# Aligned Induction

## Optimisation

### Sections

Introduction

Practicable derived alignment valency-density non-overlapping fud inducer implementation

Practicable shuffle content alignment valency-density fud inducer

Limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer

### Introduction

In order to find lower bounds on the computation time and space of implementations of the derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$, given substrate histogram $A \in \mathcal{A}_{z}$, section ‘Substrate models computation’ P634 considers the computation of the substrate models, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, by explicitly defining the (i) limited-models constraints, and (ii) layer-ordered limited-underlying limited-breadth infinite-layer substrate fuds trees. Together these constrain the computation to be a two stage process of (i) computation of finite search lists of the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds $N_A \in \mathcal{L}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}})$, where $\mathrm{flip}(N_A) \in \mathrm{enums}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}})$, and (ii) filtering subsequently applied to the search lists, $\mathrm{set}(\mathrm{filter}(\mathrm{nd},N_A)) = \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$ where $\mathrm{nd}(F) = \neg \mathrm{overlap}(F) \wedge (|W^{\mathrm{C}}| \leq \mathrm{wmax} : W=\mathrm{der}(F))$.

In particular, the section ‘Substrate models computation’ defines these searches of the limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$: (i) the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V) \in \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$, which constructs the layer-ordered fuds from sets of partition transforms of the tuple, $\{P^{\mathrm{T}} : P \in \mathrm{B}(K^{\mathrm{CS}})\}$, and (ii) the limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnubh}(U)(V) \in \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$, which constructs the fuds with non-overlapping substrate transforms of the tuples, $\mathcal{T}_{U,K,\mathrm{n}}$.

Also defined are these strong limited-models non-overlapping infinite-layer substrate fuds searches: (iii) the limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnmubh}(U)(V) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$, for which there is a non-trivial computation of the regular substrate cardinalities, and (iv) the limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted decrementing linear non-overlapping fuds infinite-layer fud tree $\mathrm{tfifdnmubh}(U)(V) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$, which constructs the fuds with strong limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds, $\mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}$, on the tuples.

In the case where the computation time and space requirements exceed available resources, a practicable inducer implementation of the derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$, must choose a subset of the substrate models. That is, (i) computation time limits imply that only a searched selection $\mathrm{select}(T_A,N_A) \in \mathcal{L}(\mathrm{set}(N_A))$, where $T_A \subset \{1 \ldots |N_A|\}$, of the traversable list $N_A$ may be computed, and (ii) computation space limits imply that only a further subset $\mathrm{select}(S_A(t),N_A)$, where $S_A(t) \subset T_A$, of these may be simultaneously represented at any step $t$ of the computation.

Given that a selection of the traversable search list, $N_A$, is necessary in some circumstances, the choice of selection can be made according to various criteria. Let $P \in \mathcal{L}(\mathcal{X})$ be a tuple of parameters. Consider a practicable derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘}$, which, given substrate histogram $A \in \mathcal{A}_{z}$, is defined such that the substrate models of its application is a subset of that of the derived alignment valency-density non-overlapping fud inducer, $\mathrm{dom}(I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘ * }(A)) \subseteq \mathrm{dom}(I_{z,\mathrm{ad,F,\infty,n,q}}^{‘ * }(A))$. That is, $\begin{eqnarray} &&I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘ * }(A) \subseteq \\ &&\hspace{2em}\{(F,I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})/I_{\mathrm{cvl}}^{ * }(F)) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}\} \end{eqnarray}$ where $I_{\mathrm{cvl}}^{ * }(F) := (I_{\approx\mathrm{pow}}^{ * }((w,1/m)) : W = \mathrm{der}(F),~w = |W^{\mathrm{C}}|,~m = |W|)$, and the power approxer $I_{\approx\mathrm{pow}} \in \mathrm{computers}$, is such that (i) $\mathrm{domain}(I_{\approx\mathrm{pow}}) = \mathbf{Q} \times \mathbf{Q}$, (ii) $\mathrm{range}(I_{\approx\mathrm{pow}}) = \mathbf{Q}$, and (iii) $I_{\approx\mathrm{pow}}^{ * }((x,y)) \approx x^y$. The practicable fud inducer is defined in terms of the alignmenter, $I_{\mathrm{a}}$, and the power approxer, $I_{\approx\mathrm{pow}}$, and so the application approximates to the application of the tractable fud inducer, $I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})/I_{\mathrm{cvl}}^{ * }(F) \approx I_{\approx\mathbf{R}}^{ * }(\mathrm{algn}(A * F^{\mathrm{T}})/\mathrm{cvl}(F))$.

