Fud Inducers

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

Sections

Practicable shuffle content alignment valency-density fud inducer

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

Practicable maximum-roll shuffle content alignment valency-density fud inducer

Practicable maximum-roll-by-derived-dimension shuffle content alignment valency-density fud inducer

Practicable excluded-self shuffle content alignment valency-density fud inducer

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

Practicable level shuffle content alignment valency-density fud inducer

Limited inducers

Practicable shuffle content alignment valency-density fud inducer

Given the single content alignment optimised next limited-underlying limited-breadth layer tuple set, $B_{\mathrm{B}} = \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}}))$, from the limited-underlying tuple set list maximiser, $Z_{P,A,A_R,F,\mathrm{B}}$, each tuple $K$ of the tuple set, $B_{\mathrm{B}}$, can be optimised in a contracted decrementing linear non-overlapping fuds list maximiser, $Z_{P,A,A_R,F,\mathrm{n},-,K}$, to construct the single content alignment optimised next limited-underlying limited-breadth layer.

The limited-layer limited-underlying limited-breadth fud tree searcher creates a path of layer-cumulative fuds of length $\mathrm{lmax}$. Define the limited-layer limited-underlying limited-breadth fud tree searcher (Haskell) \[ Z_{P,A,A_R,\mathrm{L}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L}},\{\emptyset\}) \] where the neighbourhood function returns a singleton \[ \begin{eqnarray} &&P_{P,A,A_R,\mathrm{L}}(F) = \{G :\\ &&\hspace{2em}G = F \cup \{T : K \in \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}})),\\ &&\hspace{5em}H \in \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})),\\ &&\hspace{5em}w \in \mathrm{der}(H),~I = \mathrm{depends}(\mathrm{explode}(H),\{w\}),~T = I^{\mathrm{TPT}}\},\\ &&\hspace{2em}\mathrm{layer}(G, \mathrm{der}(G)) \leq \mathrm{lmax}\} \end{eqnarray} \] If the substrate variables are pluri-variate, $|V_A| > 1$, the tree of the limited-layer limited-underlying limited-breadth fud tree searcher has a single path, $|\mathrm{paths}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}}))| = 1$, and a single leaf, $|\mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}}))|=1$.

Note that in some cases a partition transform, $I^{\mathrm{TPT}}$, may already exist in the fud, $F$, because only one variable of the tuple, $K$, need be in the fud derived variables, $|K \cap \mathrm{der}(F)| \geq 1$, and so some components of the partition of the tuple may consist of variables in lower layers of the fud, $J \cap \mathrm{der}(F) = \emptyset$ where $J = \mathrm{und}(I) \subset K$. Furthermore, if, after the first layer, a partition $I_1^{\mathrm{TP}}$ already exists in the fud, $I_1^{\mathrm{TP}} \in \mathrm{vars}(F)$, and is not a derived variable, $I_1^{\mathrm{TP}} \notin \mathrm{der}(F)$, it may sometimes be hidden by another variable $I_2^{\mathrm{TP}}$. That is, $I_1^{\mathrm{TP}} \in \mathrm{vars}(\mathrm{depends}(F,\mathrm{und}(I_2)))$. It is therefore possible that the succeeding fud, $G$, may, in some cases, contain a single derived variable, $|\mathrm{der}(G)|=1$, and consequently be independent, $\mathrm{algn}(A * G^{\mathrm{T}})=0$. This limitation is due to the separation of the optimisation into two steps, (i) tuple set list maximisation, followed by (ii) decrementing fuds list maximisation.

Implementations of the neighbourhood function that do not use partition variables, $F \notin \mathcal{F}_{U,\mathrm{P}}$, must explicitly check for uniqueness, $I^{\mathrm{TP}} \notin \{T^{\mathrm{P}} : T \in F\}$.

