# Aligned Induction

## Fud Decomposition Inducers

Haskell commentary on the implementation of Tractable and Practicable Inducers/Fud Decomposition Inducers

### Sections

Practicable summed shuffle content alignment valency-density fud decomposition inducer

Practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer

Practicable level summed shuffle content alignment valency-density fud decomposition inducer

Practicable limited-nodes summed shuffle content alignment valency-density fud decomposition inducer

Practicable label-entropy limited-nodes summed shuffle content alignment valency-density fud decomposition inducer

Practicable label-mode limited-nodes summed shuffle content alignment valency-density fud decomposition inducer

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

As shown above, the use of a shuffle histogram, $A_R$, is a practicable approximation to the independent, $A^{\mathrm{X}}$, in the practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$. This allows optimisations to avoid a two stage (i) search of possibly overlapping fuds, $\mathrm{select}(T_A,N_A) \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}$, followed by (ii) filtering of non-overlapping fuds, $\{F : F \in \mathrm{select}(T_A,N_A),~\mathrm{nd}(F)\} \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$. The optimisers do this by maximisation of the shuffle content alignment valency-density to construct fuds that approximate loosely to recursively non-overlapping pluri-partition fuds, from which an approximately non-overlapping top transform can be chosen. The same reasoning may be extended to a fud decomposition inducer. Here a shuffle histogram is constructed from each of the slices of the fuds of the decomposition. Each shuffle approximates to the independent of the contingent sample. Redefine the shuffle indices, $R_A \subseteq \{1 \ldots z_A!^{n_A}\}$, where $A \in \mathcal{A}_{z(A)}$, $z_A = \mathrm{size}(A)$ and $n_A = |\mathrm{vars}(A)|$. Redefine the scaled shuffle histogram, $A_{R(A)} = \mathrm{scalar}(1/|R(A)|) * \sum_{r \in R(A)} L_A(r)$ where $X_A \in \mathrm{enums}(\mathrm{shuffles}(\mathrm{history}(A)))$ and $L_A = \mathrm{map}(\mathrm{his},\mathrm{flip}(X_A))$. Then the scaled contingent shuffle histogram is $(A * C)_{R(A * C)} \approx (A * C)^{\mathrm{X}}$, where $(\cdot,C) \in \mathrm{cont}(D)$ and $D \in \mathcal{D}_{\mathrm{F},\infty,U_A,V_A}$.

The practicable summed shuffle content alignment valency-density fud decomposition inducer, which, given substrate histogram $A \in \mathcal{A}_z$, is constrained $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P}^{‘ * }(A) \subseteq \\ &&\hspace{2em}\{(D,I_{\mathrm{Scsd}}^{ * }((A,D))) : \\ &&\hspace{3em}D \in \mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times \mathcal{F}_{\mathrm{q}}),\\ &&\hspace{5em}\forall (C,F) \in \mathrm{cont}(D)\\ &&\hspace{7em}(I_{\mathrm{a}}^{ * }(A * C * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }((A * C)_{R(A * C)} * F^{\mathrm{T}})>0)\} \cup \\ &&\hspace{2em}\{(D_{\emptyset},0)\} \end{eqnarray}$ where $D_{\emptyset} = \{((\emptyset,\emptyset),\emptyset)\}$ and the summed shuffle content alignment valency-density computer $I_{\mathrm{Scsd}} \in \mathrm{computers}$ is defined as $\begin{eqnarray} &&I_{\mathrm{Scsd}}^{ * }((A,D)) = \\ &&\hspace{2em}\sum (I_{\mathrm{a}}^{ * }(A * C * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }((A * C)_{R(A * C)} * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F) : (C,F) \in \mathrm{cont}(D) \end{eqnarray}$ In some cases the practicable inducer optimisation may be empty. For example, the independent substrate histogram, $A^{\mathrm{X}}$, cannot have non-zero positive summation content alignment. The maximum function, $\mathrm{maxr}(I_{z,\mathrm{Scsd,D,F,\infty,q},P}^{‘ * }(A^{\mathrm{X}}))$, would therefore be undefined for some inducer domain substrate histograms. In order to have well defined maximum correlation, the practicable inducer is therefore stuffed with the empty decomposition, $D_{\emptyset} \in \mathcal{D}_{\mathrm{F}}$, in the case of empty optimisation.