Let the practicable inducer have (i) computation time limit $I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘\mathrm{t}}(A) \leq \mathrm{tmax}$, where the maximum time limit is $\mathrm{tmax} \in \mathbf{N}_{>0}$, and (ii) computation space limit $I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘\mathrm{s}}(A) \leq \mathrm{smax}$, where the maximum space limit is $\mathrm{smax} \in \mathbf{N}_{>0}$. These limits are parameters of the practicable inducer, $\mathrm{tmax},\mathrm{smax} \in \mathrm{set}(P)$. The following sections consider various definitions of practicable inducers having different selection criteria and the effect on the correlation of the maximum functions between the practicable inducer and the corresponding tractable inducer.

#### Practicable derived alignment valency-density non-overlapping fud inducer implementation

Consider an implementation of a practicable fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘}$, which, given substrate histogram $A \in \mathcal{A}_{z}$, optimises its subset of the limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, by first optimising the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}$, and then filtering for the limited-derived non-overlapping, $\mathcal{F}_{\mathrm{d}} \cap \mathcal{F}_{\mathrm{n}}$. The optimisation is implemented by means of a list maximiser. The maximisation is of a rational-valued left-total optimise function $X_{P,A,\mathrm{ad}}$ of the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds, $X_{P,A,\mathrm{ad}} \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}} :\to \mathbf{Q}$. Given (i) maximum optimise step cardinality $\mathrm{omax} \in \mathbf{N}_{>0}$, such that $\mathrm{omax} < \mathrm{tmax}$, (ii) initial subset $R_{P,A,\mathrm{ad}} \subset X_{P,A,\mathrm{ad}}$, and (iii) neighbourhood function $P_{P,A,\mathrm{ad}} \in \mathrm{P}(X_{P,A,\mathrm{ad}}) \to \mathrm{P}(X_{P,A,\mathrm{ad}})$, the list maximiser $Z_{P,A,\mathrm{ad}} \in \mathrm{maximisers}(X_{P,A,\mathrm{ad}})$ is constructed $Z_{P,A,\mathrm{ad}} = \mathrm{maximiseLister}(X_{P,A,\mathrm{ad}}, P_{P,A,\mathrm{ad}}, \mathrm{top}(\mathrm{omax}), R_{P,A,\mathrm{ad}})$ The cardinality of the elements of the list maximiser is constrained by the maximum optimise step cardinality, $|\mathrm{elements}(Z_{P,A,\mathrm{ad}})| \leq \mathrm{omax} \times |\mathrm{list}(Z_{P,A,\mathrm{ad}})|$ where $\mathrm{elements}(Z) := \bigcup \mathrm{set}(\mathrm{list}(Z))$. Note that strictly speaking this is true only in the case where cardinality of the $\mathrm{top}(\mathrm{omax})$ function in each step is less than or equal to $\mathrm{omax}$, $\forall Y \in \mathrm{set}(\mathrm{list}(Z_{P,A,\mathrm{ad}}))~(Y \in \mathrm{dom}(X_{P,A,\mathrm{ad}}) \leftrightarrow \mathbf{Q} \implies |Y| \leq \mathrm{omax})$.