If (i) each content alignment optimised next limited-underlying limited-breadth layer tuple set has cardinality less than or equal to the maximum layer breadth limit, $|B_{\mathrm{B}}| \leq \lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor$, and (ii) each tuple $K \in B_{\mathrm{B}}$ has contracted decrementing linear non-overlapping fuds list maximiser cardinality of less than or equal to the maximum tuple optimise limit, $\mathrm{pmax}$, then the cardinality of each additional layer of the fuds in the path is less than or equal to the maximum optimise step cardinality, $\mathrm{omax} = \mathrm{bmax} \times \mathrm{pmax}$. That is, $|\mathrm{der}(F)| \leq \mathrm{omax}$ where $F \in \mathrm{elements}(Z_{P,A,A_R,\mathrm{L}})$.

If the substrate variables are pluri-variate, $|V_A| > 1$, the optimised limited-layer limited-underlying limited-breadth fud $F_{\mathrm{L}}$ of layer $\mathrm{lmax}$ is the leaf \[ \begin{eqnarray} \{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}})) \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}} \end{eqnarray} \] If the optimised limited-layer limited-underlying limited-breadth fud, $F_{\mathrm{L}}$, exists, it has at least two variables, $|\mathrm{vars}(F_{\mathrm{L}}) \setminus V_A| > 1$.

Now the filtering step is computed by constructing pluri-partition transforms of the fud variables one variable at a time. Define the limited-derived derived variables set list maximiser \[ Z_{P,A,A_R,F,\mathrm{D}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{D}},P_{P,A,A_R,F,\mathrm{D}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{D}}) \] where (i) the optimiser function is \[ \begin{eqnarray} &&X_{P,A,A_R,F,\mathrm{D}} = \{(K,(I_{\mathrm{a}}^{ * }(A * G^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * G^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(G)) : \\ &&\hspace{10em}K \subseteq \mathrm{vars}(F),~K \neq \emptyset,~G = \mathrm{depends}(F,K)\} \end{eqnarray} \] (ii) the neighbourhood function is (Haskell) \[ \begin{eqnarray} &&P_{P,A,A_R,F,\mathrm{D}}(D) = \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in D,~w \in \mathrm{vars}(F) \setminus V_A \setminus K,\\ &&\hspace{2em}J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{wmax},~\mathrm{der}(\mathrm{depends}(F,J)) = J\} \end{eqnarray} \] and (iii) the initial subset is (Haskell) \[ \begin{eqnarray} &&R_{P,A,A_R,F,\mathrm{D}} = \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{2em}w,u \in \mathrm{vars}(F) \setminus V_A,~u \neq w,\\ &&\hspace{2em}J = \{w,u\},~|J^{\mathrm{C}}| \leq \mathrm{wmax},~\mathrm{der}(\mathrm{depends}(F,J)) = J\} \end{eqnarray} \] The limited-derived derived variables set list maximiser has no elements if the fud is empty, $\mathrm{elements}(Z_{P,A,A_R,\emptyset,\mathrm{D}}) = \emptyset$, or if it consists of a single partition transform, $|F|=1$.

The derived variables sets are such that none of the derived variables are nested in the depends fud variables of another derived variable in the same set, $\forall w,u \in J~(w \neq u \implies u \notin \mathrm{vars}(\mathrm{depends}(F,\{w\})))$, so that $J = \mathrm{der}(\mathrm{depends}(F,J))$. This restriction prevents unnecessary searches in the optimiser where the derived variables of the dependent fud are a proper subset, $\mathrm{der}(\mathrm{depends}(F,J)) \subset J$. However, hidden variables that are excluded in a fud are not necessarily excluded in another lower layer fud that does not contain the dependent variable.

The limited-derived derived variables set list maximiser differs from the limited-underlying tuple set list maximiser in respect of nested fud variables. The tuple set optimiser allows nested variables in a tuple in preparation for rolling in the subsequent application of the contracted decrementing linear non-overlapping fuds list maximiser, whereas the depends fuds of the derived variables of the derived variables set optimiser must be limited-models fuds, $\mathrm{depends}(F,K) \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}}$.

An upper bound on the expected cardinality of the searched may be computed given the maximum derived dimension, $\mathrm{jmax}$. The upper bound on the expected cardinality for a non-empty fud, $F \neq \emptyset$, is \[ \sum_{j \in \{2 \ldots \mathrm{min}(\mathrm{jmax},r)\}} \binom{r}{j} \] where $R = \mathrm{vars}(F) \setminus V_A$ and $r = |R|$ and $\mathrm{min}=\mathrm{minimum}$.