Each non-zero positive shuffle content alignment valency-density fud decomposition of the application of the practicable fud decomposition inducer, $I_{z,\mathrm{Scsd,D,F,\infty,q},P}^{‘}$, is related to the computation of fuds of the slices in the practicable fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, which is defined in terms of the limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,A,A_R,\mathrm{L}}$, and the limited-derived derived variables set list maximiser, $Z_{P,A,A_R,F,\mathrm{D}}$, $\begin{eqnarray} &&\forall (D,a) \in I_{z,\mathrm{Scsd,D,F,\infty,q},P}^{‘ * }(A)~(a >0 \implies \\ &&\hspace{2em}(\forall (C,F) \in \mathrm{cont}(D)~\forall F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,A * C,(A * C)_{R(A * C)},\mathrm{L}}))\\ &&\hspace{3em}\exists K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A * C,(A * C)_{R(A * C)},F_{\mathrm{L}},\mathrm{D}}))~(F = \mathrm{depends}(F_{\mathrm{L}},K)))) \end{eqnarray}$ Note that the practicable summed shuffle content alignment valency-density fud decomposition inducer, $I_{z,\mathrm{Scsd,D,F,\infty,q},P}^{‘}$, is defined such that the contingent histogram, $A * C$, is shuffled, $(A * C)_{R(A * C)}$, rather taking the contingent of the shuffled histogram, $A_{R(A)} * C$. If the contingent shuffle histogram were used then any biases for or against the alignment in the shuffle, $A_{R(A)}$, in the parent slice would be safely removed. However, the size of the contingent shuffle histogram is not necessairly equal to that of the contingent sample, $\mathrm{size}(A_{R(A)} * C) \neq \mathrm{size}(A * C)$, and so scaling is often necessary. This would not be a disadvantage if it were not the case that typically the entropy of the parent derived histogram of the contingent shuffle is greater than that of the sample, and so the contingent shuffle slice size tends to zero much more quickly than the sample slice size. Therefore the scaling factor is often large, making the contingent shuffle less effective as an approximation to the independent slice, $(A * C)^{\mathrm{X}}$.