Given (iv) that the neighbourhood function, $P_{P,A,\mathrm{ad}}$, is further constrained such that it terminates before the maximum time limit, the cardinality of the searched set is such that $|\mathrm{elements}(Z_{P,A,\mathrm{ad}})| \leq |\mathrm{searched}(Z_{P,A,\mathrm{ad}})| < \mathrm{tmax}$ where $\mathrm{searched}(Z) := \bigcup \{P(Y) : Y \in \mathrm{set}(\mathrm{list}(Z))\} \cup R$.

The domain of the elements of the list maximiser is a subset of the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds, $\mathrm{dom}(\mathrm{elements}(Z_{P,A,\mathrm{ad}})) \subset \mathrm{dom}(X_{P,A,\mathrm{ad}}) = \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}$ The subset of the substrate fuds is the further subset $\mathrm{filter}(\mathrm{nd},\mathrm{dom}(\mathrm{elements}(Z_{P,A,\mathrm{ad}}))) \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$ So the domain of the searched set is the subset of the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds search list, $\mathrm{dom}(\mathrm{searched}(Z_{P,A,\mathrm{ad}})) = \mathrm{set}(\mathrm{select}(T_A,N_A))$ where $\mathrm{flip}(N_A) \in \mathrm{enums}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}})$ and $T \subset \{1 \ldots |N_A|\}$.

The practicable inducer is implemented $\begin{eqnarray} &&I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘ * }(A) = \\ &&\hspace{2em}\{(F,I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})/I_{\mathrm{cvl}}^{ * }(F)) : F \in \mathrm{filter}(\mathrm{nd},\mathrm{dom}(\mathrm{elements}(Z_{P,A,\mathrm{ad}})))\} \end{eqnarray}$ The fuds optimise function, $X_{P,A,\mathrm{ad}}$, cannot be simply a derived alignment valency-density function, $X_{P,A,\mathrm{ad}} \neq \{(F,I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})/I_{\mathrm{cvl}}^{ * }(F)) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}\}$, because the fuds are not necessarily non-overlapping nor limited-derived, $\mathrm{nd}(F)$, where $F \in \mathrm{dom}(X_{P,A,\mathrm{ad}})$. That is, if the fud is overlapping then the derived alignment may, for example, be purely formal, $\mathrm{algn}(A * F^{\mathrm{T}}) = \mathrm{algn}(A^{\mathrm{X}} * F^{\mathrm{T}})$. This would be the case if the fud was tautological, $\mathrm{tautology}(F^{\mathrm{T}})$, which is allowed in the infinite-layer fuds, $\mathcal{F}_{\infty,U_A,V_A}$. Also, if the derived volume of the fud exceeds the maximum derived volume limit, $|W^{\mathrm{C}}| > \mathrm{wmax}$, the computation of the independent derived, $(A * F^{\mathrm{T}})^{\mathrm{X}}$, necessary to compute the derived alignment, $\mathrm{algn}(A * F^{\mathrm{T}})$, may be impracticable. The discussion below considers various limited-layer limited-underlying limited-breadth infinite-layer substrate fud optimise functions and neighbourhood functions.

### Practicable shuffle content alignment valency-density fud inducer

An optimisation of the two stage computation of the substrate models, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, must (i) first compute a possibly overlapping fuds subset of the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds, $\mathrm{select}(T_A,N_A) \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}$, and (ii) then compute a non-overlapping fuds subset of these by filtering, $\{F : F \in \mathrm{select}(T_A,N_A),~\mathrm{nd}(F)\} \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$. In some definitions of the optimisation, computation time and space limits may constrain the cardinality of the possibly overlapping fuds, $|\mathrm{select}(T_A,N_A)|$, such that none are non-overlapping. That is, computational resources may be such that the filtered subset is empty, $\{F : F \in \mathrm{select}(T_A,N_A),~\mathrm{nd}(F)\} \subset \{F : F \in \mathrm{select}(T_A,N_A),~\neg \mathrm{overlap}(F)\} = \emptyset$. In this case, the maximum function of a practicable inducer would be undefined, $\mathrm{max}(I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘ * }(A)) = \emptyset$.