The maximum derived dimension, $\mathrm{jmax}$, may be approximated from the geometric average valency $d = |R^{\mathrm{C}}|^{1/r}$, and the maximum derived volume, $\mathrm{wmax}$, \[ \mathrm{jmax} = \left\lceil\frac{\ln \mathrm{wmax}}{\ln d}\right\rceil \] Like the limited-underlying tuple set list maximiser, $Z_{P,A,A_R,F,\mathrm{B}}$, the limited-derived derived variables set list maximiser, $Z_{P,A,A_R,F,\mathrm{D}}$, has an inclusion function defined $\mathrm{top}(\mathrm{omax}) \in \mathrm{P}(X_{P,A,A_R,F,\mathrm{D}}) \to \mathrm{P}(X_{P,A,A_R,F,\mathrm{D}})$. An implementation of the derived variables set list maximiser, $Z_{P,A,A_R,F,\mathrm{D}}$, that guarantees no more than $\mathrm{omax}$ derived variables sets at each step of the optimiser list, $\forall D \in \mathrm{set}(\mathrm{list}(Z_{P,A,A_R,F,\mathrm{D}}))~(|D| \leq \mathrm{omax})$, must also have additional inclusion order criteria such as descending sum derived variables layer, $-\mathrm{sumlayer}(F,J)$.

The optimised limited-model fuds are (Haskell) \[ \begin{eqnarray} &&\{\mathrm{depends}(F_{\mathrm{L}},K) : \\ &&\hspace{5em}\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}}))\} \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}} \end{eqnarray} \]

The practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, may then be implemented (Haskell) \[ \begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray} \] where the shuffle content alignment valency-density computer $I_{\mathrm{csd}} \in \mathrm{computers}$ is defined as \[ \begin{eqnarray} I_{\mathrm{csd}}^{ * }((A,A_R,F)) = (I_{\mathrm{a}}^{ * }(A * F^{\mathrm{T}})-I_{\mathrm{a}}^{ * }(A_R * F^{\mathrm{T}}))/I_{\mathrm{cvl}}^{ * }(F) \end{eqnarray} \] In the case where the substrate histogram, $A$, is scalar or mono-variate, $|V_A| \leq 1$, the practicable fud inducer is stuffed with the empty fud, because the contracted decrementing linear non-overlapping fuds list maximiser, $Z_{P,A,A_R,F,\mathrm{n},-,K}$, in the limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,A,A_R,\mathrm{L}}$, requires a pluri-variate tuple, $|K|>1$, and so the limited-underlying tuple set list maximiser, $Z_{P,A,A_R,F,\mathrm{B}}$, requires a pluri-variate substrate, $|V_A|>1$.

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

A variation of this implementation of practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, is (i) to constrain the derived variables to intersect with the highest layer of the fud and (ii) to terminate the layer search as soon as the shuffle content alignment valency-density decreases. Define the highest-layer limited-derived derived variables set list maximiser \[ Z_{P,A,A_R,F,\mathrm{D,d}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{D}},P_{P,A,A_R,F,\mathrm{D}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{D,d}}) \] where the initial subset is (Haskell) \[ \begin{eqnarray} R_{P,A,A_R,F,\mathrm{D,d}} &=& \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{1em}w \in \mathrm{der}(F),~u \in \mathrm{vars}(F) \setminus V_A \setminus \mathrm{vars}(\mathrm{depends}(F,\{w\})),\\ &&\hspace{2em}J = \{w,u\},~|J^{\mathrm{C}}| \leq \mathrm{wmax}\} \end{eqnarray} \] The upper bound on the expected cardinality for a non-empty fud, $F \neq \emptyset$, is \[ \sum_{j \in \{2 \ldots \mathrm{min}(\mathrm{jmax},r)\}} \binom{r}{j} - \binom{r-x}{j} \] where $R = \mathrm{vars}(F) \setminus V_A$ and $r = |R|$, $X = \mathrm{der}(F)$ and $x = |X|$.