In section ‘Substrate models computation’ above, the finite limited-models infinite-layer fud decomposition tree, $\mathrm{tdfiq}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{D}_{\mathrm{F,d}} \to \mathrm{trees}(\mathcal{D}_{\mathrm{F,d}})$, is a tree of immediate super-decompositions of limited-models infinite-layer substrate fuds. The decompositions of the tree are a subset the limited-models infinite-layer substrate fud decompositions $\mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times \mathcal{F}_{\mathrm{q}}) \supset \mathrm{elements}(\mathrm{tdfiq}(U)(V))$ The limited-models infinite-layer substrate fud decompositions tree searcher chooses a sublist of a path of immediate super-decompositions from the limited-models infinite-layer fud decomposition tree. Define the limited-models infinite-layer substrate fud decompositions tree searcher $Z_{P,A,\mathrm{D,F}} = \mathrm{searchTreer}(\mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times \mathcal{F}_{\mathrm{q}}),P_{P,A,\mathrm{D,F}},R_{P,A,\mathrm{D,F}})$ where the neighbourhood function returns a singleton $\begin{eqnarray} &&P_{P,A,\mathrm{D,F}}(D) = \{E : \\ &&\hspace{2em}(\cdot,S,G,L) \in \mathrm{maxd}(\mathrm{order}(D_{\mathbf{Q} \times \mathrm{S} \times \mathcal{X}^2},\{(\mathrm{size}(B),S,G,L) : \\ &&\hspace{4em}(L,Y) \in \mathrm{places}(D),\\ &&\hspace{4em}R_L = \bigcup \mathrm{dom}(\mathrm{set}(L)),~H_L = \bigcup \mathrm{ran}(\mathrm{set}(L)),\\ &&\hspace{4em}(\cdot,F) = L_{|L|},~W=\mathrm{der}(F),\\ &&\hspace{4em}S \in W^{\mathrm{CS}} \setminus \mathrm{dom}(\mathrm{dom}(Y)),\\ &&\hspace{4em}B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A),~\mathrm{size}(B)>0,\\ &&\hspace{4em}F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,B,B_{R(B)},\mathrm{L}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,B,B_{R(B)},F_{\mathrm{L}},\mathrm{D}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\})),\\ &&\hspace{2em}M=L \cup \{(|L|+1,(S,G))\},\\ &&\hspace{2em}E = \mathrm{tree}(\mathrm{paths}(D) \setminus \{L\} \cup \{M\})\} \end{eqnarray}$ where $\begin{eqnarray} &&R_{P,A,\mathrm{D,F}} = \{\{((\emptyset,G),\emptyset)\} : \\ &&\hspace{2em}G \in \mathrm{maxd}(\mathrm{order}(D_{\mathrm{F}},\{G : \\ &&\hspace{4em}F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_{R(A)},\mathrm{L}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,A,A_{R(A)},F_{\mathrm{L}},\mathrm{D}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\}))\} \end{eqnarray}$ The computation of the slice $B$ is a tractable fud application equivalent to the application of the fud’s transforms’ histograms, $\mathrm{his}(H_L)$, multiplied by the slice derived state, $\{R_L \cup S\}^{\mathrm{U}}$, followed by reduction to the substrate, $V_A$, $\mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A) = A * \prod \mathrm{his}(H_L) * \{R_L \cup S\}^{\mathrm{U}}~\%~V_A$ The neighbourhood function $P_{P,A,\mathrm{D,F}}(D)$ returns an empty set or a singleton super-decomposition with an additional slice having non-zero positive shuffle content alignment. The order $D_{\mathbf{Q} \times \mathrm{S} \times \mathcal{X}^2}$ selects by slice size and then arbitrarily. Together the orders $D_{\mathbf{Q} \times \mathrm{S} \times \mathcal{X}^2}$ and $D_{\mathrm{F}}$ ensure that the fud decomposition is distinct. The tree of the limited-models infinite-layer substrate fud decompositions tree searcher has at most one path, $|\mathrm{paths}(\mathrm{tree}(Z_{P,A,\mathrm{D,F}}))| \leq 1$, and hence the tree has at most one leaf, $|\mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F}}))| \leq 1$. If the path exists, $\{L\} = \mathrm{paths}(\mathrm{tree}(Z_{P,A,\mathrm{D,F}}))$, it is in the limited-models infinite-layer fud decomposition tree, $L \in \mathrm{subpaths}(\mathrm{tdfiq}(U_A)(V_A,\emptyset))$.

The practicable summed shuffle content alignment valency-density fud decomposition inducer may then be implemented $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F}})),~\{D\} = Q \end{eqnarray}$

### Practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer

The highest-layer limited-models infinite-layer substrate fud decompositions tree searcher, $Z_{P,A,\mathrm{D,F,d}}$, is defined exactly the same as the limited-models infinite-layer substrate fud decompositions tree searcher, $Z_{P,A,\mathrm{D,F}}$, except that it depends instead on the highest-layer limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,B,B_R,\mathrm{L,d}}$, and the highest-layer limited-derived derived variables set list maximiser, $Z_{P,B,B_R,F_{\mathrm{L}},\mathrm{D,d}}$.