In section ‘Intractable literal substrate model inclusion’ P606 of the discussion on Tractable inducers, it is shown that the formal-abstract equality inclusion test, $A^{\mathrm{X}} * T = (A * T)^{\mathrm{X}}$, is intractable in the literal derived alignment integral-independent substrate ideal formal-abstract transform inducer, $I_{z,\mathrm{a,l}}^{‘}$, which computes the derived alignment, $\mathrm{algn}(A * T)$, for each of the formal-abstract-equal ideal substrate transforms, $\{T : T \in \mathcal{T}_{U_A,V_A},~A^{\mathrm{X}} * T = (A * T)^{\mathrm{X}},~A = A * T * T^{\dagger A}\}$. The discussion considers various alternatives before eventually concluding by defining the tractable derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$.

However, section ‘Practicable shuffles’ P663 considers the use of a shuffle histogram as practicable approximation to the independent, $A^{\mathrm{X}}$. The computation of an approximation to the formal alignment, $\mathrm{algn}(A^{\mathrm{X}} * F^{\mathrm{T}})$, is then practicable. Although there is no guarantee that a randomly chosen shuffle histogram has an alignment that is small, consider a practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, which, given substrate histogram $A \in \mathcal{A}_{z}$, is defined $\begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P}^{‘ * }(A) \subseteq \\ &&\hspace{2em}\{(F,(I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F)) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}\} \end{eqnarray}$ The scaled shuffle histogram, $A_R$, is defined $A_R = \mathrm{scalar}(1/|R|) * \sum_{r \in R} L_r$ where $X \in \mathrm{enums}(\mathrm{shuffles}(\mathrm{history}(A)))$, $L = \mathrm{map}(\mathrm{his},\mathrm{flip}(X))$, $R \subseteq \{1 \ldots z!^n\}$ and $n = |V_A|$. The shuffle indices, $R$, are in the practicable parameters, $R \in \mathrm{set}(P)$. The cardinality of the shuffle indices, $|R|$, is chosen such that the effective volume of the scaled shuffle histogram, $|A_R^{\mathrm{F}}| \leq |A^{\mathrm{XF}}|$, is practicable. In the case where the entire volume of the independent, $|A^{\mathrm{XF}}| = |V_A^{\mathrm{C}}|$, is practicable then $R = \{1 \ldots z!^n\}$ and $A_R = A^{\mathrm{X}}$. In this case the shuffle content alignment equals the content alignment.

The computation of the substrate models, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}$, in the practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, still takes place in two stages but the filtering need only test for limited-derived. That is (i) first compute the limited-layer limited-underlying limited-breadth infinite-layer substrate fuds, $\mathrm{select}(T_A,N_A) \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}$, and (ii) then compute a limited-derived subset of these by filtering, $\{F : F \in \mathrm{select}(T_A,N_A),~W=\mathrm{der}(F),~|W^{\mathrm{C}}| \leq \mathrm{wmax}\} \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}$. In terms of an implementation with a list maximiser define $Z_{P,A,A_R,\mathrm{csd}} = \mathrm{maximiseLister}(X_{P,A,A_R,\mathrm{csd}}, P_{P,A,A_R,\mathrm{csd}}, \mathrm{top}(\mathrm{omax}), R_{P,A,A_R,\mathrm{csd}})$ The limited-derived is $\{F : F \in \mathrm{dom}(\mathrm{elements}(Z_{P,A,A_R,\mathrm{csd}})),~W=\mathrm{der}(F),~|W^{\mathrm{C}}| \leq \mathrm{wmax}\} \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}$. The fuds optimise function, $X_{P,A,A_R,\mathrm{csd}}$, is related to content alignment valency-density, $(\mathrm{algn}(A * F^{\mathrm{T}})- \mathrm{algn}(A^{\mathrm{X}} * F^{\mathrm{T}}))/\mathrm{cvl}(F)$. Depending on available computational resources, the practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, may have lower correlation than the practicable derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q},P}^{‘}$.