The optimised limited-model fuds are (Haskell) \[ \begin{eqnarray} &&\{\mathrm{depends}(F_{\mathrm{L}},K) : \\ &&\hspace{5em}\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,d}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D,d}}))\} \subset \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{q}} \end{eqnarray} \]

Define the highest-layer limited-layer limited-underlying limited-breadth fud tree searcher (Haskell) \[ Z_{P,A,A_R,\mathrm{L,d}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L,d}},\{\emptyset\}) \] where the neighbourhood function returns a singleton \[ \begin{eqnarray} &&P_{P,A,A_R,\mathrm{L,d}}(F) = \{G : \\ &&\hspace{2em}G \in P_{P,A,A_R,\mathrm{L}}(F),\\ &&\hspace{2em}(F \neq \emptyset \implies ~\mathrm{maxr}(\mathrm{el}(Z_{P,A,A_R,F,\mathrm{D,d}})) < \mathrm{maxr}(\mathrm{el}(Z_{P,A,A_R,G,\mathrm{D,d}})))\} \end{eqnarray} \] where $\mathrm{el} = \mathrm{elements}$.

The practicable highest-layer shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘}$, may then be implemented (Haskell) \[ \begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,d}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D,d}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray} \] The practicable highest-layer shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘}$, assumes that there is one maximum along the layer-cumulative path of fuds. An advantage of the highest-layer fud inducer is that fuds containing frame full functional transforms, having exactly the same alignment valency-density of lower layer fuds excluding the reframes, will be excluded, avoiding the extra computation and reducing the cardinality of the maximum domain, $|\mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D,d}}))|$. Note that a computer implementing the highest-layer limited-derived derived variables set list maximiser need not recompute the previous layer highest shuffle content alignment valency-density, $\mathrm{maxr}(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{D,d}}))$, but need only to carry it to this layer.

If the inclusion functions of the tuple set list maximiser and the derived variables set list maximiser are further ordered by descending sum derived variables layer the highest-layer fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{d}}^{‘}$, must be implemented with the limited-derived derived variables set list maximiser, $Z_{P,A,A_R,F,\mathrm{D}}$, rather than the highest-layer limited-derived derived variables set list maximiser, $Z_{P,A,A_R,F,\mathrm{D,d}}$. That is, $K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}}))$. In this way reframe variables at the $\mathrm{max}$ inclusion boundary may be replaced by variables below the highest layer.

Practicable maximum-roll shuffle content alignment valency-density fud inducer

Practicable maximum-roll-by-derived-dimension shuffle content alignment valency-density fud inducer

A variation of the implementation of practicable maximum-roll shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{mr}}^{‘}$, is to include the maximum-roll-by-derived-dimension constraint by implementing with the maximum-roll-by-derived-dimension contracted decrementing linear non-overlapping fuds tree maximiser, $Z_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mm}}$. Define the maximum-roll-by-derived-dimension limited-layer limited-underlying limited-breadth fud tree searcher (Haskell) \[ Z_{P,A,A_R,\mathrm{L,mm}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L,mm}},\{\emptyset\}) \] where the neighbourhood function returns a singleton \[ \begin{eqnarray} &&P_{P,A,A_R,\mathrm{L,mm}}(F) = \{G :\\ &&\hspace{2em}G = F \cup \{T : K \in \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}})),\\ &&\hspace{5em}H \in \bigcup \{\mathrm{maxd}(\mathrm{set}(L)) : L \in \mathrm{paths}(\mathrm{tree}(Z_{P,A,A_R,F,\mathrm{n},-,K,\mathrm{mm}}))\},\\ &&\hspace{5em}w \in \mathrm{der}(H),~I = \mathrm{depends}(\mathrm{explode}(H),\{w\}),~T = I^{\mathrm{TPT}}\},\\ &&\hspace{2em}\mathrm{layer}(G, \mathrm{der}(G)) \leq \mathrm{lmax}\} \end{eqnarray} \] The practicable maximum-roll-by-derived-dimension shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{mm}}^{‘}$, may then be implemented \[ \begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{mm}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,mm}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray} \]