The practicable highest-layer summed shuffle content alignment valency-density fud decomposition inducer is implemented (Haskell) $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{d}}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F,d}})),~\{D\} = Q \end{eqnarray}$

### Practicable level summed shuffle content alignment valency-density fud decomposition inducer

The level limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}}$, of the practicable level shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,V_{\mathrm{g}},F_{\mathrm{g}}}^{‘}$, can be used to implement a fud decomposition inducer parameterised by a heritable tree of levels. Let $Z_{\mathrm{g}} \in \mathrm{trees}(\mathbf{N}_{>0} \times \mathrm{P}(\mathcal{V}) \times \mathcal{F})$ be the level hierarchy. A node $((\mathrm{wmax}_{\mathrm{g}}, V_{\mathrm{g}}, F_{\mathrm{g}}),X_{\mathrm{g}}) \in \mathrm{nodes}(Z_{\mathrm{g}})$ parameterises the node’s level fud tree searcher, $Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}}$, with (i) the maximum derived volume, $\mathrm{wmax}_{\mathrm{g}}$, (ii) the subset of the substrate, $V_{\mathrm{g}}$, and (iii) the union of (a) the level fud, $F_{\mathrm{g}}$, and (b) the level fuds from the application of recursively parameterised level fud tree searchers of the children nodes, $X_{\mathrm{g}}$. The level hierarchy has various uses including (i) the partitioning of a large substrate, for example into local regions implied by an external metric, so that the resultant fud has complete coverage, (ii) allowing multiple overlapping representations of a small substrate, (iii) hinting derived variables of the substrate that are externally known to be in alignments, and (iv) excluding mono-valent substrate variables that sometimes occur near the leaves of a decomposition.

Define the level limited-models infinite-layer substrate fud decompositions tree searcher $Z_{P,A,\mathrm{D,F,g}} = \mathrm{searchTreer}(\mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times \mathcal{F}_{\mathrm{q}}),P_{P,A,\mathrm{D,F,g}},R_{P,A,\mathrm{D,F,g}})$ where the parameters includes the level hierarchy tree, $Z_{\mathrm{g}} \in \mathrm{set}(P)$, where $Z_{\mathrm{g}} \in \mathrm{trees}(\mathbf{N}_{>0} \times \mathrm{P}(V_A) \times (\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}))$. The neighbourhood function is defined $\begin{eqnarray} &&P_{P,A,\mathrm{D,F,g}}(D) = \{E : \\ &&\hspace{2em}(\cdot,S,G,L) \in \mathrm{maxd}(\mathrm{order}(D_{\mathbf{Q} \times \mathrm{S} \times \mathcal{X}^2},\{(\mathrm{size}(B),S,G,L) : \\ &&\hspace{4em}(L,Y) \in \mathrm{places}(D),\\ &&\hspace{4em}R_L = \bigcup \mathrm{dom}(\mathrm{set}(L)),~H_L = \bigcup \mathrm{ran}(\mathrm{set}(L)),\\ &&\hspace{4em}(\cdot,F) = L_{|L|},~W=\mathrm{der}(F),\\ &&\hspace{4em}S \in W^{\mathrm{CS}} \setminus \mathrm{dom}(\mathrm{dom}(Y)),\\ &&\hspace{4em}B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A),~\mathrm{size}(B)>0,\\ &&\hspace{4em}G = \mathrm{level}(B,B_{R(B)})(Z_{\mathrm{g}}),~G \neq \emptyset\})),\\ &&\hspace{2em}M=L \cup \{(|L|+1,(S,G))\},\\ &&\hspace{2em}E = \mathrm{tree}(\mathrm{paths}(D) \setminus \{L\} \cup \{M\})\} \end{eqnarray}$ where $\begin{eqnarray} R_{P,A,\mathrm{D,F,g}} = \{\{((\emptyset,G),\emptyset)\} : G \in \mathrm{maxd}(\mathrm{order}(D_{\mathrm{F}},\mathrm{level}(A,A_{R(A)})(Z_{\mathrm{g}})))\} \end{eqnarray}$ and $\begin{eqnarray} &&\mathrm{level}(A,A_R) \in \mathrm{trees}(\mathbf{N}_{>0} \times \mathrm{P}(V_A) \times (\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}})) \to \\ &&\hspace{15em}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}) \end{eqnarray}$ is defined $\begin{eqnarray} &&\mathrm{level}(A,A_R)(Z_{\mathrm{g}}) = \bigcup \{G : \\ &&\hspace{2em}((\mathrm{wmax}_{\mathrm{g}}, V_{\mathrm{g}}, F_{\mathrm{g}}),X_{\mathrm{g}}) \in Z_{\mathrm{g}},\\ &&\hspace{2em}F_{\mathrm{h}} = \mathrm{level}(A,A_R)(X_{\mathrm{g}}),~\mathrm{wmax}_{\mathrm{g}} \in \mathrm{set}(P_{\mathrm{g}}),\\ &&\hspace{2em}F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P_{\mathrm{g}},A,A_R,V_{\mathrm{g}},F_{\mathrm{g}} \cup F_{\mathrm{h}},\mathrm{L}})),\\ &&\hspace{2em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P_{\mathrm{g}},A,A_R,F_{\mathrm{g}} \cup F_{\mathrm{h}},F_{\mathrm{L}},\mathrm{D}})),~a>0,\\ &&\hspace{2em}G = \mathrm{depends}(F_{\mathrm{L}},K)\} \end{eqnarray}$ The practicable level summed shuffle content alignment valency-density fud decomposition inducer may then be implemented (Haskell) $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{g}}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F,g}})),~\{(D,\cdot)\} = Q \end{eqnarray}$