The discussion below considers the optimisation in a practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$. The implementation of the optimisation is not restricted to a single optimiser such as the notional list maximiser $Z_{P,A,A_R,\mathrm{csd}}$.

### Limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer

Consider the limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘}$ which is implemented by means of a list maximiser $Z_{P,A,A_R,\mathrm{csd},\mathrm{p}}$ that has a neighbourhood function that constructs the fuds in the layer sequence of the paths of the limited-layer limited-derived-volume limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubhd}(U)(V) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$, described in section ‘Substrate models computation’. The fud tree constructs the fuds from tuple partition transforms but constrains the derived volume of the fuds by applying the limited-derived-volume constraint, $|W^{\mathrm{C}}| \leq \mathrm{wmax}$, at every layer, so only a subset of the limited-models infinite-layer substrate fuds is searched $\begin{eqnarray} \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{q}} \supseteq \mathrm{elements}(\mathrm{tfiubhd}(U)(V)) \end{eqnarray}$ The limited-path-models tuple partition list maximiser, $Z_{P,A,A_R,\mathrm{csd},\mathrm{p}}$, is constructed $\begin{eqnarray} &&Z_{P,A,A_R,\mathrm{csd},\mathrm{p}} = \\ &&\hspace{2em}\mathrm{maximiseLister}(X_{P,A,A_R,\mathrm{csd},\mathrm{p}}, P_{P,A,A_R,\mathrm{csd},\mathrm{p}}, \mathrm{top}(\mathrm{omax}), R_{P,A,A_R,\mathrm{csd},\mathrm{p}}) \end{eqnarray}$ The optimise function $X_{P,A,A_R,\mathrm{csd},\mathrm{p}} \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}} :\to \mathbf{Q}$, is the rational approximation to the shuffle content alignment valency-density valued total function of the limited-models infinite-layer substrate fuds, defined $X_{P,A,A_R,\mathrm{csd},\mathrm{p}} = \{(F,I_{\mathrm{csd}}^{ * }((A,A_R,F))) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}\}$ where the shuffle content alignment valency-density computer $I_{\mathrm{csd}} \in \mathrm{computers}$ is defined as (Haskell) $\begin{eqnarray} I_{\mathrm{csd}}^{ * }((A,A_R,F)) = (I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F) \end{eqnarray}$ The neighbourhood function $P_{P,A,A_R,\mathrm{csd},\mathrm{p}} \in \mathrm{P}(X_{P,A,A_R,\mathrm{csd},\mathrm{p}}) \to \mathrm{P}(X_{P,A,A_R,\mathrm{csd},\mathrm{p}})$, derived from the limited-layer limited-derived-volume limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubhd}(U)(V)$, is defined (Haskell) $\begin{eqnarray} &&P_{P,A,A_R,\mathrm{csd},\mathrm{p}}(Q) =\\ &&\hspace{2em}\{(F \cup G,I_{\mathrm{csd}}^{ * }((A,A_R,F \cup G))) : \\ &&\hspace{3em}(F,\cdot) \in Q,~\mathrm{layer}(F,\mathrm{der}(F)) < \mathrm{lmax},\\ &&\hspace{3em}G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V_A,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}}),~|P| \geq 2\},\\ &&\hspace{3em}1 \leq |G| \leq \mathrm{bmax},\\ &&\hspace{3em}W = \mathrm{der}(F \cup G), ~|W^{\mathrm{C}}| \leq \mathrm{wmax}\} \end{eqnarray}$ The definition of the neighbourhood function is stricter than the definition of the fud tree because only pluri-valent partitions are allowed, $|P| \geq 2$. This avoids the increase in capacity caused by the addition of mono-valent variables which increase the dimension but do not affect the alignment. The pluri-valent constraint also excludes empty tuples, $K = \emptyset$, because the unary partition variable is mono-valent, $|\{\emptyset^{\mathrm{CS}}\}|=1$.

The initial function, $R_{P,A,A_R,\mathrm{csd},\mathrm{p}} \subset X_{P,A,A_R,\mathrm{csd},\mathrm{p}}$, is a singleton of the empty fud, $R_{P,A,A_R,\mathrm{csd},\mathrm{p}} = \{(\emptyset,0)\}$.