Practicable excluded-self shuffle content alignment valency-density fud inducer

Another variation of the implementation of practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, is to exclude self partitions from the derived variables of the fuds of the limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,A,A_R,\mathrm{L}}$. Define the excluded-self limited-layer limited-underlying limited-breadth fud tree searcher as (Haskell) \[ Z_{P,A,A_R,\mathrm{L,xs}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L,xs}},\{\emptyset\}) \] where the neighbourhood function returns a singleton \[ \begin{eqnarray} &&P_{P,A,A_R,\mathrm{L,xs}}(F) = \{G :\\ &&\hspace{2em}G = F \cup \{P^{\mathrm{T}} : K \in \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}})),\\ &&\hspace{5em}H \in \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n},-,K})),\\ &&\hspace{5em}w \in \mathrm{der}(H),~I = \mathrm{depends}(\mathrm{explode}(H),\{w\}),\\ &&\hspace{5em}P = I^{\mathrm{TP}},~P \neq (\cup P)^{\{\}}\},\\ &&\hspace{2em}\mathrm{layer}(G, \mathrm{der}(G)) \leq \mathrm{lmax}\} \end{eqnarray} \] The rationale for excluding self partition variables is to reduce the computation necessary to process redundant variables, although note that the self partition variables will no longer appear in the top layer, $(\cup P)^{\{\}} \notin \mathrm{der}(G)$, and so cannot lift variables below during tuple building.

Also note that if all but one of the derived variables of the top decrementing linear fuds are self partition variables then the new fud $G$ will have a single derived variable, $|\mathrm{der}(G)|=1$, and hence have zero alignment, $\mathrm{algn}(A * G^{\mathrm{T}})=0$. If the top decrementing linear fuds contain only self partition derived variables then the neighbourhood function will return the given fud unchanged, $G=F$. That is, in this case no new layer is added.

The practicable excluded-self shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{xs}}^{‘}$, may then be implemented \[ \begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{xs}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,xs}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray} \]

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

Another variation of the implementation of practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, is to include the limited-valency constraint by implementing with the limited-valency contracted decrementing linear non-overlapping fuds list maximiser, $Z_{P,A,A_R,F,\mathrm{n,w},-,K}$. Define \[ Z_{P,A,A_R,\mathrm{L,w}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,\mathrm{L,w}},\{\emptyset\}) \] where the neighbourhood function returns a singleton \[ \begin{eqnarray} &&P_{P,A,A_R,\mathrm{L,w}}(F) = \{G :\\ &&\hspace{2em}G = F \cup \{T : K \in \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{B}})),\\ &&\hspace{5em}H \in \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F,\mathrm{n,w},-,K})),\\ &&\hspace{5em}w \in \mathrm{der}(H),~I = \mathrm{depends}(\mathrm{explode}(H),\{w\}),~T = I^{\mathrm{TPT}}\},\\ &&\hspace{2em}\mathrm{layer}(G, \mathrm{der}(G)) \leq \mathrm{lmax}\} \end{eqnarray} \] The practicable limited-valency shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{w}}^{‘}$, may then be implemented \[ \begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{w}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L,w}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray} \] In some cases a tuple $K$ returned by the limited-underlying tuple set list maximiser, $Z_{P,A,A_R,F,\mathrm{B}}$, will be rejected by the subsequent limited-valency contracted decrementing linear non-overlapping fuds list maximiser, $Z_{P,A,A_R,F,\mathrm{n,w},-,K}$, because there are no limited-valency tuple partitions, $\forall Y \in \mathrm{B}(K)~((|Y| > \mathrm{mmax}) \vee \neg (\forall M \in Y~(|M^{\mathrm{C}}| \leq \mathrm{umax})))$. To avoid processing a tuple which is destined to fail the limited-valency constraint, a variation of the limited-underlying tuple set list maximiser checks to ensure there is at least one limited-valency partition of the tuple. Define the checked-valency limited-underlying tuple set list maximiser \[ \begin{eqnarray} &&Z_{P,A,A_R,F,\mathrm{B,wc}} =\\ &&\hspace{2em}\mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{B}},P_{P,A,A_R,F,\mathrm{B,wc}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F,\mathrm{B,wc}}) \end{eqnarray} \] where the neighbourhood function is \[ \begin{eqnarray} &&P_{P,A,A_R,F,\mathrm{B,wc}}(B) = \{(J,X_{P,A,A_R,F,\mathrm{B}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in B,~w \in \mathrm{vars}(F) \cup V_A \setminus K,~J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{xmax},\\ &&\hspace{2em}\exists Y \in \mathrm{B}(J)~((|Y| \leq \mathrm{mmax}) \wedge (\forall M \in Y~(|M^{\mathrm{C}}| \leq \mathrm{umax})))\} \end{eqnarray} \] and the initial subset is \[ \begin{eqnarray} R_{P,A,A_R,\emptyset,\mathrm{B,wc}} &=& \{(\{w,u\},X_{P,A,A_R,\emptyset,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w,u \in V_A,~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax},\\ &&\hspace{1em}|\{w\}^{\mathrm{C}}| \leq \mathrm{umax},~|\{u\}^{\mathrm{C}}| \leq \mathrm{umax}\} \\ R_{P,A,A_R,F,\mathrm{B,wc}} &=& \{(\{w,u\},X_{P,A,A_R,F,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w \in \mathrm{der}(F),~u \in \mathrm{vars}(F) \cup V_A ,~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax},\\ &&\hspace{1em}|\{w\}^{\mathrm{C}}| \leq \mathrm{umax},~|\{u\}^{\mathrm{C}}| \leq \mathrm{umax}\} \end{eqnarray} \]

