coordinates are active, selecting only sensor 1 or only sensor 2 produces no certificate, while selecting both produces one with probability $q_1q_2>0$. The second sensor's marginal benefit is larger after selecting the first. |
|
|
| Therefore the semantic objective over individual sensor positions need not be submodular. A greedy guarantee for the source objective cannot simply be copied into the new language. The compiler instead treats a \emph{complete sound certificate} as the atomic selectable object. |
|
|
| \subsection{Packet semantics} |
| For each minimal forbidden support $C$, make a packet that prepares the needed coordinate symbols and tests all of them by their conclusive local projectors. In a binary alphabet with fixed nonzero activation probabilities, its refusal probability is |
| \begin{equation} |
| r_C(x)=\gamma_C\1\{C\subseteq\supp(x)\},\qquad |
| \gamma_C=\prod_{i\in C}q_i. |
| \label{eq:packet} |
| \end{equation} |
| Every successful packet emits \texttt{REFUSE(C)}. Its certificate is independently checkable from the grammar because $C\notin\K$, and it is sound in the calibrated ideal model. A local implementation on the minimal forbidden support is optimal for that packet's one-sided task by Theorem~\ref{thm:hereditary}. |
|
|
| Choose packets before observing the unknown input. When selected packets use separate preparations with conditionally independent outcomes given the same fixed $x$, and $\pi(x)$ is a declared design prior, their expected success is |
| \begin{equation} |
| J(\mathcal S)=\sum_x\pi(x)\left[1-\prod_{C\in\mathcal S}(1-r_C(x))\right]. |
| \label{eq:outerobjective} |
| \end{equation} |
| Repeated preparation is a charged resource. The formula does not assume that overlapping tests on a single quantum system are independent, and it does not clone an unknown supplied word. |
|
|
| \begin{theorem}[Certificate selection and an exact fast marginal] |
| \label{thm:compiler} |
| The function $J$ is normalized, monotone, and submodular. Under a cardinality budget of $b$ packets, ordinary greedy selection achieves at least $(1-1/e)$ times the optimal value. In the binary support model \eqref{eq:packet}, define |
| \[ |
| v_{\mathcal S}(x)=\pi(x)\prod_{C\in\mathcal S}(1-r_C(x)). |
| \] |
| The exact marginal of a candidate $C$ is |
| \begin{equation} |
| \Delta_C J(\mathcal S) |
| =\gamma_C\sum_{x:\supp(x)\supseteq C}v_{\mathcal S}(x). |
| \label{eq:zetagain} |
| \end{equation} |
| All candidate marginals can be computed from one upper subset-zeta transform in $O(n2^n)$ arithmetic operations and $O(2^n)$ memory. |
| \end{theorem} |
| \begin{proof} |
| For any candidate packet $C$, |
| \[ |
| \Delta_CJ(\mathcal S)=\sum_x\pi(x)r_C(x) |
| \prod_{B\in\mathcal S}(1-r_B(x)). |
| \] |
| Every term is nonnegative and decreases when $\mathcal S$ grows, proving monotonicity and diminishing returns. The classical cardinality-constrained greedy theorem\ \cite{nemhauser} gives the approximation factor. Substituting \eqref{eq:packet} proves \eqref{eq:zetagain}. The upper subset transform stores $z(C)=\sum_{X\supseteq C}v(X)$ and performs one addition for each missing coordinate in each subset, giving the stated cost. |
| \end{proof} |
|
|
| The approximation theorem and subset transform are established algorithms. The proposed coupling is the replacement of raw sensor choices by globally optimal one-sided certificate packets, with an exact probability model and source-aligned transform oracle. The $O(n2^n)$ bound is exponential in word length. It is useful for a finite dictionary and does not establish a polynomial-time method for arbitrary succinctly represented languages. |
|
|
| \subsection{A fully enumerated eight-coordinate instance} |
| The supplied program uses the at-most-two grammar on eight binary coordinates, hence 56 possible three-coordinate packets. It fixes the prior and local activation probabilities in the source code. All 27,720 three-packet choices are evaluated under the same objective. |
|
|
| \begin{center} |
| \begin{tabular}{@{}lll@{}} |
| \toprule |
| \textbf{Budget} & \textbf{Greedy expected refusal} & \textbf{Complete optimum if enumerated}\\ |
| \midrule |
| 1 & 0.2268502804 & --\\ |
| 2 & 0.3271777764 & --\\ |
| 3 & 0.4062552360 & 0.4062552360\\ |
| 4 & 0.4618133396 & --\\ |
| \bottomrule |
| \end{tabular} |
| \end{center} |
| At budget three, greedy chooses $\{1,2,3\}$, $\{1,2,6\}$, and $\{4,6,7\}$ and attains the complete finite optimum. The transform and direct marginals agree within $8.33\times10^{-17}$. This one instance verifies the implementation; it does not improve the general approximation factor. |
|
|
| \begin{figure}[ht] |
| \centering\includegraphics[width=.85\textwidth]{figures/certificate_compiler.pdf} |
| \caption{Synthetic packet selection with all preparation uses charged as separate packets. The diamond marks the exhaustive three-packet optimum.} |
| \end{figure} |
|
|
| \subsection{An executable semantic interface} |
| The minimal compiler record consists of: |
| \begin{enumerate} |
| \item \textbf{Alphabet:} the character labels and amplitudes, or calibrated local density operators and background supports. |
| \item \textbf{Meaning:} the Boolean function, sparse-repair dictionary and budget, or hereditary grammar. |
| \item \textbf{Contract:} minimum error, exact reading, or a specified false-refusal tolerance. |
| \item \textbf{Receiver:} a theorem-backed measurement and a deterministic classical decision rule. |
| \item \textbf{Certificate:} a forbidden support, a finite-field dual witness, or a recorded undecided outcome. |
| \item \textbf{Resource record:} preparations, active dimension, interactions, readout round, classical work, and transmitted metadata. |
| \end{enumerate} |
|
|
| The source manuscripts' decoder-synchronized coding interface can serialize this record. A fixed certificate-informed prior may drive a causal mixture whose predictions are reproduced at the decoder. The present work contributes the semantic probability and certificate; it does not add a new NNCP or cmix implementation. Shared calibration or grammar tables must genuinely be shared or charged in the packet. Quantum readout does not remove the source manuscripts' information-accounting constraints. |
|
|
| \subsection{A practical decision map} |
| \begin{tabularx}{\textwidth}{@{}p{.30\textwidth}X@{}} |
| \toprule |
| \textbf{Required meaning} & \textbf{Proved design consequence}\\ |
| \midrule |
| Uniform-prior Boolean guess from binary pure product signals, with $0<s_i<1$ at every coordinate & Affine meanings attain the global optimum with the fixed local Helstrom receiver. Non-affine meanings have a strict gap, even after optimizing classical postprocessing of those outputs.\\[4pt] |
| Sound exclusion of a hereditary support pattern & Parallel local background-support tests attain the full global optimum. Classical grammar membership remains to be implemented.\\[4pt] |
| Sound exclusion of a compressed sparse repair & Search for a zero-marginal spectral trade on $\im B^T$; fewer than $2^{k+1}$ active characters cannot suffice for any nontrivial refusal.\\[4pt] |
| Exact all-input phase-covariant meaning & Compute the translation period subgroup. The required active dimension is at least the number of its cosets.\\[4pt] |
| Limited number of complete certificate attempts & Use the packet objective with separately prepared uses and the exact subset-zeta marginal oracle.\\ |
| \bottomrule |
| \end{tabularx} |
|
|