### Practicable limited-nodes summed shuffle content alignment valency-density fud decomposition inducer

The limited-nodes limited-models infinite-layer substrate fud decompositions tree searcher, $Z_{P,A,\mathrm{D,F,f}}$, is a variation of the limited-models infinite-layer substrate fud decompositions tree searcher, $Z_{P,A,\mathrm{D,F}}$, in which the cardinality of the fuds of the decomposition tree is limited to the maximum fuds limit $\mathrm{fmax} \in \mathbf{N}_{>0}$. The neighbourhood function $P_{P,A,\mathrm{D,F,f}}$ is modified $\begin{eqnarray} &&P_{P,A,\mathrm{D,F,f}}(D) = \{E : \\ &&\hspace{2em}|\mathrm{nodes}(D)| < \mathrm{fmax},\\ &&\hspace{2em}(\cdot,S,G,L) \in \mathrm{maxd}(\mathrm{order}(D_{\mathbf{Q} \times \mathrm{S} \times \mathcal{X}^2},\{(\mathrm{size}(B),S,G,L) : \\ &&\hspace{4em}(L,Y) \in \mathrm{places}(D),\\ &&\hspace{4em}R_L = \bigcup \mathrm{dom}(\mathrm{set}(L)),~H_L = \bigcup \mathrm{ran}(\mathrm{set}(L)),\\ &&\hspace{4em}(\cdot,F) = L_{|L|},~W=\mathrm{der}(F),\\ &&\hspace{4em}S \in W^{\mathrm{CS}} \setminus \mathrm{dom}(\mathrm{dom}(Y)),\\ &&\hspace{4em}B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A),~\mathrm{size}(B)>0,\\ &&\hspace{4em}F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,B,B_{R(B)},\mathrm{L}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,B,B_{R(B)},F_{\mathrm{L}},\mathrm{D}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\})),\\ &&\hspace{2em}M=L \cup \{(|L|+1,(S,G))\},\\ &&\hspace{2em}E = \mathrm{tree}(\mathrm{paths}(D) \setminus \{L\} \cup \{M\})\} \end{eqnarray}$ The practicable limited-nodes summed shuffle content alignment valency-density fud decomposition inducer is implemented $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{f}}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F,f}})),~\{D\} = Q \end{eqnarray}$