Practicable level shuffle content alignment valency-density fud inducer

Another variation of the implementation of the practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, is to explicitly specify the substrate. Rather than modelling with the given substrate variables, $V_A$, level modelling is parameterised by a pair of (i) a set of variables $V_{\mathrm{g}}$, which is a subset of the substrate variables, $V_{\mathrm{g}} \subseteq V_A$, and (ii) a level fud $F_{\mathrm{g}} \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}}$, which is such that its underlying is also a subset of the substrate variables, $\mathrm{und}(F_{\mathrm{g}}) \subseteq V_A$. Here only the union of (i) the substrate variables subset, $V_{\mathrm{g}}$, and (ii) the derived variables of the given level fud, $\mathrm{der}(F_{\mathrm{g}})$, are visible to the tuple maximiser, so the substrate variables, $V_A$, are effectively replaced by the level variables, $V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}})$.

The level fud inducer allows multiple levels to be modelled in sequence, so, for example, large substrates, $V_A$, with large underlying volumes, $|V_A^{\mathrm{C}}|$, may be made practicable by (i) partitioning them into components, $V_{\mathrm{g}} \in P$, where $P \in \mathrm{B}(V_A)$, with smaller underlying volumes, $|V_{\mathrm{g}}^{\mathrm{C}}| < |V_A^{\mathrm{C}}|$, (ii) inducing a level fud on each component, $V_{\mathrm{g}}$, of the substrate partition, and then (iii) combining these level fuds in a higher level to produce a model with coverage of the whole substrate, $V_A$. Another example is to use the level fud inducer in order to exclude mono-valent substrate variables, $V_{\mathrm{g}} = \{w : w \in V_A,~|(A\%\{w\})^{\mathrm{F}}|>1\}$, which might occur near the leaves of a decomposition. Note that higher levels do not necessarily require non-overlapping level fuds.