Then the limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘}$, is defined, $\begin{eqnarray} I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘ * }(A) = \mathrm{elements}(Z_{P,A,A_R,\mathrm{csd},\mathrm{p}}) \subseteq X_{P,A,A_R,\mathrm{csd},\mathrm{p}} \end{eqnarray}$ The fud inducer is defined without the need for filtering the elements of the list maximiser because (i) the maximisation of the shuffle content alignment tends to minimise the degree of overlap, and (ii) the list maximiser limits the fud derived volume.

The limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘}$, has some limitations. First, in some cases only a subset of the limited-models infinite-layer substrate fuds is traversable $\begin{eqnarray} \mathrm{dom}(I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘ * }(A)) \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}} \end{eqnarray}$ This is because the derived volumes of the intermediate layers of the fud are limited, as well as the derived volume of the top layer.

Secondly, this limitation of the intermediate layers may restrict the derived dimensions of these layers to be considerably less than the dimension of the substrate. For example, in the case where the maximum derived volume equals the sample size, $\mathrm{wmax} = z$, then a size of $10000$ implies a maximum breadth of at most $\mathrm{bmax} = \lfloor\ln \mathrm{wmax}/\ln 2\rfloor = 13$, where the derived valency is at least two. Maximum alignment is approximately $z(n-1)\ln d$ and so the ratio of the maximum alignment of an intermediate layer to the maximum alignment of the substrate is roughly equal to the ratio of their dimensions. For example, if the substrate dimension is $n=26$, then the ratio of the maximum alignments is roughly half. This is also true for maximum alignment valency-density. Of course, this is also the case for the top layer, not just intermediate layers, but the summation alignment of the decomposition partially compensates for this.

Thirdly, as described in section ‘Substrate models computation’ P641, in a certain special case of a fud $F \in \mathrm{dom}(\mathrm{elements}(Z_{P,A,A_R,\mathrm{csd},\mathrm{p}}))$ which is such that the cardinality of the variables is $|\mathrm{vars}(F) \cup V| = \mathrm{lmax} \times \mathrm{bmax}$, it can be shown that the cardinality of the set of next layer fuds is bounded, $\begin{eqnarray} |\mathcal{N}_{U,W,X} \cap \mathcal{N}_{U,W,\mathrm{xmax}} \cap \mathcal{N}_{U,W,\mathrm{bmax}}| < ((\mathrm{lmax} \times \mathrm{bmax})^{\underline{\mathrm{kmax}}} \times \mathrm{bell}(\mathrm{xmax}))^{\underline{\mathrm{bmax}}} \end{eqnarray}$ where the implied maximum underlying dimension is $\mathrm{kmax} = \lceil\ln \mathrm{xmax}~/ \ln d\rceil$. So the upper bound on the cardinality of the neighbourhood function is also bounded $|P_{P,A,A_R,\mathrm{csd},\mathrm{p}}(\{(F,X_{P,A,A_R,\mathrm{csd},\mathrm{p}}(F))\})| < ((\mathrm{lmax} \times \mathrm{bmax})^{\underline{\mathrm{kmax}}} \times \mathrm{bell}(\mathrm{xmax}))^{\underline{\mathrm{bmax}}}$ In the case where $\mathrm{xmax} = \mathrm{wmax} = z$ the second term, $\mathrm{bell}(\mathrm{xmax})$, equals $\mathrm{bell}(z)$, which is impracticable except for very small sizes. The first term, $(\mathrm{lmax} \times \mathrm{bmax})^{\underline{\mathrm{kmax}}}$, has complexity $(\ln z)!$, but the second term, $\mathrm{bell}(\mathrm{xmax})$, has complexity $z!$. So while the limited-path-models tuple partition practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{p}}^{‘}$, strongly constrains the intermediate layer derived volume, the tuple partition cardinality is only weakly constrained.

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