### Practicable label-entropy limited-nodes summed shuffle content alignment valency-density fud decomposition inducer

A further variation of the limited-nodes limited-models infinite-layer substrate fud decompositions tree searcher, $Z_{P,A,\mathrm{D,F,f}}$, is to modify the sequence of fud search and the termination condition in order to minimise label entropy. Let the query variables $V_{\mathrm{Q}} \subset V_A$ be a proper subset of the substrate variables, $V_{\mathrm{Q}} \neq V_A$. The difference forms the label variables $V_{\mathrm{L}} = V_A \setminus V_{\mathrm{Q}}$. Here the modelling is restricted to the query variables, $V_{\mathrm{Q}}$, so that the underlying variables of the decomposition $D$ are a subset, $\mathrm{und}(D) \subseteq V_{\mathrm{Q}}$. Given a query histogram $Q \in \mathcal{A}_U$ in the query variables, $\mathrm{vars}(Q) = V_{\mathrm{Q}}$, the modelled transformed conditional product is a probability histogram if $(Q * D^{\mathrm{T}})^{\mathrm{F}} \cap (A * D^{\mathrm{T}})^{\mathrm{F}} \neq \emptyset$, $(Q * D^{\mathrm{T}} * \mathrm{his}(D^{\mathrm{T}}) * A)^{\wedge}~\%~V_{\mathrm{L}} \in \mathcal{A} \cap \mathcal{P}$ where $\mathrm{his}=\mathrm{histogram}$.

The slice histogram of the neighbourhood function is restricted to the query variables, $B\%V_{\mathrm{Q}}$, where the slice histogram is $B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A)$. The sized label entropy of the slice is defined as $\mathrm{size}(B) * \mathrm{entropy}(B\%V_{\mathrm{L}})$ If the slice is an effective singleton in the label variables, $|(B\%V_{\mathrm{L}})^{\mathrm{F}}|=1$, then the sized label entropy is zero, $\mathrm{entropy}(B\%V_{\mathrm{L}}) = 0$.

The label-entropy limited-nodes limited-models infinite-layer substrate fud decompositions tree searcher, $Z_{P,A,\mathrm{D,F,f,e},V_{\mathrm{L}}}$, is such that (i) the limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,A,A_R,\mathrm{L}}$, and the limited-derived derived variables set list maximiser, $Z_{P,A,A_R,F,\mathrm{D}}$, are restricted to the query variables, $V_{\mathrm{Q}}$, (ii) the order of fud decomposition is modified to maximise slice label entropy and then slice size, and (iii) the decomposition of a slice with zero label entropy terminates. The neighbourhood function $P_{P,A,\mathrm{D,F,f,e},V_{\mathrm{L}}}$ is modified $\begin{eqnarray} &&P_{P,A,\mathrm{D,F,f,e},V_{\mathrm{L}}}(D) = \{E : \\ &&\hspace{2em}|\mathrm{nodes}(D)| < \mathrm{fmax},\\ &&\hspace{2em}(\cdot,S,G,L) \in \mathrm{maxd}(\mathrm{order}(D_{\mathbf{Q}^2 \times \mathrm{S} \times \mathcal{X}^2},\{((e_B,z_B),S,G,L) : \\ &&\hspace{4em}(L,Y) \in \mathrm{places}(D),\\ &&\hspace{4em}R_L = \bigcup \mathrm{dom}(\mathrm{set}(L)),~H_L = \bigcup \mathrm{ran}(\mathrm{set}(L)),\\ &&\hspace{4em}(\cdot,F) = L_{|L|},~W=\mathrm{der}(F),\\ &&\hspace{4em}S \in W^{\mathrm{CS}} \setminus \mathrm{dom}(\mathrm{dom}(Y)),\\ &&\hspace{4em}B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A),\\ &&\hspace{4em}z_B = \mathrm{size}(B),~e_B = z_B * \mathrm{entropy}(B\%V_{\mathrm{L}}),~e_B>0,\\ &&\hspace{4em}B’ = B\%(V_A \setminus V_{\mathrm{L}}),~F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,B’,B’_{R(B’)},\mathrm{L}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,B’,B’_{R(B’)},F_{\mathrm{L}},\mathrm{D}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\})),\\ &&\hspace{2em}M=L \cup \{(|L|+1,(S,G))\},\\ &&\hspace{2em}E = \mathrm{tree}(\mathrm{paths}(D) \setminus \{L\} \cup \{M\})\} \end{eqnarray}$ The practicable label-entropy limited-nodes summed shuffle content alignment valency-density fud decomposition inducer is implemented $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{f,e},V_{\mathrm{L}}}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F,f,e},V_{\mathrm{L}}})),~\{D\} = Q \end{eqnarray}$