The level limited-underlying tuple set list maximiser $Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}}$ replaces the substrate variables, $V_A$, with the level variables, $V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}})$. Define the level limited-underlying tuple set list maximiser \[ \begin{eqnarray} &&Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}} = \\ &&\hspace{2em}\mathrm{maximiseLister}(X_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}},P_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}}) \end{eqnarray} \] where (i) the optimiser function is \[ \begin{eqnarray} &&X_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}} = \\ &&\hspace{2em}\{(K,I_{\mathrm{a}}^{ * }(\mathrm{apply}(V_A,K,\mathrm{his}(F \cup F_{\mathrm{g}}),A))-I_{\mathrm{a}}^{ * }(\mathrm{apply}(V_A,K,\mathrm{his}(F \cup F_{\mathrm{g}}),A_R))) :\\ &&\hspace{18em}K \in \mathrm{tuples}(V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}}),F)\} \end{eqnarray} \] and (ii) the neighbourhood function is (Haskell) \[ \begin{eqnarray} &&P_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}}(B) = \{(J,X_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in B,~w \in \mathrm{vars}(F) \setminus \mathrm{vars}(F_{\mathrm{g}}) \cup V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}}) \setminus K,\\ &&\hspace{2em}J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{xmax}\} \end{eqnarray} \] and (iii) the initial subset is (Haskell) \[ \begin{eqnarray} R_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\emptyset,\mathrm{B}} &=& \{(\{w,u\},X_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\emptyset,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w,u \in V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}}),~u \neq w,~|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax}\} \\ R_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}} &=& \{(\{w,u\},X_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}}(\{w,u\})) : \\ &&\hspace{1em}w \in \mathrm{der}(F),~u \in \mathrm{vars}(F) \setminus \mathrm{vars}(F_{\mathrm{g}}) \cup V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}}),~u \neq w,\\ &&\hspace{1em}|\{w,u\}^{\mathrm{C}}| \leq \mathrm{xmax}\} \end{eqnarray} \] An upper bound on the expected cardinality of the searched may be computed given the maximum underlying dimension, $\mathrm{kmax}$. The upper bound on the expected cardinality in the first layer, $F = \emptyset$, is \[ \sum_{k \in \{2 \ldots \mathrm{min}(\mathrm{kmax},s)\}} \binom{s}{k} \] where $s = |V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}})|$ and $\mathrm{min}=\mathrm{minimum}$. In subsequent layers, $F \neq \emptyset$, the upper bound on the expected cardinality is \[ \sum_{k \in \{2 \ldots \mathrm{min}(\mathrm{kmax},t)\}} \binom{t}{k} - \binom{t-x}{k} \] where $W = \mathrm{vars}(F) \setminus \mathrm{vars}(F_{\mathrm{g}}) \cup V_{\mathrm{g}} \cup \mathrm{der}(F_{\mathrm{g}})$, $t = |W|$, $X = \mathrm{der}(F)$ and $x = |X|$.

Then the shuffle content alignment optimised next level limited-underlying limited-breadth layer tuple set, $B_{\mathrm{B}}$, is (Haskell) \[ \begin{eqnarray} B_{\mathrm{B}} &=& \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}})) \end{eqnarray} \]

Define the level limited-layer limited-underlying limited-breadth fud tree searcher (Haskell) \[ Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}} = \mathrm{searchTreer}(\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}} \cap \mathcal{F}_{\mathrm{h}},P_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}},\{\emptyset\}) \] where the neighbourhood function returns a singleton \[ \begin{eqnarray} &&P_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}}(F) = \{G :\\ &&\hspace{2em}G = F \cup \bigcup\{\{T\} \cup \mathrm{depends}(F_{\mathrm{g}},\mathrm{und}(T)) : \\ &&\hspace{5em}K \in \mathrm{topd}(\lfloor\mathrm{bmax}/\mathrm{mmax}\rfloor)(\mathrm{elements}(Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},F,\mathrm{B}})),\\ &&\hspace{5em}H \in \mathrm{topd}(\mathrm{pmax})(\mathrm{elements}(Z_{P,A,A_R,F \cup F_{\mathrm{g}},\mathrm{n},-,K})),\\ &&\hspace{5em}w \in \mathrm{der}(H),~I = \mathrm{depends}(\mathrm{explode}(H),\{w\}),~T = I^{\mathrm{TPT}}\},\\ &&\hspace{2em}\mathrm{layer}(G, \mathrm{der}(G)) \leq \mathrm{lmax}\} \end{eqnarray} \] Note that the resultant fud of the level fud tree searcher, $Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}}$, has its underlying variables flattened to the substrate. That is, $\mathrm{und}(F_{\mathrm{L}}) \subseteq V_A$, where $\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}}))$. So it is not necessary to supply $F_{\mathrm{g}}$ along with $F_{\mathrm{L}}$.

Whereas in the limited-layer limited-underlying limited-breadth fud tree searcher, $Z_{P,A,A_R,\mathrm{L}}$, the layers of the fud increment at each step along the path, $\forall (i,G) \in L~(\mathrm{layer}(G, \mathrm{der}(G)) = i)$ where $L \in \mathrm{paths}(\mathrm{tree}(Z_{P,A,A_R,\mathrm{L}}))$, in the level fud tree searcher, $Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}}$, there is no such guarantee.

Define the level limited-derived derived variables set list maximiser \[ Z_{P,A,A_R,F_{\mathrm{g}},F,\mathrm{D}} = \mathrm{maximiseLister}(X_{P,A,A_R,F,\mathrm{D}},P_{P,A,A_R,F_{\mathrm{g}},F,\mathrm{D}},\mathrm{top}(\mathrm{omax}),R_{P,A,A_R,F_{\mathrm{g}},F,\mathrm{D}}) \] where the neighbourhood function is (Haskell) \[ \begin{eqnarray} &&P_{P,A,A_R,F_{\mathrm{g}},F,\mathrm{D}}(D) = \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{2em}(K,\cdot) \in D,~w \in \mathrm{vars}(F) \setminus V_A \setminus \mathrm{vars}(F_{\mathrm{g}}) \setminus K,\\ &&\hspace{2em}J = K \cup \{w\},~|J^{\mathrm{C}}| \leq \mathrm{wmax},~\mathrm{der}(\mathrm{depends}(F,J)) = J\} \end{eqnarray} \] and the initial subset is (Haskell) \[ \begin{eqnarray} &&R_{P,A,A_R,F_{\mathrm{g}},F,\mathrm{D}} = \{(J,X_{P,A,A_R,F,\mathrm{D}}(J)) : \\ &&\hspace{2em}w,u \in \mathrm{vars}(F) \setminus V_A \setminus \mathrm{vars}(F_{\mathrm{g}}),~u \neq w,\\ &&\hspace{2em}J = \{w,u\},~|J^{\mathrm{C}}| \leq \mathrm{wmax},~\mathrm{der}(\mathrm{depends}(F,J)) = J\} \end{eqnarray} \]

The optimised limited-model fuds are (Haskell) \[ \begin{eqnarray} &&\{\mathrm{depends}(F_{\mathrm{L}},K) : \\ &&\hspace{5em}\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{g}},F_{\mathrm{L}},\mathrm{D}}))\} \end{eqnarray} \]

The practicable level shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,V_{\mathrm{g}},F_{\mathrm{g}}}^{‘}$, may then be implemented (Haskell) \[ \begin{eqnarray} &&I_{z,\mathrm{csd,F,\infty,q},P,V_{\mathrm{g}},F_{\mathrm{g}}}^{‘ * }(A) = \\ &&\hspace{2em}\{(G,I_{\mathrm{csd}}^{ * }((A,A_R,G))) : \\ &&\hspace{5em}|V_A|>1,~\{F_{\mathrm{L}}\} = \mathrm{leaves}(\mathrm{tree}(Z_{P,A,A_R,V_{\mathrm{g}},F_{\mathrm{g}},\mathrm{L}})),\\ &&\hspace{5em}K \in \mathrm{maxd}(\mathrm{elements}(Z_{P,A,A_R,F_{\mathrm{g}},F_{\mathrm{L}},\mathrm{D}})),~G = \mathrm{depends}(F_{\mathrm{L}},K)\} \cup \\ &&\hspace{2em}\{(\emptyset,0) : |V_A| \leq 1\} \end{eqnarray} \]

Limited inducers

Of the variations described above of the implementation of practicable shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, only the practicable limited-valency shuffle content alignment valency-density fud inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{w}}^{‘}$, is potentially unrestricted. That is, the limited-valency inducer, $I_{z,\mathrm{csd,F,\infty,q},P,\mathrm{w}}^{‘}$, can perform the same search as the unlimited inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, if the maximum valency is set equal to the maximum underlying volume, $\mathrm{umax} = \mathrm{xmax}$. The other variations all have restricted functionality with respect to the unlimited inducer, $I_{z,\mathrm{csd,F,\infty,q},P}^{‘}$, no matter what the parameters.

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