### Practicable label-mode limited-nodes summed shuffle content alignment valency-density fud decomposition inducer

A similar method is to modify the sequence of fud search and the termination condition in order to minimise non-modal label size. The non-modal label size of the slice is defined as $\mathrm{size}(B) - \mathrm{maxr}(B\%V_{\mathrm{L}})$ The neighbourhood function $P_{P,A,\mathrm{D,F,f,m},V_{\mathrm{L}}}$ is modified $\begin{eqnarray} &&P_{P,A,\mathrm{D,F,f,m},V_{\mathrm{L}}}(D) = \{E : \\ &&\hspace{2em}|\mathrm{nodes}(D)| < \mathrm{fmax},\\ &&\hspace{2em}(\cdot,S,G,L) \in \mathrm{maxd}(\mathrm{order}(D_{\mathbf{Q}^2 \times \mathrm{S} \times \mathcal{X}^2},\{((m_B,z_B),S,G,L) : \\ &&\hspace{4em}(L,Y) \in \mathrm{places}(D),\\ &&\hspace{4em}R_L = \bigcup \mathrm{dom}(\mathrm{set}(L)),~H_L = \bigcup \mathrm{ran}(\mathrm{set}(L)),\\ &&\hspace{4em}(\cdot,F) = L_{|L|},~W=\mathrm{der}(F),\\ &&\hspace{4em}S \in W^{\mathrm{CS}} \setminus \mathrm{dom}(\mathrm{dom}(Y)),\\ &&\hspace{4em}B = \mathrm{apply}(V_A,V_A,\mathrm{his}(H_L) \cup \{\{R_L \cup S\}^{\mathrm{U}}\},A),\\ &&\hspace{4em}z_B = \mathrm{size}(B),~m_B = z_B-\mathrm{maxr}(B\%V_{\mathrm{L}}),~m_B>0,\\ &&\hspace{4em}B’ = B\%(V_A \setminus V_{\mathrm{L}}),~F_{\mathrm{L}} \in \mathrm{leaves}(\mathrm{tree}(Z_{P,B’,B’_{R(B’)},\mathrm{L}})),\\ &&\hspace{4em}(K,a) \in \mathrm{max}(\mathrm{elements}(Z_{P,B’,B’_{R(B’)},F_{\mathrm{L}},\mathrm{D}})),~a>0,\\ &&\hspace{4em}G = \mathrm{depends}(F_{\mathrm{L}},K)\})),\\ &&\hspace{2em}M=L \cup \{(|L|+1,(S,G))\},\\ &&\hspace{2em}E = \mathrm{tree}(\mathrm{paths}(D) \setminus \{L\} \cup \{M\})\} \end{eqnarray}$ The practicable label-mode limited-nodes summed shuffle content alignment valency-density fud decomposition inducer is implemented $\begin{eqnarray} &&I_{z,\mathrm{Scsd,D,F,\infty,q},P,\mathrm{f,m},V_{\mathrm{L}}}^{‘ * }(A) = \\ &&\hspace{2em}\mathrm{if}(Q \neq \emptyset, \{(D,I_{\mathrm{Scsd}}^{ * }((A,D)))\},\{(D_{\emptyset},0)\}) :\\ &&\hspace{5em}Q = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,\mathrm{D,F,f,m},V_{\mathrm{L}}})),~\{D\} = Q \end{eqnarray}$